Delay optimal coteries on regular symmetric networks

Computer Standards & Interfaces 23 Ž2001. 341–353 www.elsevier.comrlocatercsi

Delay optimal coteries on regular symmetric networks P.C. Saxena a , Sangita Gupta b, Jagmohan Rai c,) a b c

School of Computer and System Sciences, Jawaharlal Nehru UniÕersity, New Delhi, India School of Computer and System Sciences, Jawaharlal Nehru UniÕersity, New Delhi, India Department of Mathematics, P.G.D.A.V. College (E), UniÕersity of Delhi, New Delhi, India Received 26 October 2000; returned for revision 19 March 2001; accepted 8 May 2001

Abstract Coteries, introduced by Garcia-Molina and Barbara wJournal for the Association for Computing Machinery, 32 Ž4. Ž1985. 841x, are an important and effective tool for enforcing mutual exclusion in distributed systems. Communication delay is an important performance measure for a coterie. Fu et al. wIEEE Transactions on Parallel and Distributed Systems, 8 Ž1. Ž1997. 59x emphasize that while calculating communication delay, the actual distances between different sites in a network must be taken into account and using this idea, obtain delay optimal coteries for trees, rings and hypercubes. Also, topology of an interconnection network plays an important role in the performance of a distributed system. For certain applications, it is desirable that the degree of each node in the interconnection network is the same. Constant Degree Four Cayley Graphs introduced by Vadapalli and Srimani wIEEE Transactions on Parallel and Distributed Systems 7Ž2. Ž1996. 26x provide an ideal topology for such applications. They are regular, have a logarithmic diameter and a node connectivity of four. In this paper, we prove that no coterie on an arbitrary network can have a delay of less than half the diameter of the network and use this result to obtain delay optimal coteries on regular symmetric networks with special reference to constant degree four Cayley interconnection-network. q 2001 Elsevier Science B.V. All rights reserved. Keywords: Mutual exclusion; Coterie; Message complexity; Delay optimal coterie

1. Introduction A distributed system is a system of autonomous computers interconnected via a communication network. The topology of the underlying interconnection network plays an important role in the overall performance of a distributed system. The decision of choosing a topology depends on the application in

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Corresponding author. C-3r319C, SFS Flats, Pankha Road, Janak Puri, New Delhi, India. Tel.: q91-11-5532457. E-mail addresses: prem – [email protected] ŽP.C. Saxena., [email protected] ŽS. Gupta., [email protected] ŽJ. Rai..

hand and the network size. Some of the standard network topologies are rings, trees, hypercubes, cube connected cycles, etc. A good network topology should have low diameter as compared to the network size. It should be well connected so that it may not get disconnected easily. If it is regular Žthe same number of edges are incident on every node. and symmetric Žthe graph looks similar from whatsoever node it is viewed., it will always be an added advantage. The constant degree four Cayley graph, introduced by Vadapalli and Srimani in Ref. w23x, possesses all the qualities of a good network topology. For a given positive integer n, n G 3, the graph size

0920-5489r01r$ - see front matter q 2001 Elsevier Science B.V. All rights reserved. PII: S 0 9 2 0 - 5 4 8 9 Ž 0 1 . 0 0 0 8 0 - 0

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is n P 2 n and the diameter is as low as w3nr2x. For n s 5, for example, the diameter is 7 as compared to the graph size of 360 nodes. It is well connected, regular and symmetric. Thus, it is an ideal graph to be used as a communication network in distributed systems. The graph is discussed in detail in Section 4. A number of applications in distributed systems require that, at a time, only one process be permitted to enter the critical section, i.e., mutual exclusion be ensured. The problem of mutual exclusion in distributed systems has been tackled by using tokenbased schemes w12,16,18,19,21x, voting-based schemes w4,10,22x or through coteries w1,2,5,14,15x. Coteries were introduced by Garcia and Barbara in Ref. w9x and are a very important tool for enforcing mutual exclusion in distributed systems. A coterie on a set of nodes S is a collection of mutually intersecting, minimal, non-empty subsets of S. The performance of a coterie can be measured in terms of its message complexity, fault tolerance, availability or communication delay. It may not be possible, in general, to choose a coterie, which is optimal with respect to every performance measure. A coterie, which is optimal with respect to one performance measure, may not be optimal with respect to another. Moreover, it may not be possible to choose a coterie, which is suitable for every type of network. So, attempts have been made to obtain optimal coteries on various specific interconnection networks w6,13x. The selection of a coterie may depend on a number of factors like the application in hand, the performance measure to be optimized and the network topology. Communication delay is an important performance measure for a coterie. Fu et al. in Refs. w7,8x emphasize that while calculating communication delay, the actual distances between different sites in a network must be taken into account and using this idea, obtain delay optimal coteries for trees, rings and hypercubes. In the present article, we prove that no coterie on a connected graph can have a delay of less than half its diameter and use this result to obtain a delay optimal coterie on the regular, symmetric constant degree four Cayley graph, Gn , for every positive integer n greater than or equal to 3. The organization of the paper is as follows: Section 2 contains some basic definitions, which we need for

