Structure, Information and Communication Complexity, IIS 1 9780773591158

Table of contents :
Cover
Title
Copyright
Table of Contents
Preface
Sense of Direction: Formal Definitions and Properties
Trade-off Between Computational Power and Common Knowledge in Anonymous Rings
Orientation of Distributed Networks: Graph- and Group- Theoretic Modelling
Labeled versus Unlabeled Distributed Cayley Networks
Classifying Anonymous Networks: When Can Two Networks Compute the Same Vector-Valued Functions?
Compact Routing Methods: A Survey
Interval Labeling Scheme for Chordal Rings
Fault-Tolerant Compact Routing Based on Reduced Structural Information in Wormhole-Switching Based Networks
The Buffer Potential of a Network
Path Layout in ATM Networks

Citation preview

STRUCTURE, INFORMATION AND COMMUNICATION COMPLEXITY

Paola Flocchini, Bernard Mans, Nicola Santoro (Eds.)

STRUCTURE, INFORMATION AND COMMUNICATION COMPLEXITY

Proceedings of the 1st Colloquium on Structural Information and Communication Complexity, Carleton University, Ottawa, Canada.

CARLETON UNIVERSITY PRESS

Copyright 8 Carleton University Press, Inc. 1995 Published by Carleton University Press. The publisher would like to thank the Vice-President (Academic), the Associate Vice-President (Research), the Dean of Science, and the School of Computer Science at Carleton University for their contribution to the development of the Carleton Informatics Series. Carleton University Press would also like to thank the Canada Council, the Ontario Arts Council, the Government of Canada through the Department of Canadian Heritage, and the Government of Ontario through the Ministry of Culture, Tourism and Recreation, and the Ontario Arts Council. Printed and bound in Canada.

CANADIAN CATALOGUING IN PUBLICATION DATA Structure, information and communication complexity (International infomatics series ; 1) Revised versions of the papers presented at the 1st Colloquium on Structural Information and Communication Complexity, held at Carleton University May 18-20, 1994. Includes bibliographicalreferences. ISBN 0-88629-253-0 1. Electronic data processing-Distributed processing- Congresses. 2. Computer networks- Congresses. I. Santoro, Nicola, 1951- . 11. Flocchini, Paola 111. Mans,Bernard IV. Colloquium on Structural Information and Communication Complexity (1st : 1994 : Carleton University). V. Series.

Table of Contents Preface P. Flocchini (Milan), B. Mans, N. Santoro (Carleton) "Sense of Direction: Formal Definitions and Properties"

vii 9

P. Ferragina, A. Monti, A. Roncato (Pisa) "Trade-off Between Computational Power and Common Knowledge in Anonymous Ringsn C. Lavault (Paris-Nord) "Orientation of Distributed Networks: Graph- and Group Theoretic Modellingn

E. Kranakii, D. Krizanc (Carleton) "Labeled versus Unlabeled Distributed Cayley Networks"

N. Norris (S.Cruz) "Classifying Anonymous Networks: When Can Two Networks Compute the Same Vector-Valued Functions ?"

83

J. van Leeuwen, R.Tan (Utrecht) "Compact Routing Methods: A Survey"

M. Flarnmini (Roma I), G. Gambosi (Roma 11),S. Salomone (1'Aquila) "Interval Labeling Scheme for Chordal Rings"

111

J .Vounckx, J . Deconindr, R. Lauwereins, J .A. Peperstraete (Leuven) uFault-Tolerant Compact Routing Based on Reduced Structural Information in Wormhole-Switching Based Networks"

125

K. Diks (Hull), E. Kranakis (Carleton), A. Malinowsky, A. Pelc (Hull) T h e Buffer Potential of a Network"

149

0. Gerstel, S. ZKks (Technion) "Path Layout in ATM Networks''

Preface The ultimate goal of the research in Distributed Computing is to understand the nature, the properties and the limits of computing in a system of autonomous communicating agents. To this end, it is crucial to identify those factors which are significant for the computability and the communication complexity of prob lems. It has become quite clear that a very important role is played by those factors which can be termed as Structural Information; that is, a priori knowledge available to the entities about the structure of the system. A couple of examples: knowledge by the entities of a global consistency in the local edge labeling (sense of direction) can drastically reduce the communication complexity of several distributed problems; the careful choice of entity names and labeling of the communication links can yield routing schemes (implicit routing) which require very little storage and are yet optimal in terms of communication. The 1st CoIZoquium on Structural Information and Communication Complezit3 (SIROCCO) was held at Carleton University, Ottawa, Canada, from May 18 to 20, 1994. The purpose of this colloquium has been to start to explicitly focus on the interaction between structural information and communication complexity. Topics of interest have included: topological awareness, sense of direction, metric information (e.g. number of nodes, diameter,. ..) , implicit routing (e.g. compact, interval,...), complexity of constructing structural information, complexity of maintaining structural information. The Colloquium has been comprised of position papers (outlining open problems, research directions, etc.), presentation of current research results, and group discussions. This volume contains revised versions of nearly all the papers presented a t the Colloquium. The revisions have been based on the comments and suggestions received by the authors from referees and/or participants of the Colloquiwn. Several papers are in the form of preliminary reports on ongoing research. It is expected that more elaborate versions of these papers will eventually appear in scientific journals. We hope that this volume gives a good impression of activities in the topic of Structural Information and Communication Complexity. Unfortunately, -11g from this volume are the discussions which have been a focal point of the Colloquium contributing to its success. The friendly atmosphere of the Colloquium was enhanced by the efforts of the people involved in the local organization (Ralph Borland, Rosemary Carter, Flaminia Luccio, Mark Wineberg), the activeness of participants contributing to the presentations and discussions, and the post-mortem barbeque offered by the Urrutias. The Colloquium and these Proceedings have been supported in part by: the Natural Sciences and Engineering Research Council of Canada, the Center for Parallel and Distributed Computing (Paradise), and the School of Computer Science of Carleton University. Their support is gratefully acknowledged. Paola Flocchini, Bernard Mans, Nicola Santoro

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0. Let 6 :V x V (0,. ,n - 1 ) be the corresponding distance function; i.e., 6(x, y) is the smallest k such that 7 k ( x ) = y. The labeling X is a Chordal labeling i f i V{x, Y) € E(x): A=((x, Y)) = 6 ( x , Y) -+

..

Note that y is the function defining the cyclic ordering of the nodes; thus, different Chordal labelings arise from different ys. Further note that, if the link between p and q is labeled by d at node p, it is labeled by n - d at node q (see figure 5). When also the local names of the nodes are relative distances in the cyclic ordering, the Chordal labeling is a SV.

Theorem 4.3 Let X be a Chordal labeling and Vx, y let XisaSV.

P, (y) = 6 ( x , Y).

Then

Proof. To verify that it is a SD,consider the coding function f defined as follows: V r E P[xo],

= ( ( x o , x I ) ~( Z I ~ X ~ ) , * -(-x,m - 1 ,

xm))

It follo~vsthat

f(a, (7)) =

zm)= P., ( z m )

thus, f is consistent. Consider the folloing decoding function h:

h(X.0

( ( ~ 0YO)), ,

f (Ay,(n)))= Lo( ( x o , yo)) + f (A,, (r))

We have that

Thus, h is consistent and X is a SV. We shall call this labeling Chordal SV. Note that the set of names and the set of labels coincide: C = N = 2:.

Fig. 5. An Arbitrary Network with Chordal Sense of Direction.

20

The Chordal labeling is the natural labeling for the family of graphs called circulant graphs or chordal rings, from which it takes the name. It (.it11 obviously be defined for any graph. In the literature, the Chordal SD has bee11 extensively investigated in specific topologies. Sometimes called Distance SD,it has been studied in complete graphs [12,21,26,27,2S, 29,383 and chordal rings [2, 13,311. Its impact has been also investigated in hypercubes [6], as well as in systems of unknown topology (the arbitrary graph case) [22]. 4.3

Contracted Sense of Direction

In this Section, we will analyze a rather general class of SD based on labelings with Locally Symmetric Orientation (i.e., with both Local Orientation and Eclge Symmetry). As we will see, this class contains the traditional labelings for meshes, tori, and hypercubes, among others. Let X be a labeling with locally symmetric orientation, and let $ be the corresponding edge symmetry function. Definition 4.4 Contraction Given a sequence a E t', the contraction of a is the sequence E of labels obtained fwm a by deleting every pair of labels I and 1' such that 1 = $(P), and lexicographically sorting the resulting sequence. Definition 4.5 Contracted Labeling A labeling X with edge synmetry is contracted i . 'dx, y E V, 'dm, n2 E P[x,y]

-

That is, if X is contracted, then all the sequences of all the paths from x to 9 have the same contraction, which we shall denote it by A,,, When the local names of the nodes are the appropriate contractions, the Contracted labeling is a SV.

.

-. Then

Theorem 4.4 Let A be a Contracted labeling and 'dx, y let P, (y) = A,, X isaSD.

Proof. To verify that it is a SD, consider the coding function f defined as follows: ( ~ 0t ,i ) , ( x I , x ~ ) , -,( ~ , - i , ~ m ) )

V r E P[xo], n = (

It follows that m z o (4)= Pzo (Pm) thus, f is consistent. Consider the following decoding function h: ~ ( x oflo) , E E(xo), vr E P[l/o]o~l* = [(w, PI), ,(Pm-1, ~ m ) ] :

where o is the concatenation operator. It follows that

Atoym =

P z o (Y*

)

Thus, h is consistent and X is a SD. We shall call this labeling Contracted Sense of Direction.

Example

- Contraction in hypercubes:

Dimensional SD.

Fig. 6. Hypercube with Dimensional Sense of Direction The traditional labeling of a d - dimensional hypercube, shown in Figure 6 for d = 3, is an instance of Contracted SD where the local name &(y) is the (sorted) sequence of labels (dimensions) on the shortest path between x and y. In fact, it is a locally symmetric orientation where the edge symmetry function 7C, is the identity function. It is easy to verify that, in the hypercube, this labeling is a contracted labeling. Consider, for example, the two paths srl and a 2 from x to y, in Figure 6 with A=(al) = [3,2,3,1,3] and &(w) = [I, 293); in this case we have A,(nl) = [1,2,3]= At(~2),and &(Y) = PY(t)= 11,2931The impact of this labeling, also called Dimensional SD,in hypercubes has been extensively studied in the literature (e.g., [6, 33, 401).

Example - Contraction in Meshes: Compass SV. Each type of d-dimensional mesh (e.g. quadrilateral, hexagonal, etc.) has a natural labeling which forms a particular case of Contracted SV where the local name &(y) is the (sorted) sequence of labels on the shortest path between x and y. Consider, for example, the traditional labeling of a d-dimensional quadrilateral mesh, shown in Figure 1 for d = 2. This labeling is a locally symmetric orientation where the edge symmetry function J, is such that $(north) = south, $(east) = west, and so on. It is easy to verify that this labeling is contracted. Consider, for example, the two paths a1 and a2 from x to y, in Figure 1 with A.(nt) = [north, east, north, south] and A,(r2) = [east, north, east, west]; we have A,(al) = [east, north] = A, ( ~ 2 ) . In this case, P, (y) = [east, north], while Pu( x ) = p(& (y)) = [south, west], where p is the name symmetry function.

In the literature, the impact of this type of labelings, sometimes called Compass SZ), has been studied only for the cases of quadrilateral meshes, considered above, and of hexagonal meshes (shown in Figure 7 for d = 2) [25, 32,411.

Fig. 7. Compass SD in an hesagonal mesh. Contraction with Wraparound

An immediate generalization of the Contracted SV is the one which applies to topologies with wraparound (e.g., rings, tori, etc.). In this case, the sequences associated to paths are transformed so to use only a subset of the labels (termed "allowed directions*) and to take into account the structure of the wraparound. Let X be a labeling with locally symmetric orientation, and let .1CI be the corresponding edge symmetry function. Definition 4.6 Contraction with Wraparound ~ e t= Z { I,, I,} c t ,he,li # $(lj) f o r i # j , and let W = {wl,. ..,w,} Zm. Given a sequence of labels cr E C*,the contraction with Wraparound W and Allowed Directions 2 (shottly, ZW-contraction) of 0 is the sequence Z of labels obtained fiom the contraction E by 1. replacing any subsequence of k li s with a subsequence of wi - k +(l;), where

...,

k 2 0, li E E , wi

E W ;and 2. lezicographically sorting the resulting sequence.

Using this operation, the notions of contracted labeling and contracted SD are &.?tendedas follows.

EW

Definition 4.7 Contracted Labeling A labeling X with edge symmetry is LW contracted

Theorem 4.5 Let X be a Then X is a SV.

ZW

ifi Vx, y, Vxl, rrl E P [ x ,y]

Contmcted labeling and Vz, y let &(y) = &,Y.

The proof follows the same lines as the one of Theorem 4.4. Similarly, we can prove the following.

-

Example Contraction in Rings Each d-dimensional torus has a natural labeling which forms a particular case of 2~ Contracted SD where the local name /?,(y) is the (sorted) sequence of labels on the shortest path between z and y using only the allowed directions. Consider a ring (i.e., a 1-dimensional torus) of size n with the traditional labeling with t = { left, right) and with edge symmetry function $: right = +(left). This labeling is ZW contracted where the wrap around is W = In) &d the direction is, for example, Z = {left). Consider, for example, the two paths 1r1 and n2 from x to y, in a ring of size n = 7, with a1 = Ar(zr) = [left,left,left,left] and 02 = Al(a2) = [right,right,left,right,right];The = [right, right, right] = G.In this case corresponding EW contractions are &(y) = [right, right, right]. Example - Contraction in Tori: Compass SV Consider the 2-dimensional torus of size nl x n2 with the traditional "cornp a d assignment of the labels t = { north, south, east, west) (see Fig. 6) and edge symmetry function north = $(south), east = +(west), and so on. Clearly, the set of wraparounds is W = i n r , nz); the corresponding set of allowed directions is, for esample, = {south, west). The labeling X is a contracted labeling. Consider, for esample, the two paths rrl and a 2 from x to y (in Figure 8) with a1 = Az(rl) = [east, south, west, west, west] and 02 = A. (a2) = [north, north, east, north, north, east, east]; The contractions of a1 and 02 are = [south, west, west] and = [east, east, north, nort h, north, north]. The corresponding contractions are east, Gi = [east, east, east, south] = G.In this case, /3= (y) = [east, east, east, south].

+:

ZW

ZW

north

south

Fig. 8. A torus with Contracted Sense of Direction 4.4

Neighbouring Sense of Direction

Definition 4.8 Given a graph ( G ,A), X is a Neighbouring labeling iff: V(x,y) E E[xl, (19 w ) E E[zl, X,((x, y)) = X:((z, w ) ) iff Y = w

That is, in a neighbouring labeling, all the links ending in the same node x are labeled with the same label which we shall denote by Z(,); see Figure (9). Theorem 4.7 Let X be a Neighbouring labeling, and Vx,y E V let &(y) = Z(Y). Then X is a ST).

Proof. To verify that it is a SV, consider the coding function f with Af = C, defined as follows:

= ( ( ~ 0X ,I ) , (x1,22), ,(2,-1,xm)) .f(&(~))= ' Z m - l ((xrn- 11 ~ m ) )

V a E P[xo], A

Since, by definition of Neighbouring labeling, A=,-, lows that f (Ax(.)) = Ao(xm) thus, X is a S V .

( ( 2 , - 1, I , ) )

= I(,,)

it fol-

We shall call this labeling a Neighbouring S V . The corresponding decoding function is : ' ( t o , YO) E E(xo), Vn YO], = ((YO, Y I ) (31, , ~ 2 ) , 7 (ym-1, ym))

-

which is clearly consistent; in fact, A Property which shows an aspect of the strength of the Neighbouring SV and sets it apart from the other classes of SVs is the following:

Property 4.1 Given an anonymous system (G,X,P), if X is a Neighbouring

SV,then the Election Problem is solvable in G . Proof. Let X be a Neighbouring SV. Then each node x can acquire a unique global identifier; e.g., by asking an arbitrary neighbour for the label of the link connecting them, and assuming such a label as its identifier. In presence of a unique global identifier for each node, the Election Problem can be solved using any of the existing algorithms.

To fully appreciate this result, recall that the Election Problem is unsolvable in an anonymous unlabeled G, and that similar results do not exist for the other classes of SVs described above. In the literature, the Neighbouring SD has been studied solely in systems of unknown topology (the so called arbitrary networks) [20, 371.

Fig. 9. A complete network with Neighbouring SV.

5 5.1

Construction and Applications Construction

We will first state some basic theorems. Theorem 5.1 In an anonymous system, the neighbouring SV cannot be deterministical ly constructed. Theorem 5.2 In a (possibly anonymous) system with a leader, all the S V s considered here can be deterministically constructed.

In the following, some properties of labeling in trees are shown.

Theorem 5.3 In a tree, Local Orientation is not suficient to have a Sense of Direction. Proof. In Figure 10 it is shown a tree with local orientation. It is easy to verify that this labeling is not a sense of direction. In fact, from node A we must have f (01, a*, a3,a4) = f (as, as, a7, as); ho\vever, from m d e B, f (al, az, as, a4) # f (a5, as, a7, as), contradicting the definition of consistent local coding.

Fig.10. A tree with local orientation.

Theorem 5.4 I n a tree, Edge Symmetry is not necessary t o have a Sense of Direction. Proof. In Figure 11 it is shown a tree without edge symmetry. It is easy to verify that this labeling is a sense of direction.

Theorem 5.5 I n a tree, any Locallgl Symmetric Orientation is a Sense of Direction. We will now consider the ring topology. In rings, any SD is linearly dependent on the ring sizes. More specifically,

Theorem 5.6 Knowledge of (an upper bound on) the ring size n is necessary for SV i n rings. Proof. By contradiction. Assume that a SV which does not require knowledge of n exists for a given ring R1of even size n. By assumption, the decoding function h of the two opposite paths of distance n / 2 from a node u allows the node v to

Fig. 11. A tree without edge symmetry and SD. detect that both these paths reach the same node v. Construct now a ring Rz by doubling the size of R1 and embedding the ring R1 in its first half (see Figure 12). Since any of the previous labeling and decoding function did not require knowledge of n, v mill interpret the two nodes u and u' as the same node.

Fig. 12. Proof of Theorem 5.6.

Following the same reasoning, we have similar results for the torus and chordal rings. Theorem 5.7 Knowledge of (an upper bound on) no,nl,. ..,nd-l is necessary for SD in a toms.

Theorem 5.8 Knowledge of (an upper bound on) n is necessary for SV in chordal rings.

Although the most natural SV for rings is the so-called Ieft/right SD,discussed before, the ring can be labeled according to other Senses of Direction. Some examples are shown in figure 13. Note that both the Left/Right and the Dimensional labelings are in the Contracted class of labelings.

b) Chordal SD

a) Left/Right SD

RED

BLUE

c) Dimensional SD

d) Neighbouring SD

Fig. 13. Different Senses of Direction in the Ring.

Theorem 5.9 I n an anonymous ring i) there ezists no deterministic algorithm t o construct a SV when n is unknown ii) there e&ts no deteministic algorithm t o construct a Neighbouring SV even if n is known iii) if n is known and it is odd, a Chordal SD can be constructed i n 0( nlog n) messages i v ) when n is h o w n and it is even, t h e ~ eexists a deterministic algorithm which build either u Dimensional or a Chordal SV i n O ( nlog n ) messages. Proof. i) Particular case of Theorem 5.6 ii) This is a particular case of Theorem 5.1 iii) The proof follows directly from [I] i v ) The proof follows directly from [I] Following the same reasoning, we have similar results for the 2-dimensional torus.

Theorem 5.10 In a 2-dimensional n x n anonymous toms t h e ~ eexist no deterministic algorithm to build Compass, and Dimensional S D s , if n is even. Theorem 5.11 [41]In a Ldimensional n x n anonymous toms of even size, there ezist no deternilnistic algorithm to change a Dimensional SV into a Compass sv. 5.2

Applications

The evidence of the positive impact of particular SV for specific topologies are well known. In the case of arbitrary topologies, some results exist for two types of SDs: Chordal SV [22], and Neighbouring SD [14, 201. In the companion paper 171, we show that any SD has a positive impact on the complesity of the problems of Distributed Depth First Search (DDFS) in anonymous and non anonymous networks, Broadcast in anonymous and non anonymous networks, and Election. Some of these results are briefly reported in the following. Consider the problem of constructing a Depth First Search spanning tree of G. It is known that, with Neighbouring SD (and, thus, by Theorem 5.1, only in non anonymous networks), this problem can be solved in O(n) [37]. The presence of any Sense of Direction allows to solve this problem in O ( n )messages and time, even if the network is anonymous. More precisely:

Theorem 5.12 [7]In any anonymous network ( G ,A, P), if A is a Sense of Direction, the Depth First Search can be performed using at most 2n - 3 messages and at most 2n - 3 steps. Let us stress that, in unlabeled networks, this problem requires ROE]) messages; this bound can be easily achieved (e.g., [5]). The broadcast problem consists to arrive from an initial configuration, where esactly one processor (the initiator) has an information, to a configuration where all the processors have the information. Clearly, any single initiator DDFS algorithm is a solution to the broadcast problem. Thus, the solution mentioned above applies also to this problem.

Theorem 5.13 [7] In any anonymous network with any Sense of Direction, the Broadcast can be perfomed using at most 2n - 3 messages and at most 2n - 3 steps. Without labeling, the complexity of the bro~dcastproblem , denoted by C(B), is a(lE1)when the size is unknown. If the size of tlle network is common knowledge, for any given size there exist graphs which require a(lE1) messages 171 Tlie Election (or Leader Finding) problem consists to arrive from an initial configuration where all the processors are in the same state to a final configuration where exactly one processor is in a leader state and all the other processors

are in state lost. The Election process may be independently started by any subset of the processors. It is known that, with Chordal and Neighbouring SD,this problem can be solved in O(nlog n ) messages [20, 221. In fact, the availability of any SV allows to achieve this bound. Theorem 5.14 [7] Given (G,A), where A is a Sense of Direction, the Election problem can be solved using 3 n log n O(n) messages.