our discussion. In Section 3, we prove that no coterie on a connected graph G can have a delay of less than half the diameter of G and that for any nodes x, y belonging to G, the sets containing all nodes of G at a distance less than or equal to k from x and y, where k is the smallest integer greater than or equal to half the diameter of G, must intersect. In Section 4, we describe and explore Gn , the constant degree four Cayley graph. In Section 5, we define a coterie on Gn and show that it is delay optimal. We also give the message complexity of the coterie on Gn for values of n up to 8 and conclude that for values of n larger than 5, our coterie is an ideal choice. Though, we have restricted ourselves only to the constant degree four Cayley graph, the technique can be applied to any regular symmetric network and it is observed that the message complexity of a coterie defined using this technique depends on the underlying graph. It is further observed that, it may be possible to choose coteries on Gn with the same delay as that of our coterie but with a smaller message complexity. Defining a coterie whose quorums are subsets of the quorums of our coterie can do this. But its construction Žand existence. depends on n and differs from case to case. We give one such coterie on G 3 , as an example to support our argument. The paper has been concluded in Section 6.

2. Basic concepts A distributed system of N nodes consists of N geographically dispersed autonomous sites connected via a communication network. The sites communicate only through message passing and do not share a common memory. In graph theoretic terms, a distributed system having N nodes can be regarded as a graph G s Ž V, E . consisting of a finite nonempty set V s  Õ 1 , Õ 2 , . . . , ÕN 4 of N vertices and a set E of edges such that each edge e i j g E is identified with an unordered pair Ž Õi , Õj . of vertices. The vertices of the graph correspond to the nodes of the network and its edges correspond to the communication links. This analogy between graphs and networks is very useful. As such, we shall not make any distinction between the two concepts and shall use the terms graph and network interchangeably.

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Also, throughout our discussion we shall use the terms node, process, site, point, etc. interchangeably. A subgraph of a graph G is a graph having all its points and lines in G. A walk in G is an alternating sequence of points and edges Õ 0 , e 01 , Õ 1 , e12 , . . . , e ny 1, n , Õn , beginning and ending with points, in which each line is incident with the point immediately preceding it and the point immediately following it. A walk is closed if Õ 0 s Õn and open otherwise. A closed walk with at least three points is called a cycle in G if all its points are distinct. A walk in G is a path if all the points and all the lines in it are distinct. A path Õ 0 , e 01 , Õ 1 , e12 , . . . ,e ny1, n , Õn is generally written as the path Õ 0 , Õ 1 , . . . , Õn . The length of a path Õ 0 , Õ 1 , . . . , Õn is n, the number of occurrences of edges in the path. The distance distŽ u,Õ . between two points u and Õ in G is the length of a shortest path joining them. The diameter of a graph G is obtained by finding the lengths of the shortest path for all pairs of points in G and then choosing the longest one among them, i.e., diameter Ž G . s max  dist Ž u,Õ . N u,Õ g G 4 . A graph G is connected if there exists a path between every pair of points in G. A maximal connected subgraph of G is called a component of G. A disconnected graph is a graph with at least two components. Thus, a graph is disconnected if and only if its vertices can be partitioned into two disjoint subsets such that there is no edge in the graph, connecting a vertex in one of these sets to a vertex in the other. When, due to node or link failures, a network partitioning occurs, each partition of the network is a component of the graph representing the network. A tree is a connected graph having no cycles. A tree in which one vertex is distinguished from the others is called a rooted tree and this special vertex is called the root of the tree. A tree is said to be the spanning tree of a connected graph G, if it is a subgraph of G and contains all the vertices of G. Degree of a point Õi in G is the number of edges in G which are incident on Õi . A graph is regular if every node in the graph has the same degree. It is symmetric if the graph looks the same viewed from any of its vertices. Node connectivity of a graph G is the minimum number of nodes that must be removed in order to make G disconnected and its edge connectivity is the minimum number of

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edges that must be removed in order to make G disconnected. We have included some definitions from graph theory, which we shall need to carry on our work. An elegant description of various concepts involved in graph theory can be found in Ref. w11x. Mutual exclusion in permission-based distributed mutual exclusion algorithms can be achieved either by using voting schemes or through coteries. Whereas, the concept of voting was introduced by Thomas w22x, the concept of a coterie was introduced in Ref. w9x as a mathematical abstraction to model mutual exclusion in distributed systems. For any non-empty set U, a coterie C is a collection of non-empty subsets of U satisfying the conditions: S,T g C ´ S o T , T o S S l T / B;S, T g C

Ž Minimality Condition.

Ž Intersection Property. .

Elements of a coterie are called quorums Žor quorum sets or quorum groups.. If a node wants to perform a restricted operation, it must seek permission from each and every node of some quorum group in the assigned coterie. Since a node will give permission to only one node at a time and since any two quorum-groups in a coterie have at least one node in common, mutual exclusion is guaranteed. The message complexity of a distributed mutual exclusion algorithm is the number of messages exchanged by a process per Critical Section entry. The message complexity of a coterie-based distributed mutual exclusion is proportional to the quorum-size of the coterie. Communication delay is another important metric on the basis of which we can test the quality of a coterie. Fu et al. w7x observed that, generally, quorum size has been used to estimate communication cost of a coterie and not much attention has been paid to minimizing communication delay, which can be an important factor for the system response time. They define coteries with optimal cost in terms of communication delay, which take account of the network topology and obtain delay optimal coteries for trees, rings and hypercubes. We use the following definition of delay optimality of a coterie as given in Ref. w7x to obtain a delay optimal coterie for constant degree four Cayley graph. Let G s Ž V, E . be a network and let C be a coterie on V, the node set of G Žwe shall invariably