+

It is well known that in arbitrary unlabeled networks the Election problem requires R(e+ n log n) messages (e.g., [34]) and such a bound is achievable (e.g.,

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13. T.Z. Kalamboukis and S.L. hlantzaris. Towards optimal distributed election on chorc1;tl rings. Information Processing Letters, 38:265-270, 1991. 14. E.KorGtch, Kutten, and S. Moran. A modular technique for the design of efficient distributed leader finding algorithms. A.C.M. ToPlaS, 12(1):84-101, January 1990. 15. E. Korach, S. Moran, and S. Zacks. Tight lower and upper bounds for a class of distributed algorithms for a complete network of processors. In Proceedings of 3rd Symposium on Principles of Distributed Computing (PODC), pages 199-207, Vancouver, Canada, August 1984. 16. E. Kranakis and D. Krizanc. Labeled versus unlabeled distributed cayley networks. In this proceedings. 17. E. K r a n a b and D. Krizanc. Distributed computing on anonymous hypercubes. In Proc. 3rd IEEE symposium on parallel and distributed processing, Dallas, 1991. 18. E. Kran&, D.Krizanc, and J. van den Berg. Computing boolean functions on anonymous networks. In Proc. 17th Int. Colloquium on Automata, Languages, and Programming (ICA LP), pages 254-267, 1990. 19. E. Kranaks and N. Santoro. Distributed computing on anonymous hypercubes with faulty components. In Proc. 6th Int. Workshop of Distributed Algorithms (WDAG), 1992. 20. I. LavallCe and G. Roucairol. A fully distributed (minimal)spanning tree algorithm. Information Processing Letters, 23:55-62, Aug 1986. 21. M.C. Loui, T.A. Rfatsushita, and D.B.West. Election in complete networks with a sense of direction. Information Processing Letters, 22:185-187, 1986. see also Information Processing Letters, vo1.28, p.327, 1988. 22. B. Mans and N. Santoro. On the impact of sense of direction in arbitrary networks. In ICD CS'94, 14th International Conference on Distributed Computing Systems, pages 258-265, Poznan, Poland, June 21-24 1994. 23. B. Mans and N. Santoro. Optimal fault-tolerant leader election in chordal rings. In FTCS'94, 24th Annual Intemational Symposium on Fault- Tolerant Computing, pages 392-401, Austin, Texas, USA, June 15-17 1994. to appear. 24. S.L. Mantzaris. Almost optimal election in chordal rings. In S. Tzafestas, P. Borne, and L. Grandinetti, editors, Proceedings of Parallel and Distributed Computing i n Engineering Systems, (PDCOM'SI), pages 459466, 23-28 June 1991. 25. G.H. Masapati. Election and termination detection in specialized communication structures. PhD Thesis, Dept. of Computer Science, University of Ottawa, Ottawa, Canada, 1994. 26. G.H. Masapati and H. Ural. Effect of preprocessing on election in a complete network with a sense of direction. In Proceedings of IEEE International Conference on Systems, Man and Cybernetics, volume 3, pages 1627-1632, 1991. 27. T. Masuzawa, N. Nishikatva, K. Hagihara, and N. Tokura. Optimal f'ault-tolerant distributed algorithms for election in complete networks with a global sense of direction. In Proceedings of the 3rd International Workshop on Distributed Algorithms (WDA G), pages 171-182, Nice, France, 1989. Springer-Verlag. 28. T. Masuzawa, N. Nishikawa, K. Hagihara, and N.Tokura. A fault-tolerant algorithm for election in complete networks with a sense of direction. Systems and Computers in Japan, 22(12):11-22, 1991. 29. N. Nishikatva, T. Masuzawa, and N. Tokura. Fault-tolerant distributed algorithm in complete networks with link and processor failures. IEICE Transactions on Infomation and Systems, J74D-I(1):12-22, jan 1991. 30. N. Norris. this proceedings.

31. Yi Pan. An improved election algorithm in chordal ring networks. International Journal of Computer Mathematics, 40(3-4):191-200, 1991. 32. G.L. Peterson. Efficient algorithms for elections in meshes and complete networks. Technical Report TR-140, Dept. of Computer Science, Univ. of Rochester, Rochester, NY-14627, 1985. Choosing a leader on a hypercube. In 33. S. Robbins and K.A. Robbins. N. Rishe, S. Najathe, and D. Tal, editors, PARBASE-90, International conference on databases, parallel architectures and their applications, pages 469-471, Miami Beach, Florida, USA, 1990. 34. N. Santoro. On the message complexity of distributed problems. Journal of Computing Information Science, 13:131-147, 1984. 35. N. Santoro. Sense of direction, topological awareness and communication complexity. SIGACT NEWS,2(16):50-56, summer 1984. 36. N. Santoro, 3. Urrutia, and S. Z&. Sense of direction and communication complexity in distributed networks. In Proceedings of the 1st International Workshop on Distributed Algorithms (WDA G), pages 123-132, Ottawa, Canada, August 1985. Carleton University Press. 37. M.B.Sharma, S.S. Iyengar, and N.K. Mandyam. An efficient distributed depthfirst-search algorithm. Information Processing Letters, 32:183-186, September 1989. see also Information Processing Letters, vo1.35, p.55, 1990. 38. G. Singh. Leader election in complete networks. In Proceedings of 11th Symposium on Principles of Distributed Computing (PODC), pages 179-190, aug 1992. 39. V. Syrotiuk and J. Pachl. A distributed ring orientation algorithm. In Proceedings of the 2nd International Workshop on Distributed Algorithms (WDAG), volume 312 of LCNS, pages 332-336, Amsterdam, Holland, 1987. Springer-Verlag. 40. G. Tel. Linear election for oriented hypercube. Technical Report TR-RUU-CS-9339, Utrecht University, Department of Computer Science, The Netherlands, 1993. 41. G. Tel. Network orientation. In A. Gibbons and P. Spiralcis, editors, Lectures on Parallel Computation. Cambridge University Press, 1993. also in International Journal of Foundations of Computer Science, vo1.5, n.1, 1994. 42. J. van Leeu~venand R. Tan. Compact routing: a survey. In this proceedings. 43. A.M. Verweij. Linear-message election in hypercubes. Preprint. 44. 1%.Yamashita and T. IICameda. Computing on anonymous networks. In Proceedings of 7th Symposium on Principles of Distributed Computing (PODC), pages 117-130, 1988.

This article

mas processed

using the UTE, macro package with SICC style

Tkade-off between computational power and common knowledge in anonymous rings* P a l o Ferragina, Angelo Monti and Alessandro Roncato Dipartimento di Informatics, Universitir 'di Pisa, Italy. Emails: {ferragin, montico, roncato)@di.anipi.it

Abstract. We give an exact characterization of the M y of functions that can be computed distributively in an anonymous ring, diversifying the type of knowledge of the processors about the inpat configuration I, where I is the sequence of processor labels, (I1 = n. We consider two kinds of knowledge, namely an upper-bound M on n model R ~ )and , the exact number Y of distinct labb in I (model RL). Moreover we prove that such fo11ctios can be computed on the asynchronous R~ ( R ~exchanging ) 0 ( n M ) (O(n min {log n, V))) messages. We show that the general protocol used to attain the above results cannot be improved with respect to the message complexity, since there are functions reqnirh g n(nM) (n(nmin (log n, V)))messages, respectively. hrthermore, we investigate the problem of electing a kader in R~ and R~ using randomization.

I

Introduction

We consider the problem of computing functions in anonymous distributed ring. e that each processor has two particular registers, namely a read only We m input register and a write once output register. The input wnfiumtion I is defined as I = x o z l . . z,,-1 where xi is the value of the input register (label) of pi. Computafion of a function F on I starts with input configuration I and ends with each processor having the value F(1) in its output register. The relation between the computational power and the knowledge available to the single proassor on the input configuration has been extensively studied [2, 8, 1, 31. It has been proved that if the knowledge of the single processor consists only of the value contained in the input register, it is possible to compute only constant functions [2] while if knowledge of the single processor is extended with the size of the input configuration (that is, the size of the ring), then it is possible to compute all functions invariant under cyclic permutations of the input string. In this paper, we try to investigate the problem: what kind of knowledge is snfiicient or really necesssry to perform computations on anonymous rings ?[12]. Here, we consider an upper-bound M on the size of the input configuration I (i.e. 111 5 M), or the exact number V of distinct Labels in the input configuration.

.

* This paper has been supported in part by Progetto Finalizzato Sistemi Informatiu e Calcolo ParaUelo, CNR Italy.

The computation model with the first kind of knowledge is denoted as RM, while 'R the other one is denoted as . We solve the computation of functions in RM, considering the problem of finding proper radices in the input configuration strings, obtaining an aspchronous protocol requiring O ( n M ) messages. Instead, to compute a function in RV we derive an asynchronous protocol from the synchronous distribution algorithm of 121, obtaining an asynchronous protocol requiring O(n min {V, log n ) ) messages. Moreover we give an exact characterization of the class of functions computable in R~ and RV. In a distributed computation on an anonymous ring the main problem is the one of sgmrnetw breaking ['I,91. The state of a processor after few steps of the computation depends only on the initial configuration in a small neighborhood of that processor. If each neighborhood is replicated many times in the initial configuration, then whenever a processor sends a message, many other processors do so; hence Usuperfiuousntraflic cannot be avoided. Our tight lower bounds for R~ and R~ are obtained building symmetric initial configurations with many repetitions of local patterns [2, 141. Finally, we investigate the problem of electing a leader (i.e. a uniquely designed processor such that each processor knows if it is the leader). While the problem can be easily solved in non anonymous rings [6], in (11it has been proved that no deterministic algorithm exists for the anonymous ring even if the model is synchronous or also if the size of the ring is known. This result immediately implies impossibility results also for deterministic RM and RV. Hence, if any solution exists for these models it must use the randomization. In [7,9] randomized algorithms which always compute the leader are proposed when the size of the ring is known. Moreover, [9] proved that if the size of the ring is unknown then does not exist an algorithm with bounded probability of error. Here we show that same results hold for the R~ model. Instead, for the synchronous RM model we develop an algorithm that with high probability elects the leader.

2

Non computable functions

In this section we give some necessary conditions on the functions that are computable in R~ and RM. Since the synchronous executions are particular cases of asynchronous executions, in the proofs of the following impossibility results we consider only the synchronous mode. Given a bidirectional ring with input configuration I, we define the k-neighborhood (k 2 0 ) of pi(I) as the string z ( i - i ) m d n . . + i . . t ( i + k ) m d n . ID the case of an unidirectional ring, the k-neighborhood is given by z(i-k)modn.. .zi. First we give a result similar to one in [2]:

. .

Lemma 1 Let I and I' be two input configurutions for synchmnous RM, (or RV). If t h e n ctist i,j 0 such that, pi(I) and pj(I') have the same k-neighborhood, then the state of pi, afier k cycles of any distributed algorithm computing on I ,

>

is identical to the state of pj af(er k cycles of the same algoriihm computing on

I'. Prooft Straightforward by induction on A. 0 Given a string z,we denote by xi, i 2 1 the string obtained concatenating i times string z; and by [XI', i 2 0, the i-th cyclic permutation of z, that is [z]$= Z(d+j)mdlrl, for each 0 j < 1x1. Definitionl. A function F is invariant under cyclic permutations iff 3 ( x ) = F([x]~),for each string x and i 2 0. example: The functions addition, mu1tiplication, average, minimum, maximum over n numbers are invariant under cyclic permutations. Lemma 3 Each function computable in RM (or R ~ is)invariant under cyclic pennutations.

Proof:Let 3be a function, and A be an algorithm computing 3in synchronous RM (or RV). Consider two input configurations I and [I]', t 2 1, and let k be the number of steps required by the algorithm A to compute F on input I. It is easy to see that the k-neighborhood of pi(I), 0 i < 111, is equal to the kneighborhood of p(r+t)md(ril([l'Jt).Hence by lemma 1it must be F(I) = 3(Mt), 0 that is 3is invariant under cyclic permutations. Definition2. A function F is idempoient iff 3(xi) = 3(zi+'), for each string z and i 2 1. example: Average, minimum, maximum over n numbers are idempotent functions. Lemma 3 Each fundion computable in RM is idempotent. Proofi Let 3be a function, and A be an algorithm computing F i n synchronous

R ~ .

Consider two input configurations P and P+', f 2 1,and let k be the number of steps required by the algorithm A to compute 7 on input when M = JIl(t+l). Note that the k-neighborhood of P~(P+'), 0 5 i < IIl(t I), is equal to the k-neighborhood of pimod(lrlt)((I)t).Hence, by lemma 1 it must be F ( f ) = 0 T(P+'),that is 3 is idempotent. Given a string w, we denote by ( w ) the set of different symbols occurring in w.

+

Definition3. A function 3 is strongly idempotent iff there exists a computable function g such that for each cu, w' E {w)* and i 2 0 we have F(W~(~)+'W') = F(m)* exampIe: Minimum, maximum over n numbers are strongly idempotent.

Lemma 4 Each function computable in R~ is strongly idempotent.

Pmofi Let F be a function and A be an algorithm computing F in synchronous

RV.

Consider the function t(*)such that t ( w ) is equal to twice the number of steps to compute 3 ( w ) by A. Note that this function is computable, because we have only to emulate the behavior of A on the string w. We prove the thesis showing that F(w) = F(W~(~)+~W') for each w, w' E {w)* andiz0. Letting I = w and I' = wt("')+'w', it is straightforward to observe that the (?)-neighborhood of h ( I ) is equal to the (?)-neighborhood of pw(I0).

9

Since pl on I halts after steps giving as output 3(tu), applying the lemma 1 we can conclude that also p u on I' has to stop after steps and computes 0 the same output value. &

3 Computable functions

A string z is a substring of y if y = uzv with u, v arbitrary string. z is a pmfiz of 9 if = xu; and z is a sufiz of y if y = vz. The mdiz of z is its prefix of minimal length r such that r = zt. z is a period of z iff z = zk 2,where z' is a prefix of z.

Lemma 5 [11, 19]I/a and b a n pen'ods of z, and tat+ Ib( 5 then t h e n ezists a period c 01%of length gcd (lal, 161).

+gcd {lal, I&[],

Lemma 6 Lei z be a siring with mdiz z and 3 be a function invariant under cyclic pennutations and idempotent. Then F(z) = ~ ( [ r ] 'for ) , each i 2 0. Proofi S i c e F is idempotent we have F(x) = F(z). Moreover since 7 is invariant under cyclic pcrmntations we have F(z) = F([z]'), i 2 0. 0 Letting I = ~ 0 x 1...zn-1 be a string of integers, with i,, = marq.=o,...,,,1{zi), we have

Lemma 7 Functions that are idempotent and inwan'ant under cyclic permutabits. tions am computable in RM with 2nM messages of length log&,,

Prwfi Let 3be a function idempotent and invariant under cyclic permutations. Consider the following protocol for unidirectional asynchronous R*:

-send(input)

;z := input

-fori:=1&2M-l&

receive(y) ;t := y t ;send(y) - find b minimum period of z - output = 3 ( b ) ; halt

In the first two steps each processor accumulates the labels of the 2M-1 previous processors in the ring. Then it finds the minimum period (this can be done in linear time (111).

We show that starting with the input configuration I,processor pi, 0 i < n, finds v such that v = [z]', where z is the radix of I and thus the thesis follows hmlernma6. Suppose I = rt,t 2 1,then [q' = ( [ ~ ] 3After ~ . the first twosteps of the protocol, we have z = ([Z]')~'Z', k= 1%' 1 = 2M mod n and 2 is a prefix of ([%Ii)' (note that, [q' is a prefix of x). We can rewrite the previous relation as z = ([~]')~~+jr", where r" is a prefix of [z]'. Hence [z]' is a period of z. By contradiction, suppose that there exists a period b such that lbl < 1[z]'1. Then we have ibl+ I[z]'I < 21[z]'l 5 21[Jlil = 2111 2M < 1x1 + gcd(lbl,l[z]'J) and, by lemma 5, there exists a period c of z such that [%Ii = 8, r > 1. Note that lcl divides l[zIil and remember that [z]' is a period of x. From these observations and fiom definition [qi = ([z]')', we can mite [Ai = Ct and then I = [[q'In-' = [crtIn-' = ([c]*-')~~. Since l[cIn-'1 = lcl < I[z]'I = 1x1, it follows that z cannot be the radix of I. o Theorem 1 A jknction 3 is computable in R~ if and only i f 3 is idempotent and invariant under cgclic pemutations. Momover the computation of F can be performed with O(nM) messages. 0 Proof:Immediate fkom lemmas 2, 3 and 7. In the general we cannot infer the value of n in R ~see, section 5. In section 4 we will prove that some functions require R(nM) messages to be computed in asynchronous R ~ Hence, . the general protocol in lemma 7 is optimal with respect to the message complexity, in the asynchronous case. In the synchronous case this result doesn't hold, in fact substituting the value M to all the occurrences of n in the protocol in [2], it is possible to compute all the function in R~ with O(n log n) messages (see [5] for the proof).

Lemma 8 A function is invariant under cyclic permutations and strongly idempotent iff for each w, w' such that (w'} = { w } we have 3(w) = 3(wt).

Proof: Let 3 be a function invariant under cyclic permutation and strongly idempotent and w and w' two strings such that {w'j = {w). Since F is strongly idempotent then 3 ( ~ 9 ( ~ ) + 9 (wfg(")+g(~')) " = F(w) and 3(~~fl(~)+g(~')w9(~1+= g ( 3(wt). ~')) Note that 3 is also invariant under cyclic permutations, hence 3(w) = F(w'). If 3(w) = 3(d)when {w') = {w), then 3(w) = 3([wI1) since { w ) = ([wI1}, and ~ ( t a r ( ~ ) + ~ t= u '3(w), ) since (w~(w)+'w') = ( w ) . That is 3 is an invariant under cyclic permutations and strongly idempotent. o

Definition4. Let t be a totalorder relation on sets of integers such that, given two sets A and B if IAI > IBI then A t B. Lemma 9 Each fonction inuan'ant under cyclic pemautations and strongly idempotent is computable in Rv with O(n min (log n, V)) messages. Proof: Letting 3 be invariant under cyclic permutations and strongly idempotent, we provide a distributed protocol to compute 3 on an asynchronous unidirectional R~ requiring O(n min (log n, V)) messages. This protocol, given

for an arbitrary processor p, is a variant of the one presented in [2] for the distribution of the information in a synchronous ring. Note that in our case the ring is asynchronous, thus we provide a different condition to check the situation of "symmetry".

- ep := {input); z := 0; Active := true -While Active &

if Iep I= V then send(ep) ; output := 3(ep) ;halt

endif send(+);receive(el) send(el);receive(e2) if(ep + el) or (e2 s el) or (el = e2 =ep) * then Active := false * else send(ep U el Ue2) ;receive(ep) endif I eP I = V then send(ep) ;output := F(ep) ; send(ep);receive(el) send(el);receive(e2) % ( e p u e l Ue2 =ep) * then Active := false * &gep=epUelUe2 - end While - b e a t forever receive(set) Iset I= V then send(se2) ; output := 3(set) ;halt endif = 0 then set := set U {input) endif send(se2); o := (z 1) mod 3 - end Repeat

+

While the processor p is alive, that is Active = true, it executes the WHILE statement consisting of two phases. In the first phase p receivea the two labels (sets) el and e2 of the two living processors preceding it (possibly one of these is itself), and "comparesn el with 9 and e2 to decide its status. Since the alive processors are the ones for which el constitutes a 'local maximan (according to s), we can infer (see [2]) that at least f of the alive processors become passive, that is Active = false. Then, the consistency of the labels in the ring is maintained, in the sense that for each alive processor p, the label e, is updated in order to be equal to the set of inputlabels of the passive processors pl, ,pk that precede p and such that there is no an active processor between p and pi, i = 1,...,E. During the second phase, a processor remains active if the labels (sets) of the two processors preceding it contain at least a new input-label. I t is simple to prove that in this case, the consistency of the labels of the living processors is preserved, hence we do not need to collect the labels of the processors become passive during this phase, for this reason we update only ep without exchanging messages. Note that, at the end of the two phases alive processors have surely incremented their set of labels of at least a new input-label. Then, at most V iterations

...

can be periormed before that living processors contain all the V labels. Moreover, at least a constant fraction of alive processors die at each while-iteration (phase I), hence at most O(1og n) iterations can be perf'ormed before that a single alive processor remains in the ring. This processor has collected all the V labels. Since at each phase each processor sends a constant number of messages, we have that the total number of messages sent at each phase is O(n). From the o considerations above, we can derive the bound claimed in the theorem.

Theorem 2 A farnction 3 is computable in'R if and on$ if F is invariant trnder cyclic pennutations and strongly idempotent. Momover the computation of 3 can be performed with O(n min {log n, V)) messages.

Proofi Immediate from lemmas 2 and 4 and 9.

o

In section 4 we show that our distributed protocol is optimal with respect to the number of messages sent by the processors in the ring.

h this section we show that the protocols used in section 3 to compute functions in RM or , 'R cannot be improved with respect to the message complexity. In particular, we prove that there exist functions whose computation requires Q(nM) messages in RM and L?(n min {V,log n)) messages in R" . The lower bound technique introduced in [2], for algorithms on anonymous rings with knowledge of their exact size, may be slightly modified to work in our case.

Theorem 3 Let 3 be a function computable in R ~If .t h e n exist a and b such ) each n, m 2 1, ihen any algorithm A computing 3 that 3(an) # ~ ( a ~ - ' bfor in R~ tqtiim O(nM) messages in the worst case. Proof: Let be n 5 M. Consider the synchronized execution of A on the input configuration an and let T be the number of steps required by A to compute 3. By lemma 1 it is simple to see that the number of messages sent will be a t 1-t n T. Let consider a second execution of A on the input configuration aM-lb. Note that processor pl in the first execution has the same [?I-neighborhood of the M-I in the second execution. Thus by lemma 1 we can infer that processor p LTJ T> otherwise both processors terminate the execution computing the 13 same result, contradicting the assumption that 3 ( a n ) # 3(a"-' b).

[YJ,

Corollary 1 Any asynchronous algon'thm for computing AND, MIN, OR and

MAX fPInciions in R ~ rcpuins , O ( n M ) messages in the worst case. Proofi The proof immediatelyfollows from the theorem 3 observing that 3(On) # 3(oM-'1) for the MAX and OR functions, while 3 ( l n ) # 3(1M-10) for the

o

MIN and AND functions.

For the next results, a simple but useful observation is:if on a linear (bidirectional) array of processors with input configuration I an asynchronous algorithm A for (bidirectional) RV sends n ( M ) messages, then the algorithm A requires R(M) me85ages in RV for an input configuration I' having I as subsequence. Hence given an algorithm A for (bidirectional) RV, a lower bound for A on linear (bidirectional) arrays is a lower bound for A on (bidirectional) RV. Definition5. Given an asynchronous algorithm A on (bidirectional) RV, by Mt(z,l), where z,I 2 0 and z min{l, V), we denote the largest number w for which there exists a set B of infinitely many different sequences I of length I , each having exactly z different labels, and such that more than w messages are sent in the worst case when A is executed on a linear (bidirectional) array of processors with input configuration I. Lemma 10 For every maaimurn finding asgnchmnous algorithm A on (bidinctional) R~ and V > 1 it holds ME(1,l) 2 1 Proof= By contradiction we assume that MyA(1,l) = 0, since there exist infinitely many sequences, thus in particular we can consider V 1distinct labels a1 < < av+l such that, no message is sent during the execution of A on a processor with input value R, 1 5 i 5 V 1. It is easy to see that during the execution of A in the rings with input configurations Il = a1 .av or I2 = a1 av- 1av+l no message is sent. Since for example, the prosor p~ compufes the same output value for both the input configurations, but MAX(al .av) # MAX(a1. . a ~ - l a v + ~ hence ), it must be MC(1,l) 2 1. 0

+

...