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call it a coterie on G .. Let distŽ a,b . be the shortest distance between two nodes a and b in G. For any s g G and for any Q g C, we find the distance of s from every point of Q and denote the maximum of these distances by a Ž s,Q .. That is, a Ž s,Q . s max distŽ s,Õ . N Õ g Q4 . We obtain the value of a Ž s,Q . for every quorum Q in C, take minimum of these values and denote it by a Ž s ., i.e., a Ž s . s min a Ž s,Q . N Q g C 4 . The quorum for which a Ž s,Q . is minimum is called an optimal quorum for the node s. Note that whereas a Ž s,Q . tells us how far is the node s from the farthest node in Q, a Ž s . gives us the delay of the coterie C from node s which we denote by delayŽ s, C ., i.e.,

Condition M4 is known as the triangle inequality. If d is a metric on a non-empty set X, then the couple Ž X,d . is called a metric space and we say that X is a metric space under the metric d. Let Ž X,d . be a metric space. Let x 0 g X and let r be any non-negative number. An open sphere centered at x 0 having radius r is the set of all points in X which are at a distance less than r from the point x 0 and is denoted by SŽ x 0 ,r ., while a closed sphere centered at x 0 having radius r is the set of all points in X which are at a distance less than or equal to r from the point x 0 and is denoted by Sw x 0 ,r x, i.e.,

delay Ž s,C . s a Ž s .

S w x 0 ,r x s  x g X N d Ž x , x 0 . F r 4 .

s min  max  dist Ž s,Õ . ,Õ g Q 4 N Q g C 4 . Delay of the coterie C, denoted by delayŽ C ., is defined as the maximum of all the delays taken over all the nodes in V, i.e., delay Ž C . s max  delay Ž s,C . N s g V 4 . A coterie on G with a minimal value of delay taken over all coteries on G is called a delay optimal coterie on G. Thus, if C Ž G . denotes the set of all coteries on G, a coterie C ) is said to be a delay optimal coterie on G if Delay Ž C ) . s min  delay Ž C . N C g C Ž G . 4 . We shall need an important mathematical entity called a metric space to prove our assertions throughout this article. So, in what follows, we define a metric space and some other related concepts, which we shall use frequently. For any two sets A and B, the Cartesian product of A and B is defined as A = B s  Ž x , y . N x g A, y g B 4 . Let X be a non-empty set. A function d: X = X ™ R, where R denotes the set of real numbers, is said to be a metric on X if and only if the following conditions are satisfied: M1. M2. M3. M4.

dŽ x, y . G 0 ; x, y g X dŽ x, y . s 0 m x s y dŽ x, y . s dŽ y, x . ; x, y g X dŽ x, y . F dŽ x, z . q dŽ z, y . ; x, y, z g X.

S Ž x 0 ,r . s  x g X N d Ž x , x 0 . - r 4 and

The diameter of a subset A of a metric space Ž X,d . is defined as the supremum or the least upper bound of the set  dŽ x, y . N x, y g A4 . If this set is finite, its supremum is, obviously, equal to its maximum. We have restricted ourselves only to those concepts of metric spaces which are most essential for our work. A thorough discussion on metric spaces can be found in Ref. w20x. For any x, y in a connected graph G, if we set dŽ x, y . s distŽ x, y ., the shortest distance between the points x, y in G, d satisfies all the properties of a metric and thus, G forms a metric space under this metric. In the light of this fact, throughout our discussion, whenever we come across any connected graph, we will assume this metric structure on it and will use the symbol d for the shortest distance metric. We shall denote by Bk Ž x ., the closed sphere on the metric space Ž G,d . and by Ck Ž x ., the set  y g G N dŽ x, y . s k 4 . Thus, whereas Bk Ž x . denotes the set of all those points in G which are at a distance k or less, Ck Ž x . is the set of those points of G whose distance from x is exactly equal to k. Clearly, Ck Ž x . is a subset of Bk Ž x ., for every x g G.

3. Delay of a coterie on a connected graph In this section, two general results for connected graphs have been proved. The first relates to the minimum delay, which can be expected from a coterie on a connected graph. It states that no coterie

P.C. Saxena et al.r Computer Standards & Interfaces 23 (2001) 341–353

on a connected graph can have a delay of less than half the diameter of the graph. The second suggests how to select quorums for defining a coterie on a connected graph. It states that if k is the smallest positive integer greater than or equal to half the diameter of a connected graph, then for any two points in the graph, the sets consisting of all the vertices of the graph which are at a distance less than or equal to k from these points must intersect. The mathematical proofs of these results are given in Theorems 1 and 2. Theorem 1. Let G be a connected graph with diameter d (G). No coterie on G can haÕe a delay of less than 1 r 2 d (G). Proof. Let G be a graph with diameter dŽ G .. If possible, let C be coterie on G having delay a , a - 1r2 dŽ G ..

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(iii) Ba (x 0 ) l Ba (y 0 ) s f . If possible, let z g Ba (x 0 ) l Ba (y 0 ) z g Ba Ž x 0 . ´ d Ž x 0 , z . F a z g Ba Ž y 0 . ´ d Ž y 0 , z . F a [ d Ž x 0 , y0 . F d Ž x 0 , z . q d Ž y0 , z .

Ž d is a metric.

-aqa - 1r2 d Ž G . q 1r2 d Ž G . s d Ž G . implying thereby that the distance between x 0 and y0 is less than dŽG., a contradiction. Živ. Q x l Q x s f. This follows Ži., Žii. and Žiii.. 0 0 But this is not possible in C since it is a coterie. Thus, our assumption that a coterie C on G has a delay of less than 1r2 dŽG. leads to a contradiction. Thus, no coterie on G can have a delay of less than 1r2 dŽG.. I Theorem 2. Let G be a connected graph with diameter dŽG.. Let rŽG. s 1r2 dŽG. if dŽG. is even and 1r2 Ž dŽG. q 1. if dŽG. is odd. Then for any pair of vertices x,y g G, the sets B k Žx. and B k Žy. must intersect where k s rŽG..

Delay Ž C . s a ´ max  delay Ž x ,C . N x g C 4 s a ´ delay Ž x ,C . F a ; x g G ´ ; x g G, min  dist Ž x ,Q . ,Q g C 4 F a

˙

´ ; x g G, 'Q x g C : dist Ž x ,Q x . F a .

˙

Ž 1.

But, the existence of such a Q x for every x g G leads to a contradiction. We establish this in Ži. to Živ. below. Since the diameter of G is dŽ G ., there exist points x 0 , y 0 in G such that dŽ x 0 , y 0 . s dŽ G ., where d is the shortest distance metric on G. Let Q x 0 , Q y 0 be the quorums in C satisfying condition (1). Then (i) Q x : Ba (x 0 ). By definition of Q x , it follows that 0 0 dist Ž x 0 ,Q x 0 . F a ´ max  dist Ž x 0 ,Õ . :Õ g Q x 0 4 F a ´ dist Ž x 0 ,Õ . F a ; Õ g Q x 0 ´ Õ g Ba Ž x 0 . ; Õ g Q x 0 ´ Q x 0 : Ba Ž x 0 . . (ii) Q y : Ba (y 0 ). The proof is the same as aboÕe. 0

Proof. Let G be a connected graph with diameter dŽG. and let rŽG. be the smallest integer greater than or equal to dŽG.. Let k s rŽG.. Now, for any x g G, B k Žx. consists of the vertex x itself and all the vertices of G which are at a distance up to k from x. Let Gk Žx. denote the subgraph of G consisting of all the vertices of G contained in B k Žx. and the edges in G interconnecting them. Let x,y g G. We have to show that B k Žx. l B k Žy. / B. Consider Gk Žx. and Gk Žy., the subgraphs of G corresponding to the nodes x and y generated by B k Žx. and B k Žy., respectively. If possible, let B k Žx. l B k Žy. s B. We show that if this were the case, there cannot exist an edge in G, connecting a node of Gk Žx. to a node of Gk Žy. and vice versa. To see this, we observe the following. Ži. x cannot be connected to a node s in Gk Žy. via an edge because that would imply dŽx,s. s 1 and so, s g B k Žx.. This implies s g B k Žx. l B k Žy., a contradiction. Similarly, a node at distance less than or equal to k y 1 from x in Gk Žx. cannot be connected

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to a node s in Gk Žy. via an edge because this would imply dŽs,x. F k and so, s g Gk Žx.. This implies s g B k Žx. l B k Žy., a contradiction. That is, no node in Gk Žx. upto a distance k y 1 from x can be connected to any node in Gk Žy. via an edge. Žii. Similarly, no node in Gk Žy. up to a distance k y 1 from y can be connected to any node in Gk Žx. via an edge. Žiii. If v is a node in Gk Žx. such that dŽv,x. s k, then, in the light of what has been shown in Ži. and Žii., v cannot be connected to any node in Gk Žy. at a distance k y 1 or less from y. We show that v cannot be connected, even to a distance k node in Gk Žy.. Suppose that u g Gk Žy., dŽu,y. s k, be connected to v g Gk Žx. via an edge. Then dŽu,v. s 1. But, then dŽx,y. s dŽx,u. q dŽu,v. q dŽv,x. s k q 1 q k ) dŽG., a contradiction. Thus, from Ži., Žii. and Žiii., we find that no node in Gk Žx. can be connected to any node in Gk Žy. and vice versa. But this implies that G is disconnected, a contradiction. Thus, our assumption that B k Žx. l B k Žy. s B does not hold good. Hence, B k Žx. l I B k Žy. / B. We use the results obtained above to obtain a delay optimal coterie on the constant degree four Cayley graph Gn . In Section 4 below, we define Gn as introduced by Vadapalli and Srimani in Ref. w23x.