+

... ..

..

.

Lemma 11 If V = 2',2 2 0, then for every mazimum finding asynchronous algorithm A on (bidirectional) R ~ it, holds Me(V, V) = SZ(V log V). P r d It will be sufficient to show that if V = 2", it holds ~ t ( 2 ' + ' , 2'+') > i < z and the thesis will follow by solving the 2~6(2', 2') + 2'- - 2, 0 recurrent relation for M# and using the lemma 10. Assume that ~ 4 ( 2 (2'), = rn then there exists a set B having infinitely many sequences of length 2', 0 5 i < t,and having 2' different labels such that, more than m messages are sent in the worst case when A is used on a linear array of processors having one such sequence as input configuration. Takesl . . . , ~K~=, 5+1,si E 8, i s K, and assume without loasof generality that the maximum label in all the siYs belongs to SK. Consider now the set C = (scsi+l, 1 5 i < K 1,S K - ~ S ~sK-2sK , ,.sKsl}.We prove that at least one sequence in C of length 2'+' 'and having 2"+' distinct labels, when it is given as input configuration on a linear array, it determines an execution of A requiring more than 2m 2'-l- 2 messages. Let start the execution of A in RV with input configurations Il = sl . s ~ - 1 and I2= sl ...SK-~SK, h t of all by sending and receiving as many messages as possible within the sequences of processors with si's input configuration 1 i K, (such an execution take place when the transmission delays on the channels

1s

-

+

..


0, t h e n exists a proiocol that elects a leader in a synchronous R~ with probability 2 1- E , exchanging O(n in bits.

5)

Proof:The protocol is as follows.

- state :="alivem; i := 1 - repeat forever

5

while (i 5 (1n $)) h (stote = "aliven) do begin * flip a biased binary coin b, with probability of 1-outcome equal to p. * b = 1 then stote :="pseud+dead" & send("aliven) ; wait M steps; remove the received "alive" msg endif * i:=i+l * end while iT state =" aliven then send("leaderm);state :=" leader" ;halt endif a if state ="pseud+deadn then * wait for M steps forwarding received msgs * if "alive" not received then state :="alive" &g state := "deadn * end if state ="deadm then * receivelm) * if m = "leader" then state := "no-leader" ; halt endif * send(m) * end if - end repeat The wait steps in the protocol guarantee the synchronization of alive processors. To analyze the protocol, note that in the state aliue, a processor uses a coin to go to the next state pseudo-dead (coin = I), or to remain alive (coin = 0). In the state pseudo-dead, it checks if all the processors have died in the current phase, and in this case it becomes alive again. Let Nt be the number of processors that are alive in the phase t. We can assume 1 5 Nt n 5 M, since the situation in which all procemm die in the same phase can be checked by the protocol. In fact in this case, the dead processors become again dive, that is Nt+l = Nt. It is simple to see that Nt 2 Nr+l, Vt 2 1and No = n. We would guarantee that at least a constant fraction of the alive processors That is, the die in the current phase, but not all of them die (call this "succe~~"). number of phases to leave exactly one processor aliue is O(1og n), hence O(1og M). We denote by St the "SUCC~~S" event in phase t , that is St = {Nt Nt+I > PNi } . In, each step, let /3 = and p = be the probability of a processor to die (coin = 1). These values guarantee that the probability of success (?(St)) in each phase is greater than In fact we have:

-

A.

where Dt denotes the number of processors for which coin = 1 in phase t. P(Dt = Nt) is the probability that all the alive processors die in the current phase. P(Dt 5 PNt) is the probability that the number of dead processors is at most a fiaction j9 of Nt . It is simple to see that

P(Dt = Nt) = p N t

5

p2 ift

(5)

since Nt 2 2 when we have not yet elected the leader. Using Chernoff bound we also have:

From relation (4) to (6) we have ?(St) > f = q, independent of the phase. Let us now evaluate the number of phases m needed to have at least log+ M < In M successes, which in turn guarantee to leave only one alive processor, with probability 2 1 c. Applying a Chernoff bound, we easily find

-

S i at each atep the alive processors fires a control bit that walks in the ring until it reaches the next alive processor, each step has O(n) bit complexity,thus proving the claimed upper bounds on time and number of bits. o Note that it is possible to improve the previous algorithm attaining an a l p rithm exchanging the same number of bits but requiring O(nln $) time eomplexity. Acknowledgments

We are very grateful to b b e r t o Grossi for useful discussions and suggestions on Lemma 7.

References 1.

D. Anglain. Local and global properties in networks of processors. In Pmc. 12th

ACM Symp. on Theory of Computing, 82-93, 1980. 2. H. Attiya, M. Snir, and M. K. Warmuth. Computing on an anonymous ring. Journal of the ACM, 35(4):845-875, 1988. 3. H . L. Bodlaender, S. Moran and M. K. Warmuth. The distributed bit complexity of the ring: from the anonymous to the Nan-anonymous case. Infomation and Computation, 108(1):34-50, 1994. 4. W. Feller. An Introduction to Probability Theory and Its Applications, volume 1/2. Wiley Series in Probab'ity and Mathematicd Statistics, 1967. 5. P. Ferragina, A. Monti, and A. Roncato. Tradeoff between computational power and common knowledge in anonymous ring. Technical Report 23/93, DipartC mento di Infomatico, Univetaity of Pha,Italy.

6. G. N. Eketierickson and N. A. Lynch. Electing a leader in synchronous ring. Journ d of the ACM, 34:95-115,1987. 7. G.N. Frederickson and N. Santoro. B d i n g simrnetry in synchmnow networks, pages 26-33. Proc. 2nd Int. Workshop on Parallel Computing and VLSI, Lecture Notes in Computer Science, vol. 227. Springer-Verlag, 1986. 8. Joseph Y. Halpern and Yoram Moses. Knowledge and common knowledge in a distributed environment. Journal of the A CM, 37(3):549-587, 1990. 9. A. Itai and M. Rodeh. S i m e t r y breaking in distributed networks. Infomation and Computation, 88:60-87, 1990. 10. A. Israeli, E. Kranakis, D. Krizanc and N. Santoro. 'Ibade-offs for the weak &n problem. Ciac '94, Lecture Notes in Computer Science, vol. 778, Springer-Verlag, 167-178, 1994. 11. D. E. Knath, 3. H. Morris, and V. R Pratt. Fast pattern matching in strings. SIAM Journal of Computing, 6(2):323-350, 1977. 12. J. E. Van Leeuwen, N1 Santoro, J. Urrutia, and S. Zaks. Guessing games ond distributed computations in aynchmnous networks, pages 347-356. Proc. 14th Int. Con. on Automata, Languages and Programming, Lecture Notes in Computer Sdence, vol. 267. Springer-Verlag, 1987. 13. R. C. Lyndon and M. P. Schiitzenberger. The equation om = bn cP in a free group. Michigan Math. J., 9:289-298, 1962. 14. J. Pachl, E. Korach, and D. Rotem. Lower bounds for distributed maximum finding algorithms. Journal of the ACM, 31(4):905-938, 1984. 15. N. Santoro. Computing with time: temporal dimensions in distributed computing. PYUC.28th Allerton Confemnce on Communication, Contmt and Computing, 358567, 1990. 16. P. Vitanyi. Distributed elections in an Archimedean ring of entities. In Pmc. 16th ACM S p p . on Theory of Computing, 542447, 1984.

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This article was processed using the iBT$ macro package with SICC style

48

Orientation of Distributed Networks: Graph- and Group-Theoretic Modelling Christian Lavault

LIPN-CNRSURA 1507 (Institut Golile'e) University of Paris-Nord, Av. J.B. Climent 93430 Villetaneuse. France (On leave from Irisa-Inria Rennes. France) ([email protected] & lavaultQural507.univ-parisl3.fi) Abstract. The present paper surveys the most recent and promising results about graph-theoretic and grouptheoretic modelling, from the viewpoint of relationships between Structural Informotion (e.g. Sense of Direction) and communication complexity in distributed computing. The specific behaviour of various classes of networks (Cayley and Bore1 Cayley networks, de Bruijn and Kautz networks, etc.) is studied in terms of usual efficiency requirements, such as computability, routing, symmetry, and algebraic structure. We also address various problems arising from the definition and the application of the notion of sense of direction on several significant network topologies. A rough approximation of a "measure of densityn of networks is proposed. It leads to a conjecture about the real impact of sense of direction with respect to Leader Election.

I Introduction One of the main topics under investigation in distributed computing concerns the study and design of network topologies which have optimal efficiency with regard to several specific parameters, such as communication complexity of leader election, spanning tree construction, or broadcasting, ease of routing and message transmission, fault- tolerance, etc. To optimize the communication complexity of distributed algorithms, one introduces labellings on the network links in order to give the network Structural Information, and more precisely a ?Sense of Direction" (or 'jOrientation"). Above all, this paper is a survey on the present "state of the art" in graphtheoretic and group-theoretic modelling of interconnection networks in terms of sense of direction. Since it has developed t o a "bench-markn, we consider the Leader Election Problem (LEP) to study the effect of structural information on the communication complexity. We primarily focus on those symmetric interconnection networks which belong t o the class of Cayley networks[l3]: one can take advantage of the rich underlying group theoretic structure of the networks in order to obtain a more systematic study of their properties. Furthermore, another advantage of analysing such networks in an algebraic setting is that many properties of these networks can be proved for the class as a whole, instead of proving the property for each network independently.

The paper is organized as follows. In the Introduction, the model of distributed network, the notion of sense of direction and preliminaries are presented. In Section 2, we survey the properties generated from the very rich algebraic structure of the Cayley networks. In Section 3, we study the specific properties of another important family of networks, viz. de Bruijn and Kautz networks, in terms of various efficiency requirements. The impact of orientation on the communication complexity of the leader election problem is addressed in Section 4 for several network topologies. Section 5 is devoted to a discussion concerning the notion of symmetry and some issues about the Edge Selection Problem (ESP), the Spanning Tree construction Problem (STP), the Minimum Spanning n e e construction Problem (MSTP), and the Orientation Labelling Problem (OLP). (See [25].) Section 6 offers some conclusions and raises open problems in this domain of research. 1.1

The Model

The model is a standard point-to-point asynchronous network Af of N processes connected by m bidirectional communication links. As usual, the network topology is described by an undirected, connected graph (V, E) (devoid of multiple edges and loopfree): (V, E) is defined on a set V of vertices representing the processes of Af, and E is a set of edges representing the bidirectional communication links of N operating between neighbouring vertices. In the sequel, IVI = N is the order of the graph and IEJ= m is its number of edges (or its size), and similarly for Af, which is thus denoted by Af = (V, E), in order to simplify notation. Given a message driven algorithm A on N, it is assumed that the messages are transferred on links in FIFO order, without error, and in a finite but unbounded delay. The worst-case message complexity of A (for a given input size N) is the maximum over all networks N of order N of the largest number of messages sent in any execution of A on N. 1.2

Sense of Direction

It was first pointed out by Nicola Santoro in [IS]that the availability of a sense of direction (or orientation) improves the communication complexity of various fundamental distributed computations in networks of several topologies. The notion of sense of direction refers to this capability of a processor (or a process) to distinguish between its adjacent communication links (or its ports), according to a globally consistent scheme [18,19). In order to give a network a sense of direction, one introduces labellings on (a subset) of its links. Intuitively speaking, the labelling of a graph is understood as defining an assignment, in each node, of distinct labels to some (or all) of the edges of this node. An orientation is defined as a labelling where the labels also satisfy an additional global consistency property, which determines a local knowledge of a certain global direction thus defined within the network.

In an arbitrary distributed network Af = (V, E), a natural globally consistent labelling on the links of the network is defined as follows in [16]. Fix a cyclic ordering of all the processors. N has a global sense of direction if at each processor each incident link is labelled'according to the distance in the above cycle to the other nodes reached by this link. In particular, if a link, between two processors P and Q, is labelled by distance d at processor P, this link Is labelled by N - d at the other incident processor Q, where N = IVI. Note that such a definition requires the knowledge of the order N of the network, and it includes as special cases the definition of sense of direction for known specific topologies, such as the oriented Ring, the oriented Complete network, the oriented Chordal ring or Circulant Graph, etc. 1.3

Preliminaries

We are interested primarily in regular networks of small degree. For a given small degree, we are interested in dense networks; a dense network nf is one of large order N for a given diameter D, defined as the maximum distance between all nodes pairs in I\(. Here, the distance between two nodes refers to the smallest number of hops between these two nodes. Obviously, a dense network I\( allows the interconnection of a large number of processing elements with relatively small communication delay. Besides density, vertex symmetry (or vertex-transitivit y) is another desirable attribute of an efficient interconnection network topology. This notion of symmetry implies that for any two nodes u, v E V there exists a label preserving automorphism y E Aut(G) such that ~ ( u=) v [12, 13, 201: informally, a vertex symmetric (or vertex-transitive) network looks the same from any node. This property allows the use of identical routing algorithms at every node, and makes it possible to define a natural labelling which provides the network with a sense of direction. Many well-known interconnection networks, such as complete networks, Rings, tori, hypercubes, cube-connected cycles, nstars, e t ~,.are examples of such vertex symmetric networks. Most of them belong to the class of Cayley graphs which are connected graphs constructed from a group and a set of generators as defined in section 2. (See also [I, 7, 12, 13, 20, 211-1 In Section 3 we deal with another class of network, whose characteristic is to have the largest number of vertices for given maximum degree A and diameter D, viz. de Bruijn and Kautz networks. Though they are not vertex symmetric, these networks enjoy very interesting properties, such as having an optimal number of nodes (for small value of D or A), easy routings, an optimal fault-tolerance, the feasibility for designing efficient "consensus protocols" , symmetry of extensions, possibilities of quasi-optimal generalizations in all respects, etc. Notation. In the sequel, we use the usual terminology of group theory and graph theory. Since we only consider finite groups, the groups are mainly represented as permutation groups. The following notation is used:

2, for the ring of integers 0, 1, ...,q - 1 (modulo q), and (2,)" for the ndimensional vector space over 2, (p being a prime); S, for the symmetric group on n symbols; (S) for the group generated by the set S of generators; In for the trivial identity group consisting of the identity permutation in S,; e for the identity element of a group; and lul for the order of u E G, i.e. the smallest positive integer k such that uk = e. Given a (finite) group G and H 5 G (H is a subgroup of G), (G : H ) denotes the (finite) index of H in G, i.e. the (finite) number of cosets of H in G.

2 2.1

Cayley Networks Definition of Cayley Networks

Definitionl. Let G be a group and let S E G be a set of generators of G. The Cayley network Ns of G with set of generators S is Ns = (V, E), where V = G and E = {(u, v) / u-lv E S).We assume that S = S-l, where S-I is the set of g'l such that g E S, so that Ns can be viewed as undirected. To avoid loops in the network Ns, we assume e flS; further, if g = g-l then we identify the edges g and g-l. The Cayley network Nshas ]GI = N nodes and the degree of each node is ISI, denoted by 4 s ) . For each g E G, let [gIs denote the least number of generators from S needed to represent g (with possible repetitions). The diameter of Nsis D(S) = max{bIs / g e GIThe resulting Cayley network depends on the set S of generators. On the one hand we can choose S = G, in which case Ns is the complete network Kpy with IGI = N nodes. On the other hand, we are usually interested in "small" sets of generators with "not too big" diameter. In fact, as pointed out in [12], we would want sets S of generators minimizing the quantity D2(S) CgES Igl2. It is well known that such small S do exist, since every finite group G has a set of generators of order O(1og IGl). (We refer to [I, 2, 20, 2 11 for more indications on the importance of such groups.) Examples. To simplify notation the elements of S are listed without their inverses and multiplication of permutations is considered to the left. Throughout, we assume that n is arbitrary but fixed. The first four Cayley networks of Table 1 are arising from cyclic, abelian and dihedral groups. With cyclic groups, we obtain a variety of tori: the oriented Ring, the double Ring (with n # 2) and the d-dimensional Torus. The group of automorphisms of the n-dimensional Hypercube is denoted by r,; it is the group of bit-complement automorphisms, and r,, S (22)" The generators p{i) of rnare such that p{i}(bl, ...,bn) is the sequence of bits obtained from (b1, .. .,bn) by complementing the ith bit, while leaving the others unchanged. The last three Cayley networks are examples arising from the symmetric group S,.

Network N Group Generators Oriented Ring Cn (lJ,*--,n) Double Ring Dn ( l , , . . .n ) ,Pn d-Torus (~n)* direct product n - H ~ ~ e r c u b e r m V{I), ' P ~ o )-. , P{n) n-Star (l,k)l the group generated by the set G of generators, e denotes the identity element of the group,

and for u E G, I u I denotes the order of u in the group G, i.e. the smallest positive integer k such that uk = e. In addition, El @ G2 denotes the direct product of the groups Q1,G2; its elements consist of the pairs ( g l ,g 2 ) such that gl E 91,92 'u C2 with multiplication ( g l , g z ) ( g i , g;) = ( g l g ; , gag;). We use G 5 G' to denote that G is a subgroup of 6'.

2

Arbitrary Groups

We are interested on whether or not the introduction of the labeling CG alters the class of functions which are computable in the network. More specifically, we call the labeling Lo strong if there is a Boolean function on N = variables which is computable in the network No[&] but not computable in No. If we ignore labels then it is clear that Aut(NG)consists of all permutations 4 of Q such that for all u, v E G, u-lv E G o 4(u)-'d(v) E G . Two more groups of automorphisms that will be useful in our subsequent study are defined as follows.

and Aut**(No)= { 4 'u Aut(No) : Vu, a E G(4(u)-'$(ua) E< a >)}.

Now we have the following inequalities.

To prove that that

LG is strong it is enough to define a Boolean function f such Aut(No [ L o ] )_< S ( f ) and Aut(Ne) g S(f).

(3)

Indeed, in view of Theorem 2, f must be computable on &[CG],but it can not be computable on No since f is not invariant under all the automorphisms of

NG. In a sense, this last theorem shows that the product of a strong labeling with an arbitrary labeling is a strong labeling. 2.1

Distinguishing by network automorphism

Now we can prove the following theorem which establishes a sufficient condition for the network NG[LG]to have more computational power than the network

NG. Theorem3. If Aute*(NG)# Aut(NG) then Lo is strong.

PROOFLet # E Aut(NG)\Autm (NG).Since 4 @ Aut" (NG)there exists u, a E G such that #(ua) # q5(u)ak,for all 1 5 k < la1 . Define a Boolean function on inputs < b, : x E > as follows.

It is easy to see that f is kept invariant by all automorphisms of No[&], but this is not true for the above automorphism 4. To see this consider an input < b, : z E G > such that Qz E S(b. = b,.) and b4(,) # bO(,,). It follows that

This completes the proof of the theorem. 0 In view of Theorem 3 and inequality (2) to prove that enough to prove that Aut*(NG)# Aut(NG).

LG is strong it is

Theorem 4. Assume Gi is a sei of generators for the group Gi, with i = 1,2, G = G 1 u G 2 , G l n G 2 = 0 a n d S = S l @ & . Then

PROOF Assume on the contrary that

Let 41 E Aut(NG,).Define q5 as follows: O(ulu2)= q5l(ul)u~, where ul E Si and u2 E G2. Then for all u E 6 , with u = ulu2, ul E &,a2 E 62, and all g E G we have that if g E G 2 4 = { l u l u g if g E 01 . It follows that # E Aut(NG).Since Aut(NG)= Aute(NG)equation ( 4 ) implies E that 4l(ul)-'4l(ulg) E< g >, for all ul E h and g c G I . It follows that Aut*(n/C1). Consequently, Aut*(&, ) = Aut(NG,), which is a contradiction. This completes the proof of the theorem. 0 2.2

Distinguishing by group automorphism

Theorems. If 4 is an automorphism of the group G such that #(G) = G and #(g)#< g >, for some g E G,then Autm(NG) < Aut(&). PROOF Let us define Aut* (6) as the automorphisms of lowing condition

& satisfying the fol-

To prove the theorem we need the following precise characterization of Aut '(G). Lemma 6. The automorphisms of the Cayley network NG satisfying condition (5) are exactly the automorphisms # of the group G satisfying #(G)= G.

PROOFof Lemma 6. Let 4 be an automorphism of the group satisfying 4(G) = G. It follows from [3J[section16) that 4 is an automorphism of the corresponding unlabeled Cayley network and condition (5) is easily verified. For the other direction assume 4 is an automorphism of the network NGsatisfying condition (5). Let u E 9, g E G and put v = ug. Clearly, LG(u,v ) = g. Consequently, by (5) L G ( ~ ( u )4(ug)) , = 4(Lc(u, ug)) = 4(g)- This implies that 4(ug) = 4(u)b(g)Similarly, we can prove +(e) = e . Since G generates the group G, it is easy to show that 4 is a group automorphism. This proves the lemma. It follows from the lemma that if 4 E Aut'(G) and #(g) &, for some g E G, then 4 Aut'(NG). This completes the proof of the theorem. 0 Thus using Theorem 5 we can prove that Aut0(NG)# Aut(NG) for the star, bubble-sort and pancakesort networks previously considered. Ezample 1. LC is a strong labeling for the star, bubble sort and pancake-sort networks.

PROOF For each of the networks listed above we exhibit an automorphism 4 E Aut(G) such that d(G) = G but 4(g) 4< g >, for some g E G.

CASE1 n-Star: Consider the automorphism 4(r) = u-'ru, where u = (2,3). It is easy to check that that 4((1,2)) = (1,3)$< (1,2) > and 4(G) = G. CASE2 n-Bubble-Sort: Consider the automorphism 4(r) = u-I r u , where u = (1,2, ...,n). It is easy to check that that 4((k-1 mod n, k)) = (k, k+l modn) #< (k - 1modn, k) > and 4(G) = G.

CASE3 n-Pancake: Consider the automorphism 4(r) = p,rp,,. It is easy to check that that #(pk) = 0 pr: #< pk > and 4(G) = G. For any automorphism # E Aut(G) let Then we can prove the following theorem. Theorern7. For any automorphism g E G,then LC, is strong.

4E

Aut(G), if $(g) #< g

>, for

some

PROOF Let # be an automorphism in Aut(G). The set Gg generates G since G does. Moreover, it is trivial to check that 4(Go) = G+.Hence the result of the 0 theorem is immediate from Theorem 5.

3

Abelian Groups

Now we consider the case of abelian groups. In the most general case we have arbitrary sets of generators for such groups. A Boolean function f E BN represents a group 'H 5 SNif S(f) = ? where i,S(f) is the set of permutations on IV letters that leave f invariant on all inputs. Such groups are called representable

PI.

Theorem 8. If the group Aut (No [Lo]) is representable and Aut(NG[&I) Aut(NG) then Lo is strong. 0




It is now easy to check that for the eleven groups considered this implies that we can choose k = 1. This proves the claim. Since a theorem of Subidussi [12]implies that Aut(NG) is not abelian (the same result also follows directly from the next Theorem 13 without refering to [12])it follows that Aut(NG) # Autm(NG). This completes the proof of the theorem. 0 At this point it is interesting to note two interesting facts without proof. If A&(&) is the set of automorphisms of No fixing the identity element e of the group G then every 4 E Aut(NG)is of the form a $, for some a E S , $ E Aute(NG)(the same result holds for any of the groups Aut' (&), Autm(NG)). It is also a consequence of the definition of Autm(NG)and the proof of Lemma 11 that for a canonical set of generators of an abelian group G, AutZw(NG)= Aut*(G).We leave the details to the reader. Theorem 13. If G is a canonical set of generators for the group where S E (2,3,4,5} then

enE, Cn,

Aut (No)= Aut' (%) = @ Dn . nES

Moreover,

PROOFIn order to prove the theorem we need the following lemma. Lemma 14. Assume that G = G' @ Cn is an abelian group, G' a canonical set of generators for G', G = G'U { v ) and Ivl = 3 or lvl = 5. Moreover assume that 1. if IvI = 3 then for all g E GI 191 # 3, 2. if Ivl = 5 then for all g E G' 191 # 3,s.

PROOFof the lemma. Let 4 E Aut(NctU(,)) and suppose on the contrary that for some a E S , u E G' 4(av) = &(a)u(the case q5(av)= $(a)u-' is similar). We will derive a contradiction. First consider the case Ivl = 3. We have that for some ul, u2 E G' LJ(v}. But this implies that

If ul, u2 fZ { u ,U - l } then equation ( 6 ) implies that u = e, which is a contradiction. If ul E { u , u - l } while u2 { u ,u-l} then either ul = u which implies that u2u2 = e, or else ul = u-I which implies that u2 = e; in both cases we get a contradiction. Finally if ul, u* E { u ,u-'1 then either ul = uz = u , in which case (6) implies u3 = e (contradicting the fact that v is the unique element of order

3) or ul = u2 = u-I , in which case (6) implies u-I = e, or ul = u, u2 = u-I , in which case (6) implies u = e. This proves the lemma in the case lvl= 3. Next consider the case lvl = 5. As before there exist ul ,u2, ua, u4 E G' U {v) such that

=

U ~ U ~ U ~ Ue. ~ U

(7)

We consider five cases depending on whether or not 0,1,2,3,4 generators among the ul ,u2, ua, u4 are in the set {u, u- I). As before we use the fact that there is no generator in G' of order 3 or 5 to derive a contradiction. This proves the lemma. 0 In view of Lemma 14 it is enough to consider only groups of the form Cm@C2, with m > 2, as well as of the form Cm @ Cq,with m > 4. We show that in these cases as well Aut(NG) = Aut*(NG). This would imply that for all 16 abelian groups Cm, where S {2,3,4,5) and for any canonical set of generators G of that group Aut(NG) = Aui*(NG). First consider the case of the groups C, @ C2, with m > 2. Let u be a generator of Cm and v a generator of C2. Let 4 E AU~(N{,,.~)and suppose on the contrary that 4(uv) = +(u)u (the case #(uv) = 4(u)u-' is similar). We will derive a contradiction. It follows that d(u2v) = 4(u)ulu, for some ul E {u, u-l, v). If ul = u then 4(uku) = 4(u)uk, for all k . If ul = u-I then 4(ukv) = $ ( u ) u - ~ ,for all k. If ul = v then 4(ukv) = 4(u)uk-'v, for all k. But all these statements contradict the injectivity of 4. It remains to examine the case of the abelian groups g = C, @I C4 when m > 4. Let u, v be generators of Cm,C4, respectively. By contradiction, assume that for some a E g, and 4 E Ad(&), 4(av) = 4(a)u. But arguing as before this would imply that 4 is not 1- 1. The proof of Theorem 13 is complete. 0 The 11 abelian groups C., where S {2,3,4,5) and IS1 2 2 have a rather interesting behavior. Although Theorem 13 implies that the networks No and Nc [LG] cannot '4distinguish" the Boolean functions they can compute from their automorphism groups alone, Theorem 9 shows that in fact the labeled network &[LC] can compute more Boolean functions than the unlabeled network &. In particular, for these 11 abelian groups there exist Boolean functions which are computable on NG but such that S(f)2 Aut(NG).

en,,

4

Conclusion

We studied the labeling problem on anonymous Cayley networks and provided sufficient conditions for the labeling LC to be strong. For the case of abelian groups with canonical sets of generators we gave exact characterizations of the groups G for which the labeling LG is strong. A more general result characterizing strong labelings on arbitrary groups will undoubtedly require deeper understanding of the structure of the automorphish group of the group.

References 1. S. B. Akers and B. Krishnamurthy. A group theoretic model for symmetric interconnection network IEEE Transactions on Computers, 38(4):555 - 566, 1989.

2. H. Attiya, M. Snir, and M. Warmuth. Computing on an anonymous ring. Journal of the ACM, 35(4):845 - 875, 1988. (Preliminary version has appeared in proceedings of the 4th Annual ACM Symposium on Principles of Distributed Computation, 1985). 3. N. Biggs. Algebraic Graph Theory. Cambridge University Press, 1974. 4. E Y . Chao. On a theorem of Subidussi. Proceedings American Mathematical Society, 15:291 - 292, 1964. 5. P. Clote and E. Kranakis. Boolean functions invariance groups and parallel complexity. SIA M Journal on Computing, 20(3):553 - 590, 1991. (Preliminary version has appeared in Proceedings of 4th Annual IEEE Conference on Structure in Complexity Theory, Eugene, pp. 55 - 66). 6. W. Imrich. Graphen mit transitiver Automorphimgruppe. Monatshefte f6r Mathematik, 73:341 - 347, 1969. 7. W. Imrich. Graphs with transitive abelian automorphisms groups. In P. Erdes, A. Renyi, and V. S6s, editors, Combinatorial Theory and its Applications, volume 11. North Holland, 1970. 8. A. Israeli and M. Jalfon. Uniform self-stabilizing ring orientation. Information and Computation, 1992. to appear. 9. E. Korach, S. Moran, and S. Zaks. Tight upper and lower bounds for some distributed algorithms for a complete network of processors. In Proceedings of 3th Annual ACM Symposium on Principles of Distributed Computation, pages 199 207, 1984. 10. E. Kranakis and D. Krizanc. Distributed computing on Cayley networks. In Proceedings of the 4th IEEE Symposium on Parallel and Distributed Processing, Arlington, Texas, Dec. 1-4, 1992. 11. N. Santoro. Sense of direction, topological awareness and communication complexity. ACM SIGACT News, (16):50 - 56, 1984. 12. G. Subidussi. Vertex transitive graphs. Monatshefte ftir Mathematik, 68:426 - 438, 1964. 13. V. Syrotiuk and J. Pachl. A distributed ring orientation problem. In Jan van Leeuwen, editor, Proceedings of 2nd International Workshop on Distributed Algorithms, Amsterdam, July 1987, pages 332 - 336, Heidelberg, 1988. Springer Verlag Lecture Notes in Computer Science. Vol. 312. 14. G. Tel. Network orientation. Technical Report RUU-CS-91-8, University of Utrecht , Department of Computer Science, 1991. 15. H. Wielandt. Finite Permutation Groups. Academic Press, 1964. 16. M. Yamashita and T. Kameda. Computing functions on an anonymous network. Technical Report 87-16, Laboratory for Computer and Communication Research, Simon Fraser University, 1987. 27 page.. 17. M. Yamashita and T. Kameda. Computing on an anonymous network. In 7th Annual ACM Symposium on Principles of Distributed Computation, pages 117 130, 1988.

This article was processed using the IBTEX macro package with SICC style

82

Classifying Anonymous Networks: When Can Two Networks Compute The Same Set Of Vector-Valued Functions? Nancy Norris Department of Mathematics University of California Santa Cruz, Santa Cruz, C-4, USA, 95060 (nannoorOcis.ucsc.edu)

Abstract. An "anonymous networkn is a computer network in which all processors run the same algorithm during a computation. This paper addresses the problem of classifying anonymous networks by the functions they can compute: Let us say that two networks are "j-equivalent" if the set of vector-valued functions each can compute is the same. We will find an algorithm polynomial in the number of processors in a network for determining whether two networks are in the same f equivalence-class. Thus, classifying networks by what they can compute is not hard. Before we derive this algorithm we will characterize the set of vector-valued functions that a given anonymous network can compute, in terms of the network's toplogy. This extends results Characterizing the scalar-valued functions computable on an anonymous ring ([2]) and on an arbitrary anonymous network (1131). We wilI also develop algebraic and topological techniques for handling edge-labeled directed graphs.

1 Introduction A network is a collection of arbitrarily powerful processors connected in some configuration by communication links. This paper addresses two questions about "anonymous networks", whose processors do not have access to their ids and as a consequence all run the same algorithm during a computation. In anonymous networks the effects of network topology on network behavior are sharply displayed. There are a number of problems; for instance, leader election, which are trivially solvable on non-anonymous networks but whose solvability on an anonymous network depends on the network's topology. A network with a highly symmetrical graph may not be able, for instance, to distributively choose one processor as a leader, since processors may "look alike" to one another. One of the main computational problem facing the processors in an anonymous network is that of deducing global information-e.g., the number of processors in the network or the absolute position of any processor in the network's graph-from information obtained locally by exchanging messages with adjacent processors. The problems an anonymous network cannot solve are precisely those which require global information which cannot be obtained by "patching together" local models of the network. Thus, for instance, a network may not be

able to distributively compute the average of the input-values given to the processors, because a given processor may not be able to tell whether a message received through two of its links originated from one processor or two. We will address two problems in this paper; the problem of characterizing the set of vector-valued functions a network can anonymously compute, and the problem of classifying networks by what they can compute. The first problem can be stated as follows. Let us say that a network with n processors computes a function f (x) = ( f ( ~ ).~..,,f(x),) given input x = (xl,...,x,) if when each processor i E {I,. ..,n) is given input x i , it computes f ( x ) ~by exchanging messages with neighboring processors. The problem is to characterize the functions a given network can anonymously compute in terms of the network's topology. For this we will assume that processors can distinguish among their links, and also that they know the graph of their network but not their location in the graph. (But see [14, 151, in which the authors investigated network capability under a variety of assumptions about what processors know about the network.) The scalar-valued functions computable by a network have been characterized previously under a slightly different network model in [13]. This extended the characterization in [2] of the scalar-valued functions computable by a ring of processors. [8]considered the problem of computing boolean functions on anonymous networks. The second problem we will consider is that of classifying networks by the functions they can anonymously compute. Let us call two networks "f equivalent" if the set of vector-valued functions each can compute is the same. The question is then: Are there topological or algebraic features of networks which characterize the f equivalence-classes of networks?, and also: Is this classification problem hard? The answer, it turns out, is that the classification problem is not hard: Not only do there exist graph features which two networks share iff they are f equivalent, but these features are easy to compute. A more complete version of this work appears in [ll]. Proofs of the propositions quoted in this paper may be found there.

2

The Model

A network is a connected, directed edgelabeled graph G = (V(G), E(G), A(G)), where:

- V(G) = { 1, ...,n) (the vertices) is a set of arbitrarily powerful processors, - A(G) is a set of edge-labels, and - E(G) (the edges) is a set of tweway links between processors, given as ordered triples (v a w ) , for v and w E V(G) and a E A(G). The directedness of links is strictly for mathematical convenience and is not intended to imply that links are one-way. However, it will be convenient to describe a link (v a w ) as being directed away from v and directed towards w. In this paper we will consider networks which are also "group-graphs" ,where a graph G is called a group graph if its edge-labels satisfy the following condition:

For each vertex v E V(G) and for each a E A(G) there is exactly one edge directed away from v having label a and one edge directed towards v having label a, in G.l This edge-label condition implies that a processor can distinguish among the links incident with it. A group graph is a special case of a "monoid graph", for which the requirement is only that all edges directed away from a vertex have distinct labels and all edges directed towards a vertex have distinct labels. Computing On A Network Computations are performed distributively on a network, with each processor i E V(G) running a deterministic algorithm Ai from a set A = (AI,. ..,A,) of algorithms, where n = IV(G)I. Initially each algorithm Ai is given a letter from an input alphabet I as input. A processor begins to execute its algorithm when either it "wakes up" or when it receives a message from an adjacent processor. Algorithms run in steps. During each step of an algorithm run by a processor i, the processor may:

- Send -

-

messages through one or more of its links. Processor i can specify through which of its links a message is sent. Receive a message through one or more of its links. A processor can tell through which of its links any message is received (e.g., "this message arrived through,the link with label 'a' directed away from men). Perform a computation based on messages received, the results of computations performed in previous steps, and input. The result of a given computation can be specially marked as the output of an algorithm's run.

We will asssume that a network executes a set {A1, ...,A,) of algorithms asynchronously; that is, that messages arrive at a processor via a given link in the order in which they were sent, after a finite but unspecified delay. We take the synchronous execution of a set of algorithms by a network to be a special case of asynchronous execution. In the synchronous execution, all processors begin executing their algorithms at the same time; messages sent through a link at time t arrive a t time t 1, and computations take zero time. We will say that a network G with n processors performs an anonymous compdation if it executes the set A = {A1, ...,A, ) of algorithms, where A1 = Al = . ..= A,. Intuitively this means that the processors cannot initially distinguish among themselves, except perhaps by their input.

+

Computing ]Functions On A Network Let I and 0 be input and output alphabets and let x = (xl, ...,x,) E In.If G is a network with n processors, we will say that G is given input x if processor i gets input t i for i = 1,. ..,n. Write (G,x) for the network G with input x (or equivalently, for the graph G in which each vertex i is labeled with x i . ) We say that G computes a function f(x) =

This network model is a slight generalization of the model used by Yamashita and Kameda ([14, 13, 15)) and others, in which a link between processors v and w is of the fotm {(v, a), (w, b)), where a is processor u's name for the link and b is processor w's name for the link. A link {(v,a), (w, b)) in this model is replaced in our model by a pair of {(v, (a, b)?w), (w, (4 a), v ) ) .

~~

( f ( ~ ).~..,,f(x)n) E On given input x if there is a collection A = {A1,. .. , A n } of algorithms such that each processor i computes f ( x ) ~when given input x i , by running algorithm Ai E A. Again, we will say that G computes f (x) anonymously ifAl = A z = ... = A n . We will informally refer to "anonymous networks", meaning networks which perform anonymous computations. We will also refer interchangeably to networks and graphs, and to processors and vertices, and links and edges.

3 Graph Covers and the Edge-Label Group Before we tackle the characterization and classification problems mentioned above we need to develop some of the algebraic structure of networks. Much of this structure is similar to that used in algebraic automata theory: There is a semigroup (in our case, a group) associated with a finite-state machine or network, and the structure of the semigroup (or group) says something about the behavior of the network? Let us begin by defining this network group. Let G be a graph with n vertices. Each edge-label a E A(G) induces a permutation fa on V(G) = (1,...,n), by: f,(u) = w iff there is an edge ( v a w ) E E(G). The set (fa : a E A(G)) generates a permutation group C(G) under function composition? For instance, in the network G in Figure 1 below, fs maps vertex 1 to vertex 1, vertex 2 to vertex 3, vertex 3 to vertex 2, and vertex 4 to vertex 4. fa maps vertex 1 to vertex 2, vertex 2 to vertex 1, vertex 3 to vertex 4, and vertex 4 to vertex 3. It can be checked that C(G) is the dihedral group D8.

Fig. 1. A graph G and quotient-graph G / n For instance, see [i].[3] gives an interesting treatment of state-machines in terms of "k-algebrasn If G is a "monoid graph" then C(G) is a monoid.

.

We will write A(G)' for the set of words over A(G). If a = alas ...a, E A&)'', write ,fa for the composition fa, fa, ...fa, of fa,, ...,fan. Associated with the group &(G) is a collection of partitions of V(G), called the "block systems" of G, which are preserved by E(G). In particular, a partition A = { B l ,B2,. ..,Bk) of V(G) is called a block system of &(G) iff for each Bj E r and for each fa E E(G), the set faBi = {fabj : bj E Bi) is a block in a. Every block system a of G corresponds to a well-defined quotientgraph G/n of G, where V(G/r) = T and where ([v]a[w]) E E(G/a) for [v], [w] E A iff there are vertices v E [v] and w E [w] for which (v a w ) E E(G). (Here [v] denotes a block containing the vertex v.) For instance, the partition n = 1,4/2,3 is a block-system of the graph G pictured in Figure 1, and Gin is as pictured. Covering Maps And The Universal Cover A 'covering map' is a map from the vertices of one graph to another which preserves local graph structure. Covering maps have been used in the distributed computing literature t o describe what a processor in a network "knows" about the network (see [I] and 1141) and also to simplify proofs on fault-tolerant computing (151). For the purposes of this paper we will define a covering map to be a surjective map from the vertices of one graph to the vertices of another which commutes with the elements of their respective edge-label groups.' More formally, let GI and Gz be graphs for which A(Gl) = A(&). A surjective map p : V(Gt) -r V(G2) is called a covering map if pfa(v) = g,p(v) for all v E V(GI) and for all a E A(GI)' = A(G2)'. (Here fa and g. denote elements in &(GI) and &(G2), respectively. ) For instance, the map p : 1 -+ [2], 2 + [I], 3 [I] and 4 -. [2] is a covering map from G to G/a, where G and G/n are as pictured in Figure 1. An isomorphism is a bijective covering map, and an automorphism is an isomorphism from a graph to itself. We will say that a graph GI covers a graph Gz in case there is a covering map from GI to Gz. The following lemma is easy to prove.

-

Lemma 1. A graph G covers G / r whenever a is a block-system of G .

The universal cover U of a graph G is a tree covering G. (See Figure 2.) The universal cover of a graph is unique up to isomorphism, and is infinite unless G is a finite tree. We will usually think of U as a rooted tree with its root "corresponding" to a particular vertex in G. More formally, let @ : U -+ G be a covering map, let v E V(G) and choose u' E /3-'(v). We will write U, for the triple (U, v', 8) and think of Uu as the tree U rooted a t v'. It can be shown that if & ( v t ) = &(wt) for covering maps pl and & : U + G and for v', w' E V(U), then there is an automorphism of U mapping v' to w'. Thus U, is well-defined and is unique up to isomorphism. In [14], the authors argued that U, (or rather,

' A covering map is usually defined to be a "local isomorphism"; that is, a map from the vertices of one graph to the vertices of another, which preserves preserves the orientations and labels of the edges in a small neighborhood around each vertex. It can be shown that this definition is equivalent to the above, for group graphs.

a related object, the "view" of v ) represents precisely what v can deduce about G by exchanging messages with neighboring processors. Let US write U,x for the tree Uv with its vertices labeled with the components of x E In. In particular, if U, = (U,v1,B) then for each vertex j E V(G), all vertices in the set Bw1(j) are labeled ' z j ' . Write U ~ for X the tree U,x truncated at depth k. That is, U ~ Xis the subgraph of U v x induced by the set of all vertices distance k or less from the root? We will write U: 2 U; if there is an isomorphism from U: to U; which maps the root to the root, and write if the isomorphism preserves vertex labels. U ~ 2 X.

-

Ezample 1. For instance, if the network G in Figure 1is given input x = (y, z,z,y), then U l x 2: U q x and U2x Usx. (See Figure 2.) We also have U l x 21 U2y for Y = (2, Y, Y, 2 ) The next proposition says that U v x is determined by uZn-'x for a graph with n vertices.

Proposition2. Let G have n ueriices and let v, w E V(G), and x , y E In. Then UP-lx UE-ly i f l ~ t x . I for all k 2 0.

.-

uty

Characterizing The Functions Computable By A Network 4

We can now begin the characterization of the functions computable by an anonymous network. We have: Lemma3. Consider iwo synchronous ezecutions of an algorithm A in anony-

mous computations on a network G; the first with input x and the second with input y. If U v x 2: U,y for processors v and w E G , then at each step k of ihe first execution, processor v sends and receives the same messages and computes the same values as processor w does during the kth step of the second execution of A. The following is proved in [14]: Lemma 4. Any processor v E G can anonymously compute a graph isornophic to U ~ X for any k 0.

>

The vector-valued functions compu table by a network can now be characterized as follows. Theorems. Let G be a network with n processors. Any computable function f : In + On can be computed anonymously by G iff f satisfies: For all inputs x and y and for all processors i, j E V(G), if U ~ X 2 Ujy then f ( x ) i = f ( Y ) ~ . The distance between two vertices is the number of edges in a minimal path between them.

v:x Fig. 2. (G, x) for x = (y , z , z , y ) , and ~f x

Example&. Consider the network G in Figure 2. Since Ulx z U2y for x = (3, z, z, y) and y = (2,y, y, z), a function computable on G must at least satisfy

f ( 4 1 = f( ~ 1 2 . Proof sketch for the theorem (*) Suppose that a function satisfies the conditions given. If U ~ 2 X Ujy then by Lemma 3 above, processors i and j cannot distinguish themselves in a synchronous computation with inputs x and y , and so must compute the same output. (r) To prove this direction, note first that it does not violate anonymity to assume that each proceswr in G has available the graph of G. (That is, each processor "knows" G but does not know which processor in the graph it corresponds to.) We seek an algorithm which anonymously computes f ( x ) ~on each processor i. We will use the following fact:

Lemma6. If a network G is given input x , then each processsor in G can , x' has ihe properiy that for each anonymously compute a vector x' E I ~ where j = 1,. ..,n there ezists a vertex k E V(G) for which Ujx 2 U k x t . Furthermore, all processors in G can compute the same vector x'.

Given this lemma, it is easy to show that the following procedure, run on processor i, computes f(x): Processor i first computes u?'"-'x. Processor i then computes the vector x' described above, and finds a vertex v in its graph of G for which Utn-lx' z u?-'x. Finally, processor i computes f ( x l ) and outputs f (x')v 0

5

The Symmetries Of A Network

We now turn to our second problem; that of classifying networks by what they can anonymously compute. We will look for a small set of topological or algebraic features of graphs which two networks share in common iff the set of functions each can anonymously compute is the same. What might these graph features consist of? Plausible first guesses might be that two networks are fequivalent iff they have the same automorphism group, or the same edge-label group, or the same set of block systems. It turns out, however, that none of these is both sufficient and necessary for classifying networks. We will find, instead, that two networks are f-equivalent iff they have the same set of quotient-graph isomorphisms. Before we show this we will give an alternate characterization of the functions a network can compute, in terms of its quotient-graph isomorphisms. The relation between quotient-graphs and universal covers will perhaps become clearer if we observe that U, U, for any vertices v and w in the same block of a block-system. A further relationship between universal covers and quotientgraph isomorphisms is given in Proposition 7 below. We will need some notation:

- Any input-vector x induces a block-system ~ r xof G,where [t) = b] E

iff

Uix2 Ujx.Since srx is a block system, the quotient-graph G/rX is defined

-

as before. For instance, the input x = (y, z, z, y) induces the block-system zx = 1,4/2,3 in the graph G pictured in Figure 1. If nl and ~2 are block-systems of G, we will write sl 5 7 2 and say that x l is a refinemeni of n2, in case each block of nl is contained in a block of irz. Let 6 : G/;rl + G / r 2 be an isomorphism and let x be such that r 2 -( nx. We will write 6(x) for the vector ...,x ~ ( ~That ~ ~ is, ) 6(x) ) = ( x i , , ... zi.), where ij E 6(bJ). For instance, let rl = a2 = 1,4/2,3 for G as pictured in Figure 1. Let x = (y, z, z, y) and let 6 map the block {I, 4) to the block {2,3} and {2,3} to {1,4). Then 6(x) = (I,y, y, z).