4. The constant degree four Cayley graph Cayley Graphs w3x are based on permutation groups and include a number of graphs like star graphs, hypercubes, etc. These graphs are symmetric, regular and possess many of the desirable properties like low diameter, low degree, high fault tolerance, etc. Though these graphs are regular, the degree of nodes increases with the size of the graph. For certain applications, it is desirable that the degree of each node in the interconnection network is the same. Degree Four Cayley Graphs introduced in Ref. w23x provide an ideal topology for such applications. They are regular, have a logarithmic diameter and a node connectivity of four. These graphs are better, in terms of regularity and greater fault tolerance, than other constant degree graphs like the De Bruijn

graphs and the Moebius graphs having low vertex connectivity of two and provide a better alternative to the cube connected cycle, a Cayley graph of degree three, as they accommodate a higher number of nodes for the same degree w23x. For any integer n, n G 3, the constant degree four Cayley graph, Gn , is a graph with n P 2 n vertices and n P 2 nq 1 edges. It is a symmetric, regular graph of degree 4 and its diameter is w3nr2x, the greatest integer less than or equal to 3nr2. For any n, n G 3, the vertices of Gn are labeled as follows. Take any n symbols a1 , a 2 , . . . , a n and consider the n circular permutations a1 a 2 . . . a n a 2 a 3 . . . a n a1 a 3 . . . a n a1 a2 ... ... ... ... a n a1 a 2 . . . a ny1 of these symbols. For 1 F i F n, let A i denote the complement of the symbol a i . Each of the n places, in each of the n cycles given above, can be replaced by the corresponding complemented symbol. Thus, each of these cycles gives rise to 2 n cycles in all, including itself. Therefore, exactly n P 2 n distinct labels can be generated using this scheme, one for each vertex in Gn . For example, the 64 labels for vertices of G4 generated using this scheme are: abcd, bcda, bcdA, dabc, Dabc, cdab, cdaB, Abcd, cdAb, cdAB, abcD, Cdab, cDab, Cdab, dabC, Bcda, daBc, daBC, dAbc, DAbc, dAbC, BcdA, dABc, dABC, bcDa, bcDA, bCda, BCda, DabC, BcDa, bCDa, BCDa, abCd, abCD, aBcd, ABcd, aBcD, CdaB, aBCd, aBCD, AbcD, CdAb, cDAb, CDAb, AbCd, AbCD, ABcD, CdAB, ABCd, ABCD, cDaB, cDAB, CDaB, bCdA, bCDA, DaBc, DABc, BcDA, BCdA, DaBC, BCDA, DAbC, DABC, CDAB. Edges in Gn are obtained using four generators g, f, gy1 and f y1 defined as g Ž a1 a2 . . . a n . s a 2 . . . a n a1 f Ž a1 a 2 . . . a n . s a2 . . . a n A1 gy1 Ž a1 a 2 . . . a n . s a n a1 . . . a ny1 f y1 Ž a1 a 2 . . . a n . s A n a1 . . . a ny1 .

P.C. Saxena et al.r Computer Standards & Interfaces 23 (2001) 341–353

A vertex labeled a1 a 2 . . . a n is connected to the four vertices a2 . . . a n a1 , a 2 . . . a n A1 , a n a1 . . . a ny1 and A n a1 . . . a ny1 obtained, respectively, by applying functions g, f, gy1 and f y1 to a1 a 2 . . . a n . For example, vertex u s AbcdEg G5 is connected to the four vertices given by g Ž u . s g Ž AbcdE. s bcdEA f Ž u . s f Ž AbcdE. s bcdEa gy1 Ž u . s gy1 Ž AbcdE. s EAbcd f y1 Ž u . s f y1 Ž AbcdE. s eAbcd. The degree of each vertex in Gn is 4 and the degree of each edge is Ž4,4.. All the elements of Gn can be obtained by repeatedly applying the generators g, f, gy1 , f y1 to any one vertex, u g Gn . The procedure given below not only generates all the elements of Gn , for any integer n G 3, but also specifies the distance of newly generated elements from the source node. When g, f, gy1 and f y1 are applied to an element u g Gn , four distinct elements of Gn are generated. That is, g Ž u., f Ž u., gy1 Ž u. and f y1 Ž u. are all distinct elements of Gn for any u g Gn . But when we apply g, f, gy1 and f y1 to each of g Ž u., f Ž u., gy1 Ž u. and f y1 Ž u., we do not get 16 new vertices of Gn . This is because, although the four elements obtained by applying g, f, gy1 and f y1 to any one of g Ž u., f Ž u., gy1 Ž u. and f y1 Ž u. will be different, they may not be all different from those generated by applying g, f, gy1 and f y1 on the remaining three and from u, g Ž u., f Ž u., gy1 Ž u. and f y1 Ž u.. For example, if we apply g, f, gy1 and f y1 to g Ž u., we obtain g 2 Ž u., fg Ž u., gy1 g Ž u. and f y1 g Ž u.. Of these, at least gy1 g Ž u. s u has already been encountered. There may be other repetitions also. For n s 3, for example, g 2 Žabc. s gg Žabc. s g Žbca. s abc, i.e., g 2 Ž u. s g Ž u.. Let us call u, a level 0 element; g Ž u., f Ž u., gy1 Ž u. and f y1 Ž u. as level 1 elements; the new elements generated by applying g, f, gy1 and f y1 to each of the elements generated at level 1 as level 2 elements and so on. Elements at level i are obtained by applying the generators g, f, gy1 and f y1 to level i y 1 elements and choosing only those which have not been