We can now state a relationship between quotient-graph isomorphisms and isomorphisms of rooted universal covers:

Proposition 7. Lei G be a network. Then:

-

1. If there a n inpuis x and y and veriices i and j of G such that Uix z Ujy ihen i h e n is an isomorphism 6 : G/zx G/;ry for which 6([4) = b] and

= a(y).

2. If 6 :G/rl -r G/x2 is an isomorphism for block-systems a1 and n2, ihen for ally E In such i h ~ ~2 i 5 ~ y we , have *1 5 r q y ) and Ui6(y) Ujy for all veriices i and j for which 6([0)) = Ij].

Proof sketch for part 1 Define a map 6 as follows: b([q) = b] for [i] E 7rx and ifl E ny . For any fs E t(G), let 6fb([i]) = fb6([4) = fb(b]). Then 6 is the desired isomorphism. o This proposition will perhaps make plausible Theorem 8 (next), which characterizes the functions a network can compute in terms of its "symmetries", or quotient-graph isomorphisms. First, let us define 'network symmetry' and what we mean by saying that a function "satisfies a symmetryn. Let G be a graph and let 6 : G/x1 + Gx2 be an isomorphism of the quotientgraphs G/nl and G/Q. Then 6 induces a bijection, which we will also call 6, between the blocks of a1 and ~ 2 We . will call a triple (al, 2r2,6) a symmeiry of G if 6 :G/nl -.. G/Q is an isomorphism.

For instance, in the graph G in Figure 1, the triple (nl, q , 6 ) is a symmetry, for nl = n2 = 1/2/3/4 and 6 = (1,4)(2,3). Another symmetry of G is the triple (r3, xd, 6), where 7r3 = nd = 1,4/2,3 and 6 = ({I, 41, {2,3}) (cycle notation). We will say that a function f :In w On soiisjies a s y m m e i q (rl, Q, 6 ) if for all inputs x for which a 2 5 a x , we also have n2 5 rjcx) and f(6(x)) = 6(f(x)). insures that 6(f (x)) is defined. ) (The clause "r2 5 We have: Theorem 8 . A weitcork G computes a function f iff f saiisfies all of the symmetries of G .

Proof sketch Suppose first that f satisfies all of the symmetries of G. Let i, j E V(G) and x and y be inputs such that Uix Ujy. By part 1 of Proposition 7, there is a symmetry S = (ax, ny , 6) with x = 6(y) and ifl = b ( [ g ) . Since f satisfies S, we have f (6(y)) = 6f ( y ) . That is, f (x)i = f ( ~ ) jand , SO by Theorem 5 above, G computes f. Conversely, suppose that G computes f. Let S = ( r l , n2, 6) be a symmetry IX. If [i] = Ij]in n2 then of G and let x be an input-vector such that ~2 Uix z Ujx, by definition of ?rx, and we have f(x)i = f ( ~ ) by ~ ,Theorem 5. It can be shown that this implies that a 2 5 T ~ ( x )and , also, that 7r2 5 a x implies

that Ui6(x) Ua(lil)x.By Theorem 5, f must satisfy f(b(x))i = f ( x ) a ( ~ )since , 0 G computes f . Thus f(6(x)) = 6f(x), and f satisfies S. F'rom this we can conclude the following:

Theorem 9. Two networks are f -equivalent (that is, can anonymously compute ihe same funciions) iff ihey have ideniical symmetry-sets. Note that it is possible to talk about two sets of symmetries being equal, since a symmetry is only a bijection between block-systems, and can be described without reference to a graph or its edge-label group.

6

Generators For The Lattice Of Symmetries

How many symmetries can a network have? Unfortunately, the answer is 'quite a few'. One can show, for instance, that a Cayley graph of the group Z2x ...x Z2 has O(nlgn) symmetries, where n is the order of the group. We will see, however, that the symmetries of a network form a lattice, and that a small (no bigger than n for a graph with n vertices) set of elements of this lattice generates the lattice under the lattice-join operation. Before we show this, let us recall some facts about permutation groups. Lemma 10. [12] Fix v E V(G). T h e n is a one-to-one correspondence between ihe blocks of G containing v and the subgroups of E(G) containing the siabiliter subgroup &(G), of v. This correspondence is given by: ? -, iH(v) for each &(G)u 5 31 5 &(GI-

If 'A is a block system of G and if H(v) E sr for t(G)u 5 31 5 C(G), we will call 31 the subgroup of C(G) corresponding to n with respect t o v. We have: Lemmall. Lei X be ihe subgroup corresponding io a block-sgstem n with respect to v. Then there is a one-to-one correspondence between the left cosets of H in t(G) and the blocks of n, given by: + faX(v).

fax

(See Example 3 below .) We can also establish a correspondence between the symmetries of a network G and conjugations of the subgroups of &(G):

Proposition12. Let xl and sr2 be block-systems of G having corresponding subgoups 31 and $, respectively, with respect to a veriex v. Then:

- If6 : G / r l

+ G/12

is an isomorphism and 6(?i(v)) = faJ(v) for

fa

E C(G),

then 31fa = fa,?.

- Conversel~,if Xfa = f a 3 for some fa 6 : G/nl

+

E &(G) then there is an isomorphism G/x2 for which 6(?i(v)) = faJ(v).

(See Example 3 below.) This proposition gives us a correspondence between the symmetries of G and a subset of the set of left cosets of subgroups of E ( G ) . We have: Propositionl3. Fiz v E V(G). There is an injective map p from the set of symmetries of G info the set of lefl cosets of subgroups of E(G) coniainin.g C(G),

.

This is given as follows: Lei 6 : G/xl -. G/n2 be an isomorphism, and let TI correspond io a subgroup 7i and ir2 comspond io 3 with respect to v. Then p maps (at, irz, 6 ) to fag, where 7 ifa = f,3 and faJ(v) = 6(X(v)). (See Example 3 below.) We will call the coset p((nl, ~ 2 6)) , corresponding to a symmetry S = (nl, 7 2 , 6) the coset representative of S with respect to v. We will see next that the set of coset-representatives with respect to a vertex forms a lattice. The correspondence between coset representatives and symmetries will be used t o put a lattice structure on the set of symmetries of G. We will need the following proposition from lattice theory: Propositionl4. (['I) Let X be a sei and let L be a family of subsets of X , ordered by inclusion, such thai:

I.XEL 2.

nicIAi E L for every nonempty family {AilicI E L.

Then L is a laitice wiih ihe meei and join operations given as follows: AiAAj E Ai n Aj, and AiVAj i n { A E LIAi U Aj A), for Ai and Aj E A. This lets us conclude the following: PropositionlS. Lei G be a network, and lei C, be the sei of cosei representaiives of ihe symmetries of G, with respect to a veriex v. Then C, U 0 forms a latiice Lcv under ihe inclusion order, wiih the meet and join operaiions as given in Proposiiion 14 above. Proof sketch The proposition follows from Proposition 14 above if C3U 8 is closed under intersection. This in turn follows if the intersection of two coset representatives is a coset representative of a symmetry with respect to v. 0

Ezample 3. For G as in Figure 3, &(G) is generated by fa = (1,2)(3,4) and = (2,3) and has group elements fa, fs, fc = (1,3,4,2), fd = (1,2,4,3), f e = (1,3)(2,4), f j = (1,4)(2,3), and f, = (194). Choose v = 1. Then the symmetries and their corresponding coset representatives are as follows: fb

- Sl = (1/2/3/4, (id,f b ) ,

1/2/3/4, 61 = id) with coset representative Cl = &(G),, =

Fig. 3. Lattice of Coset Representatives

- S2 = (1/2/3/4, -

-

1/2/3/4/, b2 = (1,4)(2,3)) with coset representative C2 = fjc(G)v = {fj9 fjfb} = {fj,fp), S3 = (1,4/2,3, 1,4/2,3, J3 = id) with coset representative Cs = 3 = (id, f b , fi 9 f p ) , S4 = (1,4/2,3, 1,4/2,3, b4 = ((1,4), {2,3))) with coset representative c 4 = f a 3 = (fa, fc, fat fc), S5 = (/I, 2,3,4/, /1,2,3,4/, b5 = id) with coset representative C5 = &(G).

The lattice Lev of cosef-representatives is pictured. The lattice structure of Lev induces a lattice structure Ls on the symmetries of G, as follows: Let S1 and S2 be symmetries having coset representatives Cl l C2. and C2, respectively. Define the lattice order on Ls by Sl5 S2 iff C Define SlVS2 to be the symmetry with coset representative ClVC2, and Sl AS2 to be the symmetry with coset representative ClAC2. It can be shown that Ls is independent of the choice of vertex v in Lev. Let S = (rl,7r2,6) be a symmetry of G having coset representative fa7f with respect t o a vertex v. We will say that S is the minimal symmeiry of a vertex i with respect to v, and fa% the minimal coset of i with respect t o v, if faxis the smallest coset in L,v such that i E fa7f(v). For instance, in Example 3, the minimal symmetry of vertex 4 with respect t o vertex 1 is S2= (1/2/3/4, 1/2/3/4,6 = (1,4)(2,3)), since f/C(G) is the smallest coset representative containing the vertex 4. The set of minimal coset representatives of G with respect to vertex 1 is {Sl,S2,Sd}. We have the following result about the minimal symmetries:

Proposition 16. If G has n veriices, its set of minimal symmetn'es with respect to any veriex has at most n elements. The set of minimal symmetries of G with respect to a vertex v generates ihe lattice of symmetries under the latiice join operation. Proof sketch First of all, for each i E V(G) the minimal coset representative Ci and hence the minimal symmetry of i, is unique: For if i E faH(v) and i E fbg(v) then i E faX(v) n fbJ(v) = (fa% n fbJ)(v). By Proposition 15, f a 2 n fb.7 E Lev. Thus there are at most n minimal coset representatives, and so at most n minimal symmetries. To show that the set of minimal symmetries generates Ls, it suffices to show that the set of minimal coset-representatives generates L,v under the lattice join. Let fa% E L,v and suppose that faR(v) = {il,h , ...,it}. It can be shown that = folXSV***VfokZk, where f a i 3 i i is the minimal coset representative of vertex ij, for each i E (1,. . ,k). Hence the set of minimal coset representatives generates L,v under the lattice join. a

fox

.

As was true for the lattice of symmetries, the set of minimal symmetries of a graph is independent of the choice of a vertex v. As a corollary of the above we can conclude that the minimal symmetries of a network characterize the set of networks f-equivalent to it. In particular, we have: Corollary 17. Two networks have identical symmetry-sets i f their sets of minimal symmetries are identical. Hence, two networks compute the same set of functions iff they have the same set of minimal symmetries. Finally, we can give an algorithm for finding the minimal symmetries of a network. This algorithm is polynomial in the number of edges of the graph. Hence, classifying networks is not a hard problem. The idea behind the algorithm is fairly simple. Suppose we want to find the minimal symmetry of a vertex vertex w with respect to a vertex v. We first use an orbitfinding algorithm from computational group theory6 to construct a relation A on V(G), where A is in some sense the closest possible approximation to an automorphism of G mapping v to w. A pair (i,j) of vertices is in A if the proposed automorphism maps i to j. Next, me construct block-systexs ni a d 7r2 such that A induces a bijection 5 from the blocks of nl to the blocks of Q. Then (nl,q ,6) is the desired minimal symmetry. How are A, TI,lrz, and 6 constructed? If there did exist an automorphism a : G + G mapping v to w, it would commute with the elements of g(G), that is, it would satisfy: afa(v) = f,a(v) = fa(w) for all fa E &(G). Hence A must contain all pairs of the form {fa(v), f.(w) : fa E C(G)). That is, A is the orbit of the pair (v, w) under the action of C(G) on the set of all pairs in V(G). If A is to map each block of nl to a block of 7rz then must satisfy: If ( i , k ) and (j,k) E A then i and j are in the same block of nl. By the same argument, if E.g., see [6] or [9].

(k, i) and (k, j ) are in A then i and j must be in the same block of n2. More formally, we have: Algorithm 6.1 (For finding the minimal symmetry of a vertex w with respect to v) Step 1: Given the pair (v, w ) and the generator-set {fa : a E A(G)} for &(G), use an orbit-finding algorithm to find a relation A on V(G) x V(G), where A = {fa(v), fa(w) :f a E C(G)}* Siep 2: Construct relations R1 and R2 from A as follows: ( i ,j ) E R1 iff (i, C) and (j, k) E A, for some k E V(G), and (i, j ) E R2 iff (k, i ) and (k, j) E A for some k E V(G). Step 3: Construct partitions rl and ~2 of V(G) from R1 and R2 as follows: If (i, j)E R1then i and j are in the same block of rl, and if (i, j ) E R2 then i and j are in the same block of ~ 2 . Step 4: Define 6 : *1 + a 2 by: 6(Bi) = Cj for Bi E rl and Cj E rr2 if there are vertices i E Bi and j E Cj for which (i,j ) E A. Then ( r l , s2,6) is the desired minimal symmetry. 0 Let us argue briefly that this algorithm runs in time polynomial in the number m of edges of G: First of all, the relation A can be computed, using an orbit-finding algorithm, in time polynomial in m. The relations R1 and R2 can be constructed from A in n3 steps each: There are n2 pairs (i, j ) to check for membership, and each pair can be checked in n steps. The blocks of nl are the connected components of an undirected graph r, where V ( f ) = (1,. ..,n ) and E ( f ) = R1.Since 'l has n vertices and does not have parallel edges, these components can be found in O(n2) steps. (For instance, use a depth-first search algorithm to find a spanning tree of in O(n2) steps, where r has a t most n2 edges. ) The same argument shows that the blocks of a* can be computed in O(n2) steps. The map 6 can be computed in O(n2) steps also: In n steps the algorithm can choose a representative from each block of al. For each representative i, the algorithm can find a pair (i, j) E A in n steps and can identify the block that j belongs to in n steps. Since n 5 rn,the algorithm can be executed in p(m) steps for a polynomial

r

P*

0

This gives us the following result.

Theorem 18. Fix a veriex v E V(G). There is an algorithm polynomial in the number m of edges of G for finding ihe minimal symmetry in Lev of a veriez w , given as input ihe veriices of G and ihe set {fa : a E A(G)} of generators for W). Finally, we have: Corollary 19. Let GI and G2 be networks for which IV(Gl)I = IV(G2)1 = n, and lei m = max{lE(G1)l, IE(G2)1}, ihe maximum number of edges in GI and

G2. There is an algorithm polynomial in rn for determining whether GI and G!: compute the same set of funciions. Proof By Corollary 17, two networks have identical sets of minimal symmetries iff they have identical symmetry-sets. By Theorem 9, two networks compute the same functions iff their symmetry sets are identical. The sets of minimal symmetries of GI and GZ can be found in time polynomial in m and compared t3 for equality in time polynomial in m.

7

Extensions

There are several extensions of this work which might be worth exploring. We have investigated two. For the first, we considered the case in which &(G) is a monoid, analogous to the monoid associated with a finite-state machine. To obtain this monoid we dropped the requirement that the edge-label functions {fa : a E A(G)} be permutations of V(G), and required only that they be oneto-one partial functions. We found that all of the results about classifying networks which held for groupgraphs also held for monoid-graphs. In particular, the characterization of the functions computable by a network (Theorem 5) is independent of whether &(G) is a monoid or a group, and the algorithm given for finding the minimal symmetries of a network can be extended to monoid graphs. It might be worth saying a few words about this extension. First, the idea of a block-system and a symmetry can be extended to monoid graphs, and it can be shown that each such graph has a unique 'coarsest' block system IT, such that all other block-systems of the graph are refinements of a. It is shown next that the 'restriction' of the monoid t(G)to any block B E I? is a group &(G)lB. Finally, it is shown that the symmetries associated with the restricted group E(G)lB extend uniquely t o symmetris of the whole graph. Thus the minimal symmetries of the restriction, which can be computed as before, characterize the network. The other extension we considered was to networks which 'almost' compute the same functions, that is, which would compute the same functions if the processor ids of one of the networks were changed by a permutation. Let us say that two such networks "differ by a permutation". How hard is it to determine whether two networks differ by a permutation? Interestingly, we found that there is a polynomial transformation of this problem to the problem of determining whether two groups, given as grouptables, are isomorphic. As of this writing the best algorithm for the group isomorphism problem (see [lo]) is subexponential.

8

Acknowledgements

This work was submitted as part of a phd dissertation t o the mathematics department a t University California Santa Cruz. I would like to thank Nick Littlestone and my advisor Manfred Warmuth for the numerous engrossing conversations that led to this work, as well as for extensive help with proof-checking

and editing. I want to thank Manfred for alerting me to the utility of algebraic automata theory in this work, and Nick for suggesting that I look a t edge-label maps, as well as for a lot of notational niceities and the best counterexamples in the state.

References 1. D. Angluin. Local and global properties in networks of processors. In ACM Symposium on the Theory of Computing, pages 82-93, 1980. 2. H. Attiya, M. Snir, and M. Warmuth. Computing on an anonymous ring. 3. ACM, 35:845-875, 1988. 3. J. Richard Biichi. Finite Automata, Their Algebras and Grammars. Springer Verlag, 1989. 4. B. A. Davey and H. A. Priestley. Introduction to Lattices and Order. Cambridge University Press, 1990. 5. M. J. Fischer, N. A. Lynch, and M. Merritt. Easy impossiblilty p r o o t for distributed consensus problems. In Proceedings of The Fourth Annual ACM Symp. on Principles of Distributed Computing, pages 59-70, Minaki, Ontario, 1985. 6. Christoph M Hoffmann. Group Theoretic Algorithms and Graph Isomorphism. Springer Verlag, 1982. 7. W. M. L. Holcombe. Algebraic Automata Theory. Cambridge Studies in Advanced Mathematics, Cambridge University Press, 1982. 8. E. Kranakis, D. Krizanc, and J. van den Berg. Computing boolean functions on anonymous networks. In Proceedings of The International Conference on Algorithms, Languages and Programming, pages 254-267,1990. 9. Eugene Luks. Lectures in polynomial-time computation in groups. Technical Report CIS-TR-90-21, Department of Computer and Information Science, University of Oregon, Eugene, OR, 1990. 10. Gary Miller. On the nlgn isomorphism technique. In Proceedings of the Tenth ACM Symposium on the Theory of Computing, pages 51-58, 1978. 11. Nancy Norris. Classifying anonymous networks: When can two networks compute the same vector-vallued functions? Technical Report UCSGCRL-94-01, Department of Computer and Information Science, University of California a t Santa Cruz, 1993. 12. Helmut Weilandt. Finite Permutation Groups. Academic Press, 1964. 13. M. Yamashita and T. Kameda. Computing functions on an anonymous network. Technical Report LCCR 87-16, Simon Fraser University, Vancouver, Vancouver! British Columbia, 1987. 14. M. Yamashita and T. Kameda. Computing on an anonymous network. Technical Report LCCR 87-15, Simon Fraser University, Vancouver, Vancouver, British Columbia, 1987. 15. M. Yamashita and T. Kameda. Computing on anonymous networks. In Proc. 7th ACM Symp. on Principles of Distributed Computing, pages 117-131, Ontario, 1988.

This article was processed using the BTEX macro package with SICC style

Clompact Routing Methods: A Survey Jan van Leeuwenl and Richard B. Tan2 Department of Computer Science, Utrecht University Padualaan 14, 3584 CH Utrecht, the Netherlands (janQcs.ruu.nl) Department of Computer Science, U trecht University Pzdualaan 14, 3584 CH Utrecht, the Netherlands ([email protected]) and Department of Computer Science University of Sciences & Arts of Oklahoma Chickasha, Oklahoma 73018, U.S.A. ([email protected])

Abstract. We give a short survey on various compact routing methods used in communication networks. The routing schemes considered are Interval Routing, Prefix Routing and Boolean Routing. Various known characterizations of networks that have such routing schemes are presented. A few open problems in this area also are given.

1 Introduction In a parallel computer, as more processors are added to increase the computing power, the underlying communication network needs to scale favorably along with the expansion. As the amount of storage space at each processor is limited, the expansion of the network should not put undue burden locally by requiring excessive space for communication purpose. The routing methods used should also be simple and dynamically adjustable with the growth. The underlying network structure can be quite arbitrary, so the communication methods should not rely on any fixed topology. More and more emphases are given to this type of universal routing on arbitrary network (see, for example, [16], [12], [ll]). This gives rise to a need of simple compact routing rnetliods that are scalable with the growth of networks and independent of any unde~lyingtopology. For instance, the INMOS T900C Transputer [12] uses one such method called the Interval Routing, which was introduced in [18]and [14]. In this paper we survey a few of the available methods.

* This research

was partially supported by EC Cooperative Action IC-1000 (project ALTEC: Algorithms for Future Technologies) and by the Netherlands Organization for Scientific Research (NWO) under contract NF 62-376 (NFI project ALADDIN: Algorithmic Aspe~tsof Parallel and Distributed Systems).

Communication Model

1.1

We shall model the interconnection network as a (finite) graph and phrase the terminologies accordingly. Let G =< V, E > be a connected graph with vertet set Vof size n and edge set E. Vertices (nodes) carry unique identifiers (addresses) taken from some ordered domain, and edges (links) are assumed to be bidirectional. We shall concentrate on distributed models, where processors have access only to their own local memory and communicate with each other by sending messages. Each message contains headers that typically include the source and destination addresses of the processors. In order to route a message m from processes i to j, a path from i to j must be identified to transport m. Traditionally a path of shortest distance or cost is used, but there are other variants. The path information must be stored somehow at each intermediate node to allow progress of the message from its source to destination. Typically the necessary information is stored in a routing table with n entries, one entry for every possible destination and one table a t every node. As the message m arrives at each intermediate node, it checks if it is indeed the intended destination target. If so the message is processed, otherwise the local routing table is consulted for an appropriate link to further relay the message. The message m thus travels in a series of hops until it reaches its final destination. Normally the routing is implemented by a special unit termed the router that is associated with each processor. It is thus the function of the router to decide if it should keep the message for local processing or pass it on further to another neighboring router via an appropriate link. The routing method can be described as follows:

1 2 3 4

5 6 7 8

procedure SEND(id, dest, m) ( m is a message to be sent from current node id to node dest) if id = dest t h e n process rn else find appropriate link x out of node id via which dest needs to be sent send m over the link x id := the node that receives m over link x SEND(id, dest, m) end if endiprocedure SEND ) Figure 1

The original sender source of a message m will then invoke the protocol SEND(source, dest, m) and sends the message along onto its proper destination dest . 1.2

Compact Routings

With the expansion of the networks, the above method of keeping routing tables becomes untenable, as at each node, the table size is O(n) (measured in log n-

size words). A more compact way of representing the tables is needed. Now, in practice, the degree d (the maximum number of edges incident to a node) of a graph is usually much smaller than the size of the graph. Thus, if we arrange the routing table according to the edges, then we have a more compact table, with only d rows of entries, though each entry may now consist of more than one node-label. Searching the table for a node-label will be less efficient however, as there is no special ordering in general. We can make the tables even more compact if we can represent the group of nodes associated with an edge in a simple way. Some kind of a relation needs to be established among the nodes that belong to the same group. In this survey, we look at some of the relations that have been used to compactify such groups of nodes. The first relation we study is the Interval Labeling Scheme. The idea here is to label the nodes in the graph such that all the nodes belonging to the same edge form a (cyclic) interval. This is presented in section 2. Section 3 covers a special case of Interval Labeling called Linear InZerval Labeling . In section 4, we look at Prefix Labeling , where nodes having a maximum common prefix belong to the same group. Section 5 introduces the Boolean routing, which uses certain boolean predicates as its relation. Finally in section 6 we discuss extensions of the above relations to multi-label relations. Throughout we discuss the known results in each scheme and list some open problems. We assume that the networks do not fail and messages do arrive eventually. We also concentrate mainly on results pertaining to networks with optimal shortest path or minimum cost routing.

2

Interval Routing

The idea behind an Interval Labeling Scheme (ILS) is to group the nodes belonging to the same outgoing link in a cyclic interval (modulo n ) . This is done by first labeling all the nodes in the graph with some unique integer in [O..n- 11. Each link is then labeled with a unique interval [a,b). Wrap-around of intervals is allowed, so if a > b then [a,b) = { a , a + I, ...,n - 1,0,...,b - 1). The set of all such intervals associated with the edges of a node form a parlition of the cyclic interval [O..n). The routing protocol is exactly as procedure SEND presented in Figure I, with line 4 modified to : 4

find link x at node id with label [a, b) such that dest E [a,b)

An ILS is valid if for all nodes i and j of graph G, messages sent from i to j via the procedure SEND eventually reach j and get processed. Figure 2 gives an example of a valid ILS. It is not a priori clear that there is a valid ILS for every graph. Of course, one cannot just label the graph arbitrarily, as the route may contain a cycle and any message caught in the cycle can never reach its proper destination. There is an O(n2) algorithm for checking if a given ILS is valid ([14]). A valid ILS is termed an Interval Routing Scheme (IRS) for short.

Figure 2

A preliminary version of Interval Routing was introduced by Santoro and Khatib (181.They showed that any tree and ring admits a valid ILS. By constructing a valid ILS on a spanning tree of a graph, one can thus impose an IRS on the graph also, but then not all the links are being used. Van Leeuwen and Tan [14] extended the result to general graphs where all the links are used. However, the IRS constructed is not necessarily optimal (shortest path). As the union of all the disjoint intervals spans the cyclic interval [O..n), we can reduce the routing table further, by listing only the left end-points of the intervals. Also, as there is no overlapping of intervals, the path specified by the IRS is unique. This gives rise to deterministic schemes and allows for efficient searching. Sort the interval labels and arrange the tables accordingly to the sorted order. One can then use a modified binary search to search quickly the desired link for each dest. We now mention some known results. 2.1

Uniform Cost Links

Suppose by optimum we mean shortest path. This can be considered as assigning positive uniform cost to each link and then finding the minimum cost. The following graphs (with uniform cost links) have optimum (shortest path) IRS. Santoro and Khatib [18] nees

Rings van Leeuwen and Tan 1151

Meshes Complete bipartite graphs Complete graphs Grids with column-wrap-around Fhigniaud and Gavoille [7] Unit circular graphs

Not unexpectedly, we have also some negative result.

RuEiEh [17] There are networks with no optimum (shortest path) IRS. On the other hand, as mentioned earlier, we at least have the following. van Leeuwen and Tan [14]Every network has a valid ILS (but not necessarily optimum), using all edges of the graph. It is not clear exactly what type of graphs admit an optimum IRS.

OPEN PROBLEM: Characterize the graphs (with uniform cost links) that allow an optimum (shortest path) IRS. Not surprisingly, it is believed that the above problem is actually N P Complete. Things look more positive if we abandon the restriction to uniform cost links. 2.2

Dynamic Cost Links

Suppose a graph is given and one is allowed to label the nodes appropriately. It is only reasonable that the names of the nodes remain fixed over time, to avoid confusion, for example. But over time, it is only to be expected that the cost of the links may vary. The cost of the links are some non-negative numbers and this is termed dynamic cost links. Can one still relabel the links of the graphs accordingly (with no change of the labels of the nodes) to allow for an optimum (minimum cost) IRS? Fkederickson and Janardan [8]came up with a very nice way of characterizing the type of graphs with dynamic cost links that allow optimum IRS.But they did this over a stricter class of IRS. An ILS is termed strict if no interval assigned to the edges of a node contains the label of the node itself. Thus, for example, a node with label 5 cannot have an edge that is labeled with the interval [3,7), as the interval contains the node label. (Note however, that the foregoing example is a legal interval for the regular ILS as we have defined earlier.) Of course, this is only a conceptual difference and in no way affects the size of the routing table. Frederickson and Janardan [8]A graph G (with dynamic cost links) has an optimum (minimum cost) strict IRS if and only if G is an outer-planar graph (a planar graph with all the nodes on one face). The corresponding result for a regular IRS is almost similar, with an extra graph. Bakker, van Leeuwen and Tan [2] A graph G (with dynamic cost links) has an optimum (minimum cost) IRS if and only if G is an outer-planar graph or Ks (complete graph of order 4). 2.3

Other Results

We now state some further results and areas of research concerning IRS.

Chordal Rings have been studied by Flammini, Gambosi and Salomone in [6], where they present some positive and negative results concerning existence

of optimal (shortest path) IRS. Non-deterministic IRS We have introduced ILS as a deterministic scheme by insisting that the intervals do not overlap, which allows us to use the efficient binary search. In practice it might be better to assign a complete interval [a,b) to a link instead of just a single left end-point and allow overlaps. With this non-detemzinistic scheme, a node label may belong to several links. A message for a destination j might then be routed by sending it over a random link whose interval contains j. This gives greater flexibility in distributing the t r a c in the network. This is a suggestion for further study mentioned in [14]. We state it here as an open problem. OPEN PROBLEM: Study the above non-detemzinistic ILS schemes.

3

Linear Interval Routing

Linear Interval Labeling Scheme (LILS) is just a restriction of ILS where the intervals have no wraparound. Thus an interval such as [5,1) is definitely not allowed. It is also a special case of Prefiz Labeling Scheme, to be discussed in the next section. As expected, the class of graphs that has a LIRS is smaller than IRS;but i t still contains some interesting networks, such as the hypercubes. 3.1

Uniform Cost Links

The following classes of graphs (with uniform cost links) have optimum (shortest path) LIRS.

Bakker, van Leeuwen and Tan [4] Complete Graphs Hypercubes n-Dimensional Grids Rings if and only if size of rings 5 4 n-Dimensional Tori q ! , d i if and only if di 5 4 Trees if and only if they contain no T-graph (see Figure 3) as a subgraph

Figure 3

Kranakis, Krizanc and Ravi [13] Complete r-partite Graphs K,,,,, ,,...,,,, r 2 2, ni 2 1 (A graph where every pair of distinct groups of nodes is connected as a complete bipartite graph) fiaigniaud and Gavoille [7] Unit Interval Graphs It is known that there are quite many type of graphs that do not have optimum (shortest path) LIRS, such as Cube-Connected-Cycle, Star-graph, ... (see [4] and [7] for a list); but a complete characterization is still elusive.

OPEN PROBLEM: Characterize the graphs (with uniform cost links) that have optimum (shortest path) LIRS or strict LIRS. 3.2

Dynamic Cost Links

Again the situation is more pleasant for networks with dynamic cost links.

Bakker, van Leeuwen and Tan [4] A graph (with dynamic cost links) has an optimum (minimum cost) LIRS if and only if it is a Centipede. A centipede is defined recursively as one of the following two graphs (see Figure 4):

Figure 4 or a centipede joined by another centipede, where by joining we mean that the head (h) of the centipede is identified with the tail (i) of another centipede that is "attached" to it. Figure 5 gives an example of a centipede.

Figure 5 Bakker, van Leeuwen and Tan [2] A graph (with dynamic cost links) has an optimum (minimum cost) strict LIRS iff it is a Line ( a tree of degree 2).

4

Prefix Routing

A Prefiz Labeling Scheme (PLS)is based on the notion of source or path routing. This type of routing assumes that the address in each message explicitly or implicitly specifies a particular path, for example cuny!mcsun!ruuinf! .... Thus the address is a sequence of names consisting of strings of characters with suitable

separators. When a message is to be relayed, the next name in the sequence, i.e. a prefiz of the address, is extracted and the routing table is consulted for the next link in the path. The idea behind PLS is to group nodes together on a link by the maximum common prefix. This is done by labeling each node with a string, over some alphabet E,as name. Each link also is labeled with a unique string, possibly by e, the empty string. When a message arrives a t a node with destination dest, the routing table is consulted for a link label that is the maximum length prefix of the address. For example, if the destination address is abc and the link labels available are e, a, ab, ca, then the link label ab will be selected as it is a prefix of abc of maximum length. Of course, each link must be properly labeled so that for any address there is always a maximum length prefix. The routing protocol is also similar to procedure SEND (Figure I), with appropriate modification to line 4. 4 find link x at node id with the label that is maximum length prefix of dest

Figure 6

A PLS is valid if procedure SEND works correctly, i.e. all messages arrive at their proper destination eventually. Figure 6 shows a valid PLS.Again for short, we call a valid PLS a Prefix Routing scheme (PRS). The feasibility of such a scheme is shown in the following result. Bakker, van Leeuwen and Tan [4] There is a valid PLS for any dynamically growing network, i.e. insertions of links and nodes are allowed, with adaptation cost of O(1). We now look a t known results on optimality. 4.1

Uniform Cost Links

Unfortunately most graphs do not have an optimum (shortest path) PRS. The following graphs (with uniform cost links) have optimum (shortest path) PRS . Bakker, van Leeuwen and Tan [4] Trees

Rings iff size of rings 5 4 Complete Graphs Complete Bipartite Graphs As with all the previous routing schemes, we have the following unsatisfactory state. .

OPEN PROBLEM: Characterize the graphs (with uniform cost) that have optimum (shortest path) PRS. 4.2

Dynamic Cost Links

Again assume that the labels of the nodes remain fixed but the cost of the links may vary non-negatively. In the previous schemes (ILS and LILS) we only considered fixed networks, i.e. there are no insertions and deletions of nodes or links. We now consider this for PRS first.

Bakker, van Leeuwen and Tan [4] A fixed (with no insertion or deletion of nodes and links) network with dynamic cost links has an optimum (minimum cost) PRS iff its biconnected components are of size 5 4. As PRS is a dynamic scheme, with arbitrary insertion and deletion of nodes and links (as long as this does not cause the network to be disconnected), we also consider dynamic networks.

Bakker, van Leeuwen and Tan [4] A dynamic (with insertion and deletion of nodes and links and no disconnection of the network) network (with dynamic cost links) of more than 4 nodes has an optimum (minimum cost) PRS iff it contains no cycle of length > 3. 4.3

Remarks

PLS is a naturally dynamic scheme. Insertion and deletion of nodes and links (without disconnecting the network) are easy to do with constant adaptation cost. Only the neighboring nodes and links that are affected need to adjust their routing tables. There is no massive update of the whole network involved. Unfortunately, in general the address label can be quite large, of O(Diameter . log(Degree)). We note also that if the alphabet E consists of only one symbol, then PLS is exactly LILS. Thus LILS is a special case of PLS also. There are quite a few variants of PLS. For example, the Path Labeling Scheme, where the destination address is shortened as it is relayed by stripping off the prefix. (This is fairly similar to the technique used in UUCP.) See [3] for more details.

5

Boolean Routing

In the IRS and PRS,to route a message we check for a link label that satisfies a certain condition. This relation can be expressed as a boolean predicate. This is the idea behind a Boolean Labeling Scheme. Destinations in the network are

grouped together to share the same link if they satisfy a certain boolean predicate on the labels of the nodes. Strings of bits are assigned as labels to nodes and predicates are attached to the links based on these labels. Elementary boolean functions such as V, A, 5,...are used to form the predicates. For example, one can represent a LILS by setting the predicate pi(dest) t (ai 5 dest) A ~ ( a i + ~ dest) for each link li that has the LILS label of [ai, u ~ + ~When ) . a message arrives, each predicate is checked in turn until a satisfied one is found. The appropriate line 4 in procedure SEND (Figure 1) is modified as follows: 1,

4


0). Thus the standard LS is just 1-LS. Multi-label schemes were considered by van Leeuwen and Tan in [15]to handle certain graphs that do not have 1-ILS.For example, 1-LILS is quite restrictive: a ring of more than 4 nodes has no optimum 1-LIRS; but a ring of any size has an optimum 2-LIRS. Thus the class of graphs increases as we increase the number of labels. Of course, trivially any graph has an optimum (n-1)-LS for any scheme LS (ILS, LILS, PLS, BLS): just label a link with all the node labels that need to be sent optimally via it. This boils down to the traditional complete routing table. Unfortunately, nothing much is known about optimum scheme for any RS with k > 1. As characterization results on optimum (shortest paths) RS on graphs (with uniform cost links) are non-existent, in the following we only classify graphs with dynamic cost links. Let k - RS be the class of graphs (with dynamic cost links) that have optimum (minimum cost) routing scheme using k-RS. Then obviously we have k - 7ZS E (k 1) - R S for any k-RS and k - LZRS k -ZRS. Only just little bit more is known.

+

Frederickson and Janardan [8] k-ZRSc(k+l)-ZRS

Bakker, van Leeuwen and Tan [4] k - L Z R S c (k + 1) - LZRS ~-Z'RSC(~+~)-LZRS 1 - U R S # 1-Z'RS

Bakker, van Leeuwen and Tan [3] k-LZRSCk-PRS k-Z'RSc(k+l)-PRS 1

-zns # 1 -pns

OPEN PROBLEM: Characterize k - R S for k > 1 for IRS, LIRS and PRS.

References 1. E. M. Baliker, Combinatorial Problems in Information Networks and Distributed

Data-structuring, Ph. D. Thesis, Dept. of Computer Science, Utrecht University,

(1991). 2. E. M. Bakker, J. van Leeuwen and R. B. Tan, manuscript. 3. ElM. Bakker, J. van Leeuwen and R. B. Tan, Prefix Routing Schemes in Dynamic

Networks, Tech. Rep. RUU-CS-90-10, Dept. of Computer Science, Utrecht University (1990). Also in: Computer Networks and ISDN Systems 26 (1993) pp. 403-421. 4. E. M. Bakker, J. van Leeuwen and R. B. Tan, Linear Interval Routing Schemes, Tech. Rep. RUU- CS-91-7, Dept. of Computer Science, Utrecht University (1991). Also in: Algorithms Review 2 (2), (1991) pp. 45-61. 5. M. Flammini, G. Gambosi and S. Salomone, Boolean Routing, Proc. Tth International Workshop on Distributed Algorithms (WDA G'93), Springer-Verlag LNCS 725 (1993) pp. 219-233.

6.

M. Flammini, G. Gambosi and S. Salomone, Interval Labeling Schemes for Chordal

Rings, Tech. Rep. No. 52, Department of Pure and Applied Mathematics, University of L'Aquila (1994). Also in this proceeding. 7. P. Fraigniaud and C. Gavoille, Interval Routing Schemes, Tech. Rep. No. 94-04, Laboratoire de 1'Informatique du Parallasme, Ecole Normale Supirieure de Lyon (1994). 8. G. N. Frederickson and R. Janardan, Optimal Message Routing Without Complete Routing Tables, Proc. 5'" Annual ACM Symposium on Principles of Distributed Computing (1986) pp. 88-97. Also as: Designing Networks with Compact Routing Tables, Algorithmica 3 (1988) pp. 171-190. 9. G. N. Frederickson and R. Janardan, Efficient Message Routing in Planar Networks, SIAM Journal on Computing 18 (1989) pp. 843-857. 10. G. N. Frederickson and R. Janardan, Space Efficient Message Routing in cDecomposable Networks, SIAM Journal on Computing 19 (1990) pp. 164-181. 11. H. Hofestiidt, A. Klein and E. Reyzl, Performance Benefits from Locally Adaptive Interval Routing in Dynamically Switched Interconnection Networks, Proc. 2nd European Distributed Memory Computing Conference (1991) pp. 193-202. 12. The T9000 Transputer Products Overview Manual, Inmos (1991). 13. E. Kranakis, D. Krizanc and S. S. Ravi, On Multi-Label Linear Interval Routing Schemes, in: J. van Leeuwen (Ed.), Proc. 19'" International Workshop in GmphTheoretic Concepts in Computer Science (WG'93), Springer-Verlag LNCS '790 (1993) pp. 338-349. 14. J. van Leeuwen and R. B. Tan, Routing with Compact Routing Tables, Tech. Rep. R UU- CS-83-16,Dept. of Computer Science, Utrecht University (1983). Also as: Computer Networks with Compact Routing Tables, in: G. Rozenberg and A. Salomaa (Eds.) The Book of L, Springer-Verlag, Berlin (1986) pp. 298-307. IS. J. van Leeuwen and R. B. Tan, Interval Routing, Tech. Rep. RUU-CS-85-16,Dept. of Computer Science, Utrecht University (1985). Also in: Computer Journal 30 (1987) pp. 298-307. 16. D. May and P. Thompson, Transputers and Routers: Components for Concurrent Machines, Inmos (1990). 17. P. RuWEka, On Efficiency of Interval Routing Algorithms, in: M.P. Chytil, L. Janiga, V. Koubek (Eds.), Mathematical Foundations of Computer Science 1988, Springer-Verlag LNCS 324 (1988) pp. 492-500. 18. N. Santoro and R. Khatib, Routing Without Routing Tables, Tech. Rep. SCS-TR6, School of Computer Science, Carleton University (1982). Also as: Labelling and Implicit Routing in Networks, Computer Journal 28 (I), (1985), pp. 5-8.

This article was processed using the BT)jX macro package with SICC style

110

Interval Labeling Schemes for Chordal Rings Michele Flamminil, Giorgio Gambosi2 and Sandro Salomone3 Dipartimento di Inforrnatica e Sistemistica, Universitk di Roma "La Sapienzaw, via Salaria 113, 1-00198 Rome, Italy, and Dipartimento di Matematica Pura ed Applicata, Universiti di L'Aquila, via Vetoio loc. Coppito, 1-67010 L' Aquila, Italy (flamminiQsmaq20.univaq.it) Dipartimento di Matematica, Universitk di Roma "Tor Vergataw,via della Ricerca Scientifica, 1-00133 Rome, Italy (gambosiQmat.utovrm.it) "ipartimento di Matematica Pura ed Applicata, UniversitL di L'Aquila, via Vetoio lac. Coppito, 1-67010 L'Aquila, Italy ([email protected]) -

Abstract. In this paper, the problem of devising interval routing schemes for chordal ring networks is considered. Both positive and negative results are provided concerning the existence of optimal interval routing schemes for specific classes of chordal rings. Moreover, results are also provided for schemes which are not optimal, in terms either of non minimal space occupation or of routing messages along paths which are not of minimal length.

1

Introduction

Routing messages between pairs of processors is a fundamental task in a d i s tributed network. A network of processors is modeled as an (undirected) connected graph G = (V, E), where V is a set of n processors and E is a set of pairwise communication links. Assuming some cost function on the network edges exists, it is important to route each message along a shortest path from its source to the destination. This can be trivially accomplished by referring, at each node v , to a complete routing table which specifies, for each destination u, the set of links incident to v which are on shortest paths between u and v. Such a table has 8 ( n ) [log dl-bits entries (d-bits if all shortest paths between each pair of processors have to be represented), where d is the node degree. Since in the general case such tables are too space consuming for large networks, it is necessary to devise routing schemes with smaller tables, possibly accepting the possibility of broadcasting messages along paths which are not of minimal length. Research activities focused on identifying classes of network topologies where the shortest path information at each node can be succinctly stored, assuming In the following, a l l logarithms will be assumed with base 2.

that suitably "short" labels can be assigned to nodes and links at preprocessing time. Such labels are used to encode useful information about the network structure, with special regard to shortest paths. In the ILS (Interval Labeling Scheme) routing scheme ([lo], (111, [12]) node labels belong to the set (1, ...,n} , assumed cyclically ordered, while link labels are pairs of node labels representing disjoint intervals on {1,...,n). To transmit a message m from node vi to node vj, m is broadcast by vi on the (unique) link e = (vi, vk) such that the label of vj belongs to the interval associated to e. With this approach, one always obtains an optimal memory occupation, while the problem is to choose node and link labels in such a way that messages are routed along shortest paths. In [lo], [ll], [12] it is shown how the ILS approach can be applied to optimally route on particular network topologies, such as trees, rings, etc. The ILS approach has also been used in other papers as a basic building block for routing schemes based on network decomposition and clustering ([2], [3], [6], [ I ,[8], [g]). In [ll], the approach has been extended to allow that more than 1 interval is associated to each node; in particular, a 2-ILS scheme - that is a scheme associating at most 2 intervals for each edge - is proposed for two dimensional doubly wrapped grids. In this paper we consider the problem of devising interval routing schemes on networks whose underlying topology is a chordal ring. In particular we present both positive and negative results on the existence of interval routing schemes in specific classes of chordal rings. Moreover, some results concerning the existence of non optimal interval routing schemes (i.e. schemes which do not route messages along shortest paths) are provided. The paper is organized as follows: in section 2 we give some preliminary definitions; in section 3 we provide (positive and negative) results on the existence of interval routing schemes in classes of chordal rings; in section 4 we derive some upper bounds on the number of intervals necessary to interval routing messages in some classes of chordal rings; in section 5 some results are given for non optimal schemes.

2

Definitions

Let us first provide some preliminary definitions. Let G = (V, E) (JVI = n) be a graph with weighted edges. Given a node v f V, we will denote as I(v) E the set of edges incident to v. For each node u and each edge e E I(u), let us denote as s(u, e) the set of nodes v optimally reachable from u through that edge (i.e. such that e belongs to some shortest path from u to v ) . We denote by path represenlation a function d : V x E u 2" such that: 1. if e E I(u) then d(u,e) C s(u,e); 2. if e @ I(,) then d(u, e) =undefined; 3. for each u E V, UeEI(U)d(~,e) = V - {u). Definition 1. A path representation is Overall if Vu E V, d(u, e) = s(u, e).