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encountered so far. This process terminates, for each n, at level w3nr2x, generating a minimal spanning tree T Ž u. of Gn . T Ž u. has u as its root vertex and its depth is equal to dŽ Gn . s w3nr2x. For each i, 0 F i F w3nr2x, the vertices at level i of this tree are the vertices of Gn at a distance i from the node u. The subtree Ti Ž u. of T Ž u. of depth i consists of all vertices Õ g Gn such that dŽ x,Õ . F i. Fig. 1a shows the spanning tree T Žabc. of G 3 generated using this procedure, with abc as the root vertex and Fig. 1b shows T2 Žabc., a subtree of T Žabc. of depth 2. The number of new elements generated at level i in Gn depends on n because orders of the permutations g, f, gy1 and f y1 depend on n. For example, at level 2, the number of new elements generated is 8 in G 3 , 9 in G4 and 10 in G5 . Table 1 shows the number of new elements generated at different levels for different values of n and the total number of elements in Ti Ž u. for different i. These values have been obtained by using the procedure described above.

Fig. 1. Ža. The tree T Žabc. of G 3 rooted at the vertex abc. Žb. T2 Žabc., a subtree of T Žabc. in G 3 .

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Table 1 Number of new elements generated at various levels for different values of n, 3 F n F 8 Level n )

0

1

2

3

4

5

6

7

8

9

10

11

12

Total nodes NsnP2n

3 4 5 6 7 8

1 1 1 1 1 1

4 4 4 4 4 4

8 9 10 10 10 10

9 18 22 23 24 24

2 21 43 47 51 52

– 10 54 92 106 110

– 1 24 118 206 220

– – 2 72 272 430

– – – 16 174 589

– – – 2 40 430

– – – – 2 151

– – – – – 26

– – – – – 1

24 64 160 384 896 2048

Having explored Gn , we now proceed to define a coterie on Gn , for every n, n G 3.

5. A delay optimal coterie on the constant degree four Cayley graph, Gn In this section, we define a coterie on the constant degree four Cayley graph, Gn , and using the results proved in Section 3, show that it is delay optimal. We also give the message complexity of the coterie on Gn for values of n up to 8. Finally, we present another coterie on G 3 , which has the same delay but smaller message complexity. To begin, note that the diameter of Gn is equal to w3nr2x, the greatest integer less than or equal to 3nr2 w23x. For any positive integer n, dŽ Gn . can be expressed as

°6 m 6mq1 d Ž G . s~ ¢66 mm qq 34 n

if n s 4 m if n s 4 m q 1 . if n s 4 m q 2 if n s 4 m q 3

Let r Ž Gn . be the smallest integer greater than or equal to dŽ Gn .. For the sake of notational convenience, we write k s r Ž Gn .. Therefore, k s r Ž Gn . s

½

1r2 d Ž Gn . if d Ž Gn . is even 1r2 Ž d Ž Gn . q 1 . if d Ž Gn . is odd

Then,

°3m 3m q 1 d Ž G . s~ q2 ¢3m 3m q 2 n

if n s 4 m if n s 4 m q 1 . if n s 4 m q 2 if n s 4 m q 3

.

For any x g Gn , consider Bk Ž x . s  y g G: Ž d x, y . F k 4 . Bk Ž x . is precisely the set of all vertices in the graph Gn at a distance less than or equal to k, i.e., it is the closed ball in the metric space Ž Gn ,d . centered at the point x g Gn . Let C be the collection of Bk Ž x ., x g Gn . We show that C is a delay optimal coterie on Gn . Theorem 3. C s {Bk (x) N x g Gn } is a coterie on Gn , k s r (G). Proof. Clearly, Bk Ž x . is non-empty for every x g Gn . Moreover, since Gn is connected, it follows from Theorem 2 that Bk Ž x . l Bk Ž y . / B; x, y g Gn . Also, since Gn is a regular symmetric graph, Bk Ž x . will contain the same number of nodes for every node x g G. That is, for any x, y in Gn , < Bk Ž x .< s < Bk Ž y .<. Since two sets containing an equal number of elements cannot be properly contained in each other, we conclude that no element of C can contain another element of C as its proper subset. Thus, C is a coterie. I For any x g Gn , Bk Ž x . is, in fact, the set of all vertices of Tk Ž x ., the tree of depth k in Gn with x as the root node. Fig. 2 shows two intersecting quorums T3 Žabc. and T3 ŽBCA. of the coterie C on G 3 . We show that for any x g Gn , Bk Ž x . is an optimal quorum and use this result to show that the coterie C defined above has delay k s r Ž G .. Lemma 1. The quorum Bk (x) g C is an optimal quorum for x g Gn .

P.C. Saxena et al.r Computer Standards & Interfaces 23 (2001) 341–353

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Lemma 3. No coterie on Gn can haÕe a delay of less than k s r (Gn ).

Fig. 2. Intersecting quorums T3 Žabc. and T3 ŽBCA. in G 3 .