Definition 2. A path representation is Muliiple(p) if Vu, v E V, u # v Definition 3. A path representation is Single if Vu, v E V, u # v I{e

E I(u) I v E d(u,e)}l= 1.

Note that in the Single case, for each u E V, sets d(u, e) (e E I(u)) partition set V - {u). According to the definitions above we will denote by kdingle (respectively k-Multiple@) and k-Overall) Scheme an interval labeling scheme that, using a Single (respectively Multiple(p) and Overall) path represent ation, requires at most k intervals per edge. Furthermore, for the sake of brevity we will denote as 1Interval a scheme that associates with each edge a unique interval (independently from the path representation considered). The path representations defined above merely correspond to our informal definitions of the various schemes, and generalize the ILS scheme to the case in which multiple shortest paths are represented. They are defined in order to emphasize in all cases the paths that are represented, as each set d(u, e) is the subset of s(u, e) for which link e is activated.

Definition4. A chordal ring [I] is an undirected graph &(C) = (V, E), where C E (2 ,...,n - 21, V = {vo, ...,v ~ - ~and } E = E l U & with El = { ( ~ i , ~ ( ~ + ~I i) = ~ ~0, d..n. ,)n - 1) and E2 = {(vi, ~(i+~)modn) I i = 0, .-.,n- 1, j E C}. Edges in

E2

are called chords.

By adding chords we can obtain classes of denser graphs, from rings up to complete graphs. In the following we will drop the mod notation implicitly assuming that all operations are mod n and, given a node vi, we will denote its label by li.

Optimal schemes

3

Some of the results provided in [4] [5] state the difficulty of devising minimumspace interval routing schemes. In this section, we consider some restrictions which allow polynomial constructions in the case of edges with uniform weights. Similar results for the ILSScheme, for the general case of different weights, have been presented in [6], [lo], [Ill, [I210 In the following, we will show the existence of optimum schemes (that can be derived in polynomial time) for particular types of chordal rings. In all cases, the interval scheme is obtained by assigning integer labels in increasing order, starting from one node and traversing edges in El along a chosen direction, i.e., by assigning labels ( i k) mod n to node Vi, for some 0 k 5 n - 1. We will denote a scheme derived by such an assignment as cyclic. The following fact is immediate.


'i), b = [lj+L+l+l,~ + L + J + Iand ) 14 = [li+~:j+i 1 lj-Lg!)* 3. Fact 3.1 applies. The intervals associated with edges at node vi are 4 =

5.

4. Let us consider a generic node vi and let B = Associate with the outcoming edges el = (vi, vi+i), e2 = (vi, vi-I), es = (vi, vi+tl), e4 = (vi, vi-k,), es = (vi, v ~ + ~and , ) e~ = (vi, v ~ - of ~ node ~ ) vi the following intervals: - I1 = ['i+l, li+llJ]; - I2 = [li-L+j)li-l]; - I = = [1i+L3J+19 fi+19Jkl+l~j];

- 4 = [lj-L9Jkl-L+J 9'i-L+l-lI; - b = [5.+L9J.il+L+J+1)&l:~l; - ' 6 = [ ' ~ + L + J +1j-L8Jk1-L~1111. ~P The result derives easily by observing that fact 3.1 holds also in this case. 5,6. Immediate. By observing that fact 3.1 holds in both cases. The following theorem introduces some negative results for chordal rings. Theorem 7. The following results hold:

# 0, k 2 4) such that then is no 1-Single Scheme for &({El); 2. then exist pairs k,n (n mod k = 0, k 2 4) such that there is no 1-Multiple(2) or 1-Overall Scheme for &((k));

1. there exist pairs k,n (nmod k

3. there exist pairs k, n such ihaf there is no 1-Muliiple(2) or 1-Overall Scheme for Rn({2))* Proof. We shall prove each case separately:

1. Let us notice first that, if a 1-Single Scheme exists for a chordal ring R = &({k)), LfJ > 2, k mod 2 = 0, k 2 4, then such a scheme must be cyclic (by lemma 5). It is easy to see that for some n, k which satisfy the constraints above, a cyclic 1-Interval Scheme cannot be derived for R. Infact, let 6 = and assume 6 odd. Starting from node vi and performing L$J 1 clockwise (counterclockwise) moves along chords, node v, (v,) is optimally reached, w h e r e x = i + [$Jk+k ( y = i - L i J k - k ) . I f y - ( 2 - k ) > 1 (that is the remainder of f is greater than 1) the unique shortest path t o node v, (v,) is < ~ iVi+k, , Vi+2k, ...,V= > (< Vi, ~ i - k , vi-~k, .,vy >). In such a case, nodes v,-k, V, are optimally reached from vi through edge (vi, vi+k), while node V, is optimally reached from vi through edge (vi, viWk).But, since z- k < < 2, a t least two intervds are necessary to label edge (vi, vi+k). 2. It is immediate that, for k 2 4, k mod 2 = 0 and n large enough, it must be si = si for all i , thus implying (by lemma 5) that any scheme for R must be cyclic. In such a case, for any scheme, the unique two shortest paths from vi to Vi+k+l are < Vi, Vi+l, Vi+l+k > and < Vi, Vi+k, Vi+k+l >. Hence, if more than one shortest path must be represented, the interval associated with edge (vi, vi+l) must contain labels li+ and li+k+l, but not label li+k. This implies that at least two intervals a t edge (vi, v ~ + are ~ ) necessary; 3. Let us consider the chordal ring R8({2)). If more than one shortest path has to be represented, starting from node vo and following edge el = (vo,vl) it is possible to optimally reach nodes vl and us. Hence, if there is a unique interval associated with el, labels 11, I3 must be adjacent. W.1.o.g. let us assume Il < b. The unique interval associated with edge e2 = (vo, v2) must only contain labels 12,13, 14. The only orderings where all such labels are in a single interval both for el and e2 are l1k4l2and lr131214. In both cases, labels of nodes in s(v6, (us, uo)) are not in a single interval. Since in this case Multiple(2) is coincident with Overall, the theorem holds.

+

LtJ

..

4

Upper Bounds

In this section, we provide some upper bounds on the number of intervals that are necessary to assign labels to edges in chordal rings in the case in which only one shortest path between each source-destination pair must be represented. In the following theorem we further investigate the relation between remainder of f and the existence of interval schemes for chordal rings &({k)). Let Rcm(f) be the remainder of f .

Theorem 8. There crisis a 2*rnin{L+ J +I,[$ J +1)-single Scheme for &({k)), 1 2 maz{k - r, r 1), when r = &m(f ).

LtJ+

+

Proof. As stated in theorem 6 if n mod k = 0 then there exists a 1-Single Scheme for R ( { k ) ) . ~ s s u m enow n rnodk # 0 and let us assign cyclic labels to nodes. W.1.o.g. let us consider node vi and let us assign edge labels as in the proof of theorem 6. Clearly, by the above assignment, not all nodes reachable through such edges are on a shortest path. There are cases in which, as stated in theorem 7, some of the nodes whose labels are contained in I3 are optimally reachable through edge e4. Let b = LfJ. Starting from node vi and performing ~fJ clockwise (counterclockwise) moves along chords, node v, (vy) is reached, where t = i J![ k (y = i - Lijk). Let d be the distance between v, and vy along the external face of the ring. There are two cases:

+

1. b is even. In this case, recalling that all operations are mod n, it results d = i - b$ - (i 64) = n - bk. Thus, 0 < d < k and the shortest way to reach nodes adjacent to v, (v,) is to reach v, through the edge e3 (e4) and then moving along the external face. Thus, one interval is still sufficient at edge e3 (e4); 2. b is odd. In this case d = n-(b-l)k and, as aconsequence, k < d < 2C. Thus, another chord can be traversed from node v, (v,) to optimally reach node v,+k (vy-k) and, in case, some adjacent nodes (see figure 1). Furthermore, if y - k - t > 1, that is r > 1, this shortest path is unique and two intervals must be associated with edge e3 (e4).

+

Let us consider this second case. The same considerations can be applied to node v,+k. Starting from such a node and traversing chord (v,+k, v=+*~) node ~ = + 2 ris reached such that vy < v,+2k < vy+k. Hence, if y - x - k > 2 or z 2k y > 2 one more interval must be added to edge e3. By hypothesis, this can be done at most JL: - 1 times, that is the maximum number of chords that can be traversed starting from node vs+k before completing one lap from vi, that is reaching some node vj , i < j < i k. However, the worst case occurs when node v,+k is at a same distance from vy-k and v, . In such a case, after L$J moves the path starting from edge e3 becomes longer than the path starting from edge e4. Thus, interval I3can be divided in at most min{L$J, [QJ}more intervals. Furthermore, since the same considerations can be applled to edge el, that is at most min{LgJ, LQJ) subintervals of I3 are constituted by the labels of nodes optimally reachable through e4, the number of intervals necessary to code shortest path information is 2 mini[& J 1, [$J 1).

+ -

+

+

+

Theorem 8 emphasizes the strict dependence between Rem(H) and the number of intervals necessary to code shortest path information. In theorem 6 it is shown that if Rem(f) = 0 then there exists a 1-Single Scheme for This result still holds if Rem(f) = 1. In this case, in fact, the distance between nodes v, and v,-k is equal to 1. Hence, starting from node v i , L ~ J 1 moves are necessary to reach node v,-k from both edges es, e4 (L!J clockwise moves

a({))).

+

Fig. 1. Rn({k)), n mod k # 0

+

along the chords plus 1 move along the external face in the first case, and 1iJ 1 counterclockwise moves along the chords in the second one). Analogously, the same holds if Rem(f) = k - 1 and b is even. Furthermore, as seen in the previous theorem, the worst case is given by the case Rem(t) = 1tJ . The result in theorem 8 holds if, for any node vj, the set of nodes optimally reachable by traversing chords does not include any node vj, i < j < i k. This is stated by the condition LfJ + 1 maz(k - r, 1+ r). The following theorem provides an upper bound on the number of intervals necessary to label edges in the class of chordal rings &({kl, k2)), k2 mod k1 = 0, when only one shortest path between any sourcedestination pair is to be represented. Notice that, by theorem 6, if n mod k2 = 0 there exists a 1-Single Scheme for &({kl ,k2}).

>

Theorem 9. T h e n exists a o(% min{ L+J &({kl, La)), k2 modkl = 0, nmod kz # 0,

+

+ I,[+ J + 1))-Single Scheme for + 1 2 rnoz(k2 - r, r + 1).

Proof. Let us assign labels to nodes and edges as in the proof of the theorem 6. Let b = 1f-J and assume b odd. As seen in the proof of theorem 8, since y - k2 < x C2 (that is v,+k, follows v,-k, in the total ordering of nodes), one more interval must be added to es. Starting from node v,+k, and traversing the new intervals can be created (see figure 2). chords of length kl, at most

+

[&

The theorem derives by the same considerations in the proof of theorem 8 and by observing that for each new interval added to ea in such a proof at most [%J new intervals must be added in this case.

Fig-2. Rn((kr, kt)), n mod kz # 0, k2 mod kl = o

5

Relaxing shortest paths constraints

In this section we investigate the relation between the number of intervals necessary to code path information and the length of the longest path. In particular, we will provide schemes for classes of chordal rings, that, by relaxing the constraint on the minimal length of the represented paths, associate a unique interval with each edge, thus resulting in an optimal space complexity. In this case ([3] [7] [8] [9]), the efficiency of a routing scheme is measured not only in terms of memory occupation, but also in dependence from the stretch factor (SF),defined as the maximum ratio (among all pairs of nodes u, v) between the length of the path represented by the scheme and the distance between u and v. Notice that the definitions of Single, Multiple(p) and Overall path representations can be trivially extended to consider set of paths satisfying a given stretch factor. In all of the following cases a cyclic labeling is considered. Furthermore, we will assume that 13j > 2, where k is the longest chord (i.e. that there is significative amount of chords in the ring).

Theorem 10. Then exists a l-Single Scheme for &({kl, k2}), n mod k2 = 0, kz = kl 1, with a stretch factor SF 5

+

i.

Proof. Let us consider a generic node vi. First of all, observe that a cyclic labeling of nodes implies that a unique interval at each edge is not sufficient to code shortest path information. Starting from node vi and traversing edge (vi ,v ~ + ~nodes ,)

vi+kl, Vi+2kl can be optimally reached, while node Vi+k, is optimally reachable through (vi, v ~ + ~ ,Thus, ). a unique interval is not sufficient for edge (vi, v ~ + ~ , ) . Let us associate the following intervals with edges el = (vi , vi+l), e2 = (vi, vi-l), e3 = (vi, vi+kl), el = (vi, vi-k,), e5 = (vi, vi+k,) and e6 = (vi, vi-k2):

In such an assignment, starting from node vi, node vi+2t1 is reached through es and the difference between the length of this path and the length of the shortest one (the one through e3) is equal to 1. The same holds for node vi+ak,, and so on. Thus, if d is the length of the shortest path the length of the path chosen by the scheme is at most equal to d 1. Hence, SF 5 and, in the worst case for d = 2, SF 5 $.

+

+

9

Corollary 11. There exists a 2 min(l& J 1, L ~ +J 1)-Single Scheme for &({kl, kz}), k2 = kl 1, J 1 2 max{ k2 - r, r I), with a stretch factor

SF 5 q.

+

le +

+

Proof. As in the proof of theorem 10, let us associate unique intervals with edges (vi, v ~ + ~ thus , ) , obtaining a stretch factor at most equal to $ for nodes optimally reachable through chords ( ~ ~ , v ~ +For ~ , )what . concerns chords of length k2, we may label such chords by not considering chords of length kl, that is by considering the &({k2}) subgraph. By theorem 8 , 2 min{L+J 1,l%J 1) information. intervals are necessary at edges (vi, ~ i + ~ to , ) code shortest

+

+

Theorem 12. Then exists a l-Single Scheme for &({kl, k2}), k2 = k1 + 1, wiih o stretch factor SF 5

v.

Proof. Let us assign edge labels as in the proof of theorem 10 and let us consider node vi, By such an assignment, for the nodes optimally reachable through e3 the stretch factor is at most Let us estimate the stretch factor for nodes whose labels are in the interval associated with (vi, ~ i + ~and , ) that are optimally reachable through (vi, v~,~,).Clearly, by theorem 10, if n mod k2 = 0, it results SF 5 q . Assume now that n mod k2 # 0 and let b = 1i;J. Let us consider node vi and nodes v, and vY,x = i LtJk2, y = i L4Jk2and let d be the distance between v, and v, along the external face. There are two cases:

4.

+

-

1. b is even = n-bk2. Thus, 0 < d < kg and no more In such a case d = i-b$-(i+b%) chords of length k2 can be traversed to reach any node between v, and v,. It may happen that a chord of length kl is traversed, but such a chord will

directly connect nodes v, and v,. Thus, starting from v, (v,) nodes between v, and u, are optimally reached through (v,, v,+l) ((v,, vy- 1)). Hence, the interval associated with es (es) is sufficient t o reach all those nodes with a stretch factor still less than $. 2. b is odd In such a case, nodes v,+k,, v,+r, (v,-k, ,vy-ka) are optimally reached from node v, (v,) through chords k1 and k2. Notice that x k2 > y - k2, which implies that the path chosen a t node vi to reach those nodes is longer than the shortest one. Note that node v, (vy) is optimally reached by traversing chords. Moreover, by the assignment of edge labels, starting from node v,, nodes between v, and v,+k, are optimally reached. Thus, starting from vj, the path to reach nodes between v, and V ~ + L + J has length at most

+

'

9

9+

LvJ,while the shortest one has length 9+ 1. Thus, SF 5 w . The 4+l

difference between this two paths becomes smaller as their len&h.becornes greater, hence, the worst case is = 1, so SF

9

9.

Theorem 13. Then erisis a 1-Single Scheme for R,,({kl, k2}), 2k1 > k2, with a stretch factor SF

w.

Proof. First of all note that, as observed in the proof of theorem 10, a unique interval a t each edge is not sufficient to code shortest path information. W.1.o.g. let us consider node vi and let us associate with el = (vi, vj+l), e2 = (vi, viol), e3 = (vi, vi+kI), e4 = (vi, vi-k,), es = (vi, vi+k,) and e6 = (vi, vi-kg) the following intervals:

Let us consider the stretch factor induced by the assignment of a unique interval to es. Starting from node v; the shortest path to reach node vi+2kl has length 2 while the path to vi+2r, starting from es has length minil (i+ 2k1 ) (i kz), 2 (i k2 El - (i 2kl))}, and the difference between the shortest path and the path chosen by the scheme is at most min{(2kl - k2 - 1,k2 - kl) and becomes smaller as their length increases. Thus, the stretch factor is maximal when the length of the path represented by the scheme is minimal, that is equal to 2. In such a case it results SF < 2+k?-kX. L Let us consider now the interval associated with es. Notice that it may h a p pen that some node, whose label is contained in such an interval, is optimally reachable through e ~ by , traversing L&J 1 chords (moving counterclockwise).

+

+ + +

+

+

+

In order t o reach such a node through es (i.e. moving clockwise), a t most L&J moves on the chords, plus at most 1 % moves ~ along the external face, plus one possible move on the chord kl, are necessary, while the shortest path requires exactly L&J 1 moves. Thus,

+

Recalling that

> 2 it results

Let us finally consider the interval associated with el. Notice that it may happen that some node vj, i 1 < j i 191, can be optimally reached by a shortest path through es. In the worst case, such a path can have length as small as 2, while, in the case j = i L ~ Jmoving , from vi to vj through el requires 1 % steps. ~ This provides a stretch factor SF f 1 % ~ %. The theorem derives by observing that, since kl < kz, >1

+ +

.:.:a:.

a:.ffi. :.

link 0

ROUTE (header) { header-1 = header >> number-oi'bi ts-of'rt-3 header-2 = header & number-oi'bits_ofqartrt2; link1 = INTERVAL-ROUTE-l(header-1); li ok2 = INTERVAL_ROUTE_2(header-1); retum(link1 & link2);)

Fig. 14. a) The splitted-header interval routing technique. Each header is splitted into several parts. For each part a series of possible Links is given by classical interval tables. Combination gives the final outgoing link. b) Implementation of the splitted-header interval routing mechanism for a header consisting of 2 parts. Extension to more parts is straightforward.

This algorithm seems quite complex at first sight though a very simple implementation erists. We give a Glike notation in figure 14b. The routines INTERVALBOUTEJ/Z are routines implementing classical interval routing and returning the outgoing link associated with the interval the header belongs to. Since these outgoing links must possibly represent multiple links we choose for the following notation: The links are represented by a word having as many bits as the number of links. Each link is defined by having a 1in the corresponding bit and 07selsewhere. A choice between several outgoing links is then logically represented by setting 1's in the corresponding bits and 0's in the bits for links

not belonging to the options. The actual link can then easily be found by determining which bit is set for all header-parts. In fact, if we use the same notation to indicate the links for a single link, a simple bit-wise and-operator gives the outgoing link. We will now apply the splitted-header technique to our deadlock-tree routing algorithm. The choice of splitting the header is straightforward. The least significant part of the header determines the x- position, the most significant part indicates the y-position of the destination node. We then must define two interval tables, a first one for determining possible outgoing links according to the horizontal position of the node and a second one for determining possible outgoing links according to the vertical position of the node. We illustrate this in figure 15 for the fault-free routing algorithm.

Fig. 15. a) The definition of the liuks. b) The classical intervals translated to two interval tables, one for the x-coordinate and one for the y-coordinate.

If we put the fault-tolerant routing algorithm in this notation we remark that the amount of routing information is very compact. Indeed, the number of intervals

is proportional to the number of failure rectangles and hence to the number of failures. A second very interesting feature of this routing scheme is the complexity of the algorithm calculating the intervals. Though the interval tables must be globally changed the complexity of this algorithm remains proportional to the number of failures. Indeed to set the interval tables a t some node it is sufficient to know the position of this node in the system (reduced structural information) and the position relative to the failure rectangles (actual network information).

4

Conclusion

In this paper we presented a routing scheme which uses both actual network information and implicit information. This results in a fault-tolerant routing algorithm which remains nearly optimal and is yet very compact due to the interval routing technique. The complexity of the algorithm calculating the i n t e r d s is proportional to the number of failures in the network. This makes this algorithm very well suited for massively parallel computers. Because wormhole switching gains importance in (massively) parallel machines and the buffering mechanism in these networks is mostly fixed we demonstrated how to adapt the routing to become a deadlock-free routing scheme. Unfortunately the number of intervals then becomes proportional to the square root of the number of processors in the system. Therefore we introduce an extension to interval routing, which allows the routing information to remain extremely compact. Again the complexity of the algorithm calculating this information is proportional to the number of failures in the network. This routing scheme has been presented for the specific case of a two-dimensional mesh. Yet it can be easily extended to the more general case of k-ary n-cube topologies. In the future we will elaborate the algorithm to support this more general network topology. We will also study the problem of load-balancing in these networks and try to combine i t with compact routing information as well as iqiured networks. The relation between adaptive routing and compact routing must be clarified, especially in the presence of failures.

References van Leeuwen, R.B. Tan, Prefix routing schemes in dynamic networks, Technical Report RUU-CS-90-10, Rijksuniversiteit Utrccht, March 1990 2. A.A. Chien, J.H. Kim, Planar-Adaptive Routing: Low-cost Adaptive Networks for Multiprocessors, Proc. 19th ACM Intl. Symp. on Computer Architecture, 1992, pp. 268-277 3. W.J. Dally, C.L. Scits, The torus routing chip, Distributed Computing, 1, 1986, pp. 187-196

1.

E.M. Bakker, J.