Proof. Let x g Gn . Consider B k Ž x .. Now, distŽ x, Bk Ž x .. s max dŽ x,Õ . N Õ g Bk Ž x .4 . For any Õ g Bk Ž x ., dŽ Õ, x . F k and there exist points in Bk Ž x . which are at a distance k from x. Therefore, max  d Ž x ,Õ . N Õ g Bk Ž x . 4 s k ´ dist Ž x , Bk Ž x . . s k. Now, consider Bk Ž y . different from Bk Ž x . in C. So, there must exist Õ 0 g Bk Ž y . such that Õ 0 f Bk Ž x . because otherwise Bk Ž y . s Bk Ž x .. But Õ 0 f Bk Ž x . implies dŽ x,Õ 0 . ) k. This implies, dist Ž x , Bk Ž y . . s max  d Ž x ,Õ . N Õ g Bk Ž y . 4

Thus, we find that, whatever positive integer n, n G 3 we may choose, no coterie on Gn can have a delay of less than k. Since the coterie C has delay n, we deduce that Theorem 4. C is a delay optimal coterie on Gn . Proof. Follows from Theorem 3, Lemma 2 and Lemma 3. I

G d Ž x ,Õ 0 . ) k. Thus, we have shown that, for any x g Gn , distŽ x, Bk Ž x .. s k and distŽ x, Bk Ž y .. ) k whenever Bk Ž y . / Bk Ž x .. ´ Bk Ž x . is an optimal quorum for x g Gn in C and I distŽ x, Bk Ž x .. s k. Lemma 2. delay(C) s k. Proof. For any x g Gn , delay Ž x ,C . s min  dist Ž x , Bk Ž y . . N y g Gn 4 s dist Ž x , Bk Ž x . . s k Thus, delayŽ x, C . s k ; x g Gn ´ max  delay Ž x ,C . N x g Gn 4 s k ´ delay Ž C . s k.

Proof. For any positive integer n, n G 3, consider Gn . Case I: n s 4m. In this case dŽ Gn . s 6 m. Therefore, by Theorem 1, no coterie on Gn can have a delay of less than 1r2 dŽ Gn . s 3m s k. Case II: n s 4m q 1. In this case, dŽ Gn . s 6 m q 1. Therefore, no coterie on Gn can have a delay of less than 1r2 dŽ Gn . s 3m q 1r2. So, minimum possible delay for a coterie on Gn is 3m q 1, which is equal to k. Case III: n s 4m q 2. In this case, dŽ Gn . s 6 m q 3. Therefore, no coterie on Gn can have a delay of less than 1r2 dŽ Gn . s Ž3m q 1. q 1r2. So, minimum possible delay for a coterie on Gn is 3m q 2,which, again, is equal to k. Case IV: n s 4m q 3. In this case, dŽ Gn . s 6 m q 4. Therefore, no coterie on Gn can have a delay of less I than 1r2 dŽ Gn . s 3m q 2 s k.

I

5.1. Message complexity of C A quorum Bk Ž x . in the coterie C on Gn is obtained by collecting all the points in Gn which are at a distance less than or equal to k from x. As mentioned earlier, the number of new elements generated at each level is different for different values of n. Therefore, although for a fixed value of n, every quorum in C has the same size, the number of elements in a quorum of C is different for different Gn . That is, the message complexity of the coterie C is different for different values of n. So is the delay of C. Table 2 shows the delay and the message complexity of our coterie on Gn for values of n up to 8. The size of the graph and its diameter is also

P.C. Saxena et al.r Computer Standards & Interfaces 23 (2001) 341–353

350

Table 2 The delay and message complexity of the coterie C on Gn for 3F nF8 n

Ns nP2 n

dŽ Gn . s w3n r2x

Delay, k

MC s < Bk Ž x .<

ŽMCrN .= 100

3 4 5 6 7 8

24 64 160 384 896 2048

4 6 7 9 10 12

2 3 4 5 5 6

13 32 80 177 196 421

54.16 50 50 46.09 21.87 20.55

given therein. A look at Table 2 shows that as n increases, the message complexity of our coterie decreases considerably. Whereas, the message complexity of the coterie on G 3 is 13, which is equal to that of the majority coterie, it is 50% of the graph size for G4 and G5 and falls from 46.09% of the graph size for G6 to 21.87% of the graph size for G 7 and to 20.55% of the graph size for G 8 . Thus, we deduce that if, in a distributed system, the constant degree four Cayley graph is being used as the underlying communication network, our coterie on Gn is an ideal choice for values of n larger than 5, as it is delay optimal and has reasonably low message complexity. For example, on G 8 Žconsisting of 2048 nodes., our coterie has delay 6 and a message complexity of 421 whereas the majority coterie has delay 8. The technique we have used for obtaining the coterie on Gn can be used to obtain a delay optimal coterie on any regular symmetric graph. The delay of the coterie will be r Ž G ., with r Ž G . as defined earlier. The message complexity of the coterie, of course, depends on the graph. For example, for the n-dimensional folded Petersen graph w17x, having 10 n nodes, the coterie will have delay n and a message complexity of 4 n. Although the coterie C, defined on Gn , is delay optimal, it may not be optimal with respect to message complexity and it may be possible to find a coterie on Gn , which has a message complexity smaller than that of C. This is because, in order to define a quorum for x g Gn , we collected all the nodes at a distance less than or equal to k from x. It was not necessary at all. We know that for any graph G, the delay of a coterie on it cannot be less than

k s r Ž G .. Had we chosen all nodes at a distance F k y 1 from x and just one node from among those, which are at a distance k from x, we would have still got the delay to be equal to k. But, there is no guarantee that the sets thus formed will be mutually intersecting. If they intersect, we get a coterie with a delay equal to the coterie C defined for Gn and having a message complexity smaller than C. If they do not intersect, we may try Žas a quorum for x . the set consisting of all nodes at a distance F k y 1 from x and two nodes from among those which are at a distance k from x and so on. Thus, if any coterie on Gn is delay optimal, its message complexity will lie between < Bky 1Ž x .< q 1 and < Bk Ž x .<. We show by an example that it may actually

Fig. 3. Ža. f-cycles in G 3 . Žb. g-cycles in G 3 . Žc. G 3 .

P.C. Saxena et al.r Computer Standards & Interfaces 23 (2001) 341–353

be possible to define a delay optimal coterie without including all the nodes in Ck Ž x . in the quorum for a node x. Consider G 3 . It has 24 elements. Each f-cycle in G 3 is of length 6 and each of its g-cycle is of length 3. The node set of G 3 can be partitioned into four disjoint f-cycles. It can also be partitioned into eight disjoint g-cycles. Fig. 3a shows all the f-cycles of G 3 , Fig. 3b shows all its g-cycles and Fig. 3c the complete graph with f-cycles in bold lines. To define a coterie on G 3 , we observe the following. Ž1. Each f-cycle contains six nodes and each node of G 3 belongs to exactly one f-cycle. Ž2. An f-cycle containing x consists of exactly two nodes which are at a distance 1 from x, exactly two nodes at a distance 2 from x and exactly one node at a distance 3 from x. Let us denote by x 1 and x1 , the nodes at a distance 1 from x in the f-cycle containing x and by x 2 and x2 , the nodes at a distance 2 from x in the f-cycle containing x. Ž3. A g-cycle and an f-cycle in G 3 cannot have more than one point in common. The six nodes of a given f-cycle lie in six different g-cycles of G 3 ŽFig. 4.. We define a coterie C on G 3 as Cs  QŽ x .< x g G 34 where for any x g G, QŽ x . consists of g-cycles at x, x 1 and x1 and the nodes x 2 and x2 . It is easy to check that the 24 quorums thus generated are, in fact, intersecting. Fig. 5a shows how the quorum for

351

Fig. 5. Ža. Quorum QŽ x . for a typical node x in G 3 . Žb. Quorums QŽCAB. and QŽabc. in the coterie C defined on G 3 .

a typical node x in G 3 looks and Fig. 5b shows that the quorums for the nodes abc and ABC Ždistance 4 apart!. intersect.

6. Conclusion

Fig. 4. g-cycles of the nodes in an f-cycle of G 3 .

In this paper, it has been shown that no coterie on a connected graph G can have a delay of less than half the diameter of the graph and that if r Ž G . denotes the smallest positive integer greater than or equal to half the diameter of G, then for any pair of vertices in G, the sets consisting of all the vertices of the graph which are at a distance less than or equal to r Ž G . from these points must intersect. These results have been used to obtain delay optimal coteries on a class of regular, symmetric graphs—the constant degree four Cayley graphs. The constant degree four Cayley graphs have all the properties a good network topology should possess. The coteries defined on these graphs, besides being delay optimal, have low message complexity, particularly for large

352

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values of n. Thus, we deduce that if, in a distributed system, the constant degree four Cayley graph is being used as the underlying communication network, our coterie on Gn is an ideal choice for values of n larger than 5, as it is delay optimal and has reasonably low message complexity. Though we have restricted ourselves only to Gn , the technique Žused for obtaining the coterie C on Gn . can be used to obtain a delay optimal coterie on any regular symmetric graph. The delay of the coterie will be r Ž G .. The message complexity of the coterie, of course, depends on the graph. The coterie C defined on Gn , however, is not optimal so far as its message complexity is concerned and it is possible to choose coteries with the same delay and smaller message complexity. This fact has been illustrated by actually defining a coterie, with smaller message complexity, on G 3 .

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Acknowledgements

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The authors are thankful to the reviewers for their valuable comments. Their suggestions have helped us to put the paper in its present form.

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Prof. P.C. Saxena is a professor of computer science at the School of Computer and System Sciences, Jawaharlal Nehru University, New Delhi, India. He has supervised nine PhDs in the area of Database Management, Networking and Multimedia. He has guided 55 M.Tech. Dissertations. His research interests include DBMS, Data Communication, Networking, Distributed Computing and Multimedia.

P.C. Saxena et al.r Computer Standards & Interfaces 23 (2001) 341–353

Dr. Sangita Gupta is an assistant professor at the School of Computer and System Sciences, Jawaharlal Nehru University, New Delhi, India. She received a Bachelor in Engineering degree from Delhi Institute of Technology, Delhi, in 1991 and M.Tech. and Ph.D. in computer science from the School of Computer and System Sciences, Jawaharlal Nehru University, New Delhi, in 1995 and 1998, respectively. Her research interests include Distributed systems, Networks and Multimedia Databases. Jagmohan Rai is a senior lecturer in the Department of Mathematics, P.G.D.A.V. College ŽEve., University of Delhi, New Delhi, India. He received a M.Sc. degree in Mathematics in 1982 and a M. Phil degree in Mathematics Žspecialization: Operator Theory. in 1985, both from the University of Delhi. Currently, he is pursuing a Ph.D. in computer science from the School of Computer and System Sciences, Jawaharlal Nehru University, New Delhi, India. His research interests include Distributed Mutual Exclusion, Networks, Distributed Computing and Multimedia.

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