4. W.J. Dally, C.L. Seitr, Deadlock-Free Message Routing in Multiprocessor Interconnection Networks, IEEE 'lkans. on Computers, vol. 36, no. 5, May 1987, pp. 547-553 5. ReEssu, R. Knecht, Intel Paragon XP/S Architecture and SoftwareEnGoament, Proceedings of Supercomputa 93, Mannheim, June 1993. 6. G.N. Frederickson, R. Janardan, Separator-Based Strategies for Efficient Message Routing, Proc. of the 27th IEEE annual Symp. on Foundation of Computer Sdence, 1986, pp. 428-437 7. G.N. Frederickson, R. Janardan, Designing Networks with Compact Routing Tables, Algorithmica, 3, 1988, pp. 171-190 8. C.J. Glass, L.M.Ni, The flun Model for Adaptive Routing, Proc. 19th Intl. Symposium on Computer Architecture, IEEE CS Press, Order No. 2940, 1992, pp. 278-287 9. C.3. Glass, L.M.Ni, Fault-Tolerant Wormhole Routing in Meshes, Proc. FTCS 23, France, 1993, pp. 240-249 10. K.D. G nther, Prevention of Deadlocks in Packet-Switched Data 'Bansport Systems, BEE 'haasactions on Commdcations, vol. 29, no. 4, April 1981, pp. 512524 11. Grand Challenges 1993: High Performance Computing and Communications, report by the Federal Coordinating Council for Science, Engineering, and Technology, Committee on Physical, Mathematical, and Engineering Sciences, c/o National Science Foundation, 1993 12. h o s , The T9000 Tkansputer Products overview manual, INMOS SGSThompson, first edition 1991 13. C.E. Leiserson and d.,The Network Architecture of the Connection Machine CM5, Thinking Machines Corporation, November 9, 1992 14. D.H. Linder, J.C. Harden, An Adaptive and Fault Tolerant Wormhole Routing Strategy for k- ary n-cubes, IEEE 'Ikans. on Computers, vol. 40, no. 1, January 1991, pp. 2-12 15. D. May, P. Thompson, Tkansputers and Routers: Components for Concurrent Machines, INMOS Ltd, April 4, 1990 16. L.M. Ni, P.K. McKinley, A Survey of Wormhole Routing Techniques in Direct Networks, IEEE Computer, February 1993, pp. 62-76 17. Parsytec GmbH, Technical Summary Parsytec GC, Vasion 1.0. Parsytec GmbH, 1991. 18. D. Peleg, E. Upfal, A 'lkade-Off between Space and Efficiency for Routing Tables, Journal of the ACM, vol. 36, no. 3, July 1989, pp. 510-530 19. P. Ru idka, On Efficiency of Interval Routing Algorithms, in Mathematical Foundations of Computer Science, M.P. Chytil, L. Janiga, V. Koubek (Eds), 1988, vol. 324 of Lecture Notes in Computer Science, pp. 492-500 20. N. Santoro, R. Khatib, Labelling and Implicit Routing in Networks, The Computer Journal, vol. 28, no. 1, 1985, pp. 5-8 21. M. Shumway, Deadlock-Free Packet Networks, Proc. 2nd Conf. of the North American Transputer Users Group, October 1989, Durham, NC, IOS Press 1990, pp. 139-177 22. H. Sullivan, T. Bashkow, D. Klappholz, A large scale, homogeneous, fully distributed parallel machine, Proceedings 4th Symposium on Computer Architecture, March 1977, pp. 105-124 23. F. Tiedt ,Parsyt ec GCel Supercomputer, Technical Report, Parsytec GmbH, 1991.

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24. J. van Leeuwea, R. B. Tan, Interval Routing, The Computer Journal, vol. 30, no. 4,1987, pp. 298-307 25. J. Vounckr, G. Deconindr, R. Cuyvers, R. Lauwereins, J .A. Peperstraete, Network Fault- Tolerance with Interval Routing Devices, IASTED Intl. Symp. Applied Informatics, France, May 1993. 26. J. Vounckx, G. Deconinds, R. Cuyvers, R. Lauwereins, J.A. Peperstraete, Multiprocessor Routing techniques, Deliverable 03.1.1/L of ESPRIT Project 6731, July 1993

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The Buffer Potential of a Network Krzysztof ~ i k s Evangelos ' ~ r a n a k i Adam i~ Malinowskil Andrzej Pelc3 Instytut Informatyld, Uniwersytet Wanzawski, Baacha 2, 02-097, Warszawa Carleton University, School of Computer Science, Ottawa, ON, KIS 5B6 Wpartement d'Informatique, UQAH, Hd,QuCbec J8X 3x7

In the standard model of anonymous networks [I, 4, 31 it is assumed that each processor is permanently attached to its neighbors in such a way that when a bit arrives the processor knows the neighbor it came from. This is equivalent to assuming that a processor of degree d has d buffers, one corresponding to each neighbor, for storing the bits arriving from respective neighbors. The following question arises: "What network computation can be performed if the number of buffers at a node is less than the degree of the node?" This question is considered in (21 for the case of anonymous rings of size N. It is assumed that each processor has exactly one buffer. Unlike in the standard ring, when a bit arrives at the buffer the processor cannot distinguish the originating neighbor; when a bit is sent, it is transmitted to both neighbors. In addition, buffers are assumed to be of FIFO type. The following result is proved in [2]

f : {0, 1IN + {O, 1) is computable in the single-bufler ring of size N if and only i f f is i n v a ~ a n tunder cyclic shifts and refEections of the inputs. In both the synchronous and asynchronous model the computation uses O(NZ)one-bit messages. Moreover, in the synchronous case such a function can be computed in O ( N ) time.

Theoreml. A boolean function

Similarly as in the case of the ring, we can define, for any network N, a corresponding wireless network AfWirer,,,. In such a network, all neighbors of a node v are partitioned into subsets corresponding to a common buffer at v. Bits coming from neighbors of a given subset are received in the respective buffer (without the possibility of distinguishing originators). The following problem remains open: What classes of boolean functions are computable in wireless networks and what is the complexity of such computations?

It should be noted that some functions are clearly computable, for example the OR function in any wireless synchronous network. Let the bufer potential of N,denoted by @(Af), be the maximum number of buffers needed at a node such that the network n/,ipere,, is computationally equivalent to N (This is interpreted very liberally to mean computing the same class of functions and/or with the same bit-complexity.) The results of [2] show that @(RING) = 1 both for the synchronous and asynchronous ring. The following problem seems to be very challenging:

Determine the value of @(N)for various networks, such as meshes, hypercubes, or even arbitrary networks.

References 1. H. Attiya, M. Snit,and M. Warmuth. Computing on an anonymous ring. Journal of the ACM, 35(4):845 - 875, 1988. (Preliminary version has appeared in proceedings of the 4th Annual ACM Symposium on Principles of Distributed Computation, 1985). 2. K. Diks, E. Kranakis, A. Malinowski, and A. Pelc. Anonymous wireless rings. Theoretical Computer Science. (to appear). 3. E. Kranakis, D. Krizanc, and J. van der Berg. Computing boolean functions on anonymous networks (extended abstract). In M. S. Paterson, editor, Proceedings of ICALP 90, volume 443. Springer Verlag Lecture Notes in Computer Science, 1990. (Full version of the paper to appear in the Journal of Information and Computation). 4. M. Yamashita and T. Kameda. Computing on an anonymous network. In 7th Annual ACM Symposium on Principles of Distributed Computation, pages 117 130, 1988.

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150

Path Layout in ATM Networks Oman Gerstel and Shmuel Zaks Department of Computer Science Technion, Haifa, Israel ORIG,[email protected]

Abstract. Models and design methods are developed for the layont of virtual paths in ATM networks. We define exact measures for determining whether a given layout uses the network efficiently, does not over-utilize the routing tables, and guarantees short setup time for every connection request. We then present schemes for designing the layout of virtual paths for relatively simple networks (chains, rings and meshes) and extend the results to the dass of tree networks, and discass the optimality of these schemes.

1 Introduction The Asynchronous Tkansfer Mode (ATM)[lg, 51 is the trassmission, switching, and multiplexing technique chosen by CCITT for B-ISDN.Due to the future importance of fast, broadband, integrated networks, ATM has been extensively discussed in recent years. ATM is based on small fixed size packets, which are called cells. Because of the very high requirements for switching rates, the routing of the cells must be done by a dedicated hardware, implying very simple routing algorithms. The routing scheme chosen in ATM is based on two fixed labels in the header of each cell (VCI and VPI). These labels serve as indices into the routing tables of the switches, and they determine the route that a cell will take in the network. The VCI label determines the specific connection (termed virtual channel or V C ) to which the cell belongs, and hence all cells which belong to a certain connection have the same VCI, while cells that belong to different connections have different VCIs, even if these connections share the same route in the network. The concept of the virtual path (VP) is a later concept, whose main role is to reduce various network and connection management overheads and to avoid overflowing the VC tables of the network switches. This path multiplexes many VCs that share the same route, requiring a single VP entry in every intermediate switch, instead of a separate VC entry in the original scheme. This is achieved by a layered routing scheme, in which the VCI of the cell is ignored as long as the VPI is non empty. Only when the VPI is empty, is the VCI considered, to allow the separation of the different connections. The above discussion hints that the VP is essentially an internal network concept ,whose main concern is to keep the VC routing tables s m d . From this stems the different life span for VCs and VPs: A VP is typically a fixed route,

which remains in the network for long periods, as opposed to the VC that is created at call setup and destroyed at the tear-down of the call. We note that a different model for high-speed routing was suggested in [3,4, 6, 8, 91. The architecture issues concerning these two models are quite distinct, though they both try to deal with the case of high-bandwidth lines, and control over a huge capacity of traffic.

A fuller discussion on B-ISDN and ATM can be found in [5, 191. A clear and precise description of the exact routing mechanism in ATM can be found in [ll]. Previous work on VCs and VPs was mainly empirical, based on heuristic approaches [2, 181, while this work and a few additional works [15, 7, 17, 161 attempt to analytically address the problem. While the simple solution of having a VP between every pair of switches is ideal for fairly small networks it is not suitable for bigger networks, which a likely to be constructed in the future: in certain network topologies, some of the VP routing tables may be filled with N(N - 1) entries, in a network of N switches, and since the VPI is restricted to 12 bits, ~ N ( N- 1) 5 212 or N 91 if we allow to fill the VP table up to its total capacity, which is not likely to be the case (see [15] for a comprehensive discussion).

4

< -

This work focuses on techniques for designing the layout of VPs in a given communication network. To this end, we first define (in Section 2) the essential characteristics of a good layout (aaother contribution of this work). This layout may serve as an initial setup of VPs in a network, and may still be extended dynamically, according to the needs of the network, while keeping the "goodn performance of the initial setup. We a h define an easier problem - the 1-m VP layout that helps to solve our layout problem.

-

We then examine (in Section 3) several simple yet practical networks, namely chains, rings and meshes, show how to construct a layout of VPs for them, and discuss the optimality of these constructions. The techniques are extended in Section 4 to the class of tree networks, and to general networks. We conclude by summing up the results and suggesting further research directions in Section 5. The h u e of sense of direction (first introduced in [22]), refers, generally speaking, to any global knowledge a processor in a distributed network has, in addition to the only necessary local informationthat consists of the number of its neighbors (e.g., identities of networks, structure of the network, the knowledge of 'left' and 'right' in a chain or ring configuration, and routing tables). The scheme presented in the solution for the chain, ring and mesh networks requires us to let each node have a sense of direction, in a form of a routing table (in either software or hardware), which will enable it to know where to direct the various messages, going on the various VPs that pass it. Thus, translating our schemes to software or hardware units for performing the routing, amounts to installing - besides the physical high-speed links - this knowledge in the switch.

2

Problem Definition

Before devising schemes for designing a VP layout, one has to define precisely the characteristics that make such a layout a good one, in terms of its performance. We argue that the following properties are central in these networks:

Full reachability: Each switch can be reached by any other switch, using a route which is composed of a concatenation of VPs - this allows to connect every pair of parties, regardless of the switch that connects them to the rest of the network. Restricted VP count: The number of VPs that are used by any VC should be relatively small. This fact is important to decrease the call setup time, since the VC routing tables must be updated only when a VP ends (and the VC must be routed into another VP). Since the update of these tables is done by a network layer facility - in software - the fact that this facility must be used only in a small number of switches along the route, substantially speeds up the process Short underlying route: The chosen route must also be short in terms of the underlying communication network, to efficiently utilii it. Fault Tolerance: The VP layout must keep the above properties even in f m of hardware failures in switches and in communication lines I . Restricted load: As mentioned above, the VP routing tables are limited in size, and the layout must utilize them in a way that not too many VPs go through a switch.

.

We now present an exact mathematical model and formal definitions of the above properties, based on graph theory (see [12] for the basic terminology). For the sake of simplicity we shall not present here the exact formal definitions, but more intuitive ones. Exact definitions as well as a detailed discussion, may be found in [IS]. In our model we have an underlying communication network, which consists of switches and I i i between them. This network is modeled by a simple undirected graph G = (V,E). Let PG be the set of all simple paths in G. The virtual path layout VPL is represented by a a graph Gp = (V,Evp)and a function I :Eyp + PG where the vertices of Gp are the same as of G, and each edge e E Evp represents a VP between two switches. Each VP is mapped to a simple path in G by Z . We term this path the induced path of the VP. Dehition 2.1 The load L(e) on a link e E E is equal to the number of VPs z E EVP that include e in their induced paths (namely e E Z(z)). The load L(VPL)ofthe layout i s the rnazimurn load on any of the links.

This property is out of the scope of this paper, and we present it here for the completeness of the list of desired properties of a VP layout.

In certain situations the application results in considering the load on the vertices in this graph-theoretic model. In such situations we use the following definition. Dewtion 2.2 The load t (v) on a vertex v E V is equal to the number of VPs x E Evp that include v in their induced paths. The load L(VPL) of the layout is the maximum load on any of the vertices. Except for the NP-complete result mentioned at the end of this section, and the algorithm described in Section 4.1, the load measure discussed in this paper refers to edges rather than vertices. Underlying this defmition is the assumption that a VP routing table exists in every port adaptor, following the switch architecture of [ll].Also we assume that each VP is a bidirectional route, comprised of two unidirectional routes in opposite directions, (this assumption is a&epted in the literature (e.g. 1111) as it substantially simplifies connection management). The rest of the discussion applies, with small changes, also when there exists a singe table in the whole switch, or when the tables in the port adaptors of a switch are identical - see 1151 for a full discussion.

Definition 2.3 Lef tC, 2 1 be a real number called the stretch factor. Also define the hop count H(P) for a simple path P in Gp , as the number of VPs that form the path. The hop count Hg(v, w ) is the minimum hop count H(P) for all paths P connecting u and w in Gp , whose induced path Z(P) is shorter than J, x dG(u,w ) (dG(v, w) being the shortest distance between v and w in G). w e define x g (G)= m ~ , W p v ( oztj ) (v, w). Ezcrmple: Consider the ATM network with VPs in Figure 1, in which there are three VCs, one between user z l and user 22 (we denote it by VC(21,22)), one between y l and y2, and one between 21 and 22. All VCs are composed of a concatenation of VPs (e.g. VC(yl,y2) is composed of VP(a, b) and VP(b, g)). Since each VC is composed of complete VPs, there is no way to create a VC between a and c, despite the fact that the physical network itself is connected (hence %+(a, c) = 00). Note that if # < $ then Xtb(a, e) = oo, since the shortest path do(a, e) = 3 and the path induced by the VPs (a, b), (b, d), (d, e) E Evpis of iength 5 (since P = ((a, b), (b, d),(d, e)) andZ@) = Z((a, b)),Z((b, d)),Z((d, 4)= (a, c, b, c, d, 4 ) ; If, however, $ > then H+(a, e) = 3. The load of the edge (c, d ) is L((c, d)) = 2, since two VPs path through it. As to the load on vertices, L(c) = I((a, b) ,(a, d) , (e,f)3l = 3 and t ( d ) = I{(b,d),(e,d),(e,f)ll= 3-

9

Note that 31g(v, w ) is the minimum number of VPs that may be used to form a VC between v and w, such that the length of that VC will not be too large with respect to the minimal route between v and w (the meaning of "too largen depends on the stretch factor). Finally we define the desired VP layout by:

-

physical link

._--_.__... virtaalcbarrnel 88x2

a n d m

o

dw

e

Fig. 1. An example for VP/VC layout Definition 2.4 A VPL is feasible with respect to and X$ (G)) h.

(G,h, $) if Gyl is connected

Remark 2.1 A feasible VPL will be denoted by VPL($, h, G), or when $ = 1 and IVI = N (and G belongs to a fixed family of graphs): VPL(h, N). Definition 2.5 A VPL is optimal i f it is feasible and its load L(VPL) is minimal

amongst all other feasible VPLs. So far we have discussed the case where it is required to connect every switch to every other switch - we term this access pattern "m-rn access". We have found useful to limit o d v e s first to a more restricted access pattern, namely when all switches may connect only a single chosen switch (called the root);we term this access pattern 1-m accessn. The VPL for m-m access will be denoted by VPL,while a VPL for 1access will be denoted by VPL1- . Definition 2.3 and 2.4 must be redefined in a straightforward way for VPL1- .

Remark 2.2 Finding a VPL1the following fads:

is easier than finding a VPL-

as hinted bp

I . Every feasible VPLis also a feasible VPLI, but the reverse is not true. 2. The load L(VPLM) of an optimal WLis never less than the load of an optimal VPL1* ,

Besides its methodical value as an easier problem to be tackled first, VPL'also has its own practical importance, as it may prove useful for server networks,

where data flows from a center to different destinations and vice versa. An example for this is a video conterencing server which has connections to all users who are currently engaged in a video conference [20];Another example is an interactive TV station - which is engaged in many separate sessions with different users Assuming an unbounded stretch factor (i.e. t,b = oo), and load on vertices, the following decision problem was shown in [17] to be NP-complete: INSTANCE: An undirected graph G = (V, E), a vertex r E V, integers L,h > 0. QUESTION: Doe8 there exist a feasible VPL1- such that I: ( VPL1+" (G, oo,h, r)) 5 L ? Unless otherwise stated, we will assume a stretch factor tC, = 1. This stretch factor does not allow any inefficiency in the usage of the underlying communication network namely for every pair of switches there must exist a path in Gp with less than h VPs, which is also a shortest path in G.

-

*.

-

3 Simple Networks: Chains, Rings and Meshes In this section we present constructions for virtual path layout for the simple networks with a topolgy of chain, ring and mesh. The discussion involves load of edges; the modifications to load of vertices, based on similar arguments, are omitted. 3.1

-

Chains and Rings VPLLm

Let G be a ring of N switches, with a root r. Due to the stretch factor (=I), the ring may be seen as two separate chains of switches, each of size since each switch may access the root through the shortest route only Therefore it s&ces to concentrate on a chain of N switches with the root at one of the ends of the chain. Let h = 2, construct VPL1- in the following way: First divide the chain into f l equd sections of size C d the switch that is closest to r in section i the "pivot of section i". Connect r to all pivots by VPs; Connect each pivot i to all the switches in section i; See Figure 2 for a graphic description. It is clear that r can be reached from every other switch by using two VPs (hence h = 2): one VP to the pivot of the section, and one VP from the pivot to r. It is also clear that this route is shortest in terms of the underlying chain (hence t,b = 1).

'.

9,

n.

This is not to be confused with a multicast service, where all destinations receive the same data &om a given source, while here we discuss separate streams of data from a service center. if N is odd then the switch which is furthest of r has two alternatives, but we ignore this detail since it does not change the discussion substantially

< 2m,

The load of the layout satisfies L(VPL) since at most flVP pass through an edge to the pivot of the current section, and at most fiVPs, from the pivots to r. As demonstrated by Figure 2, the results can be extended to VPL1- with any h, with load L(VPL) hJV1Ih (See [15] for a formal description).


1, and A the mutimum degree of a switch (and let = 1). For every VPLwith h hops, N* there exisis an edge e E E(G)with load L(e) =

+

A-h

Proof: We extend the optimality proof by defining the set of edges E' which are adiacent to the pivot. Clearly every pair of switches that belong to different subtrees (w.r.t. the pivot), must use a V P that uses an edge from E'. It is easy to see that the switches can be divided into two distinct sets Sl,S2 such that ISl I N and IS2I 5 4N.Us& Lemma 3.1 and considerations similar to those

>

of Theorem 3.1, we get L-t 2 ,*N*

32 A h

4.3

= Rc7).N* doh

General Networks

We cannot use the previous scheme for general networks, since there does not necessarily exist a set of switches, through which all the rest are connected. The new construction scheme is based on a technique of [I], for construction of "regional routing schemes'' (used for regular routing problems). This scheme is based on a clustering algorithm that divides the vertices of a graph into overlapping clusters, with three important properties:

- For each vertex v there is a cluster that includes v, and a l l other vertices that are not further than a given distance d from v (d is a parameter of the algorithm). - The distance between every pair of vertices in each cluster is not more than a fixed factor xd. - Each vertex is included in not too many clusters. In each cluster we choose a center as a pivot, and connect all the vertices in it to the pivot by W L ~ We . repeat this scheme for increasing parameter d (until a the algorithm yields one cluster which includes the whole network). Clearly, each pair of vertices belong to at least one common cluster, and are connected t o its pivot. For a minimal $ each pair uses the smallest common cluster. As shown in [15], this scheme yields a VPLwith load L(VPL) = O( h k log N N*+*) and stretch factor $ < 8t.

5

Summary and Extensions

In this paper we considered a new problem of designing the layout of virtual paths in a given network. We defined the characteristics that are important for such a layout, and an exact model for it. Basically, it is desired to design a path layout that satisfies constraints that limit the number of hops to get from point t o point in the network, but, in the same time, also limit the number of paths that go through any edge or vertex (most results deal with load on edges). We presented a construction scheme for a fairly simple network first, and extended it to a more general network. The solutions were for the case of stretch factor 1. Several design strategies (recursive and g~eedy)were suggested. We also discussed lower bounds for such schemes, and showed that our constructions are quite close to that bound, and presented a scheme, applicable to every graph. There is much work t o be done in finding better schemes for general networks, and in extending the existing schemes for supporting fault tolerance in the network. Of crucial importance is the study of schemes with stretch factor greater than 1. The characteristics of a good layout may be extended to the case when the expected volume of connections is known between every pair of switches.

References 1. B. Awerbuch and D. Peleg, Routing with polynomial communication-space tradeoff, SIAM Journal on Discnzte Math. 5, 2,1992, pp. 151-162. 2. S. Ahn, R.P. Tsang, S.R. Tong and D.H.C. Du, Virtual path layout design on ATM

networks, INFOCOM'94, pp. 192-200. 3. S. Bitan and S. Zaks, Optimal linear broadcast, Journal of Algorithms, 14, 1993,

pp. 288-315. 4. S. Bitan and S. Zaks, Optimal linear broadcast routing with capacity limitations,

International Symposium on Algorithms and Computation (ISAAC), Hong Kong, December 1993, pp. 287-296. 5. J.Y. Le Boudec, The asynchronous transfer mode: a tutorial, Computer Networks and ISDN Systems, 24, 1992, pp. 279-309. 6. C. T. Chou and I. S. Gopal, Linear broadcast routing, Journal of Algorithms, 10,4, 1989,pp. 490-517. 7. I. Cidon, 0. Gerstel and S. Zaks, A scalable approach to routing in ATM networks, 8th International Workshop on Distributed Algorithms (WDAG), Netherlands, October 1994, pp. 209-222. 8. I. Cidon and I. S. Gopal, PARIS: An approach to private integrated networks, International Journal of Digital and Analog Cabled Systems, 1, 2, April-June, 1988, pp. 77-86. 9. I. Cidon, I. S. Gopd and S. Kutten, New models and algorithms for future networks, P d i n g s of the 7% Annual ACM Symposium on principle^ of Distributed Computing (PODC), Toronto, Canada, August 1988, pp. 75-89. 10. R.. Cohen, B. Patel, F. Schaffaand M. Willebeek-LeMair, The sink tree paradigm: connectionless trafficsupport on ATM LANs, IBM Research Report, IBM Research Division, Watson Research Center, March 1993.

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