Optimal Algorithms for Dissemination of Information in Some. Interconnection Networks. Universitat-GH Paderborn Paderborn.

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1 Optimal Algorithms for Dissemination of Information in Some Interconnection Networks Juraj Hromkovic, Claus-Dieter Jeschke, Burkhard Monien Universitat-GH Paderborn Fachbereich Mathematik-Informatik 4790 Paderborn Keywords: gossiping, broadcasting, lower bounds, communication in interconnection networks Abstract The problems of gossiping and broadcasting in one-way communication mode are considered for some prominent families of graphs. The complexity is measured as the number of communication rounds in the gossip and broadcast algorithms. The main result of the paper is the precise estimation of the gossip problem complexity in cycles. To obtain this result a new combinatorial analysis of gossiping in cycles is developed. This analysis leads to an optimal lower bound on the number of rounds, and also to the design of an optimal algorithm for gossiping in cycles. The optimal algorithm for gossiping is later used to design new, eective algorithms for gossiping in important families of interconnection networks (cube connected cycles; butterfly networks). Further, a new, eective algorithm for broadcasting in Shue-Exchange networks is developed. On the leave of Comenius University, Bratislava 1

2 Introduction In this paper we shall investigate the problem of information dissemination in some prominent undirected families of graphs (parallel architectures). Assume each node (processor) in a graph (network) has some piece of information. We shall study how eciently information can be spread in some given graphs. Specically we investigate the "gossip" problem and the "broadcast" problem studied already in several papers (see [HHL88], [EM89]). To solve the gossip problem for a graph G we have to nd such communications via undirected edges of G such that after these communications each node of G has learned the cumulative message (all pieces of information originally distributed in all nodes of G). To solve the broadcast problem for a given graph G and a given node v of G we have to nd such communications that all nodes in G learn the piece of information distributed in v. Here the complexity of gossiping and broadcasting is measured as the number of rounds of communications. The message complexity of the gossip problem has been investigated in the early seventies [BS72, HMS72]. In this paper we shall mainly investigate the "telegraph communication mode" (called also "one-way" mode), where in each round, each node (processor) is active only via one of its adjacent edges (links) and the communication is one-way, i.e. each processor can either transmit or receive, but not both. There is no bound on the length of the information code submitted in one round via one edge, i.e., the whole cumulative message can be also submitted in one round by a node which knows it. We shall also consider the "telephone communication mode" (called also "two-way" mode) where in each round, each node is active only via one of its links and the communication is two-way, i.e. two communicating nodes exchange all the messages they know to that point. Let r(g) (r 0 (G)) denote the necessary and sucient number of rounds for gossiping in G in the one-way (two-way) mode. For a given G and a node v in G we denote by b v (G) (b 0 v(g)) the 2

3 complexity of broadcasting for v and G in the one-way (two-way) mode. We dene b(g) = maxfb v (G)jv is a node in Gg, b 0 (G) = maxfb 0 v(g)jv is a node in Gg. One can easily see that b(g) = b 0 (G) for each graph G. We call attention to the fact that the "census" problem, where one node has to learn the whole cumulative message, has the same complexity as broadcasting (to solve the census problem one can take the rounds of a broadcast algorithm in a reverse order with messages owing in the opposite direction). We shall use also another denition for the complexity of broadcasting. mb(g) = minfb v (G)jv is a node in Gg. We note that both denitions were already used in literature, and to see such a graph G that mb(g) < b(g) one can consider a chain of 4 nodes. The most important previous work for gossiping has been for the complete graph K n [ES79, EM89]. Our aim here is to study gossiping and broadcasting in several types of graphs, some of them representing prominent classes of networks. This research direction was already considered in [BHMS89, FP80, BP88, HHL88, LP88], where among others the following two results were proved: (i) For any tree T : r(t ) = 2mb(T ) (note that r(g) 2mb(G) for any graph G because one can solve the broadcast problem by cummulating the cummulative message in one node u and then broadcast from the source node u) (ii) For any cycle C n of n nodes: r(c n ) n=2 + O( p n) (the algorithm presented in [BHMS 89] is not optimal) Our results are divided in three sections. In Section 1 we give a new, eective algorithm for broadcasting in shue-exchange networks (SE), i.e., we show that b(se k ) 2k. Broadcasting in bounded degree variants of the hypercube have been studied in several papers [BP88, LP88, St89]. Broadcasting can be done on Cube 3

4 Connected Cycles CCC k within d5k=2e? 1 rounds (straight forward algorithms), on buttery networks, BF k, within 2k + 5 rounds [St89] and on DeBruijn networks, DB k, within 1:5k + 1 rounds [BP88]. Here the index k always denotes the "dimension" of the network. For the shue-exchange network, SE k, only the straightforward bound 3k was known (see also [LP88]). In [LP88] a new network, the folded shue-exchange network, of degree 3 was dened on which broadcasting can be done within 2k + 1 rounds. We note that the trivial lower bound b(se k ) 2k? 1 follows directly from the distances among nodes in SE. The main result of our paper is presented in Section 2. We prove a lower bound on gossiping in cycles (note that the only lower bound known up to now was the trivial lower bound n/2) and we improve the upper bound from [BHMS89]. We show that (a) r(c n ) = n=2 + d p 2ne? 1 for n even, (b) n d p 2n? 1=2e? 1 r(c n ) n d p 2(n + 1)e? 1 for n odd. We note that the upper bound and the lower bound in (b) dier at most by 1. To achieve the results (a) and (b) we develop a combinatorial technique enabling not only to prove high lower bounds on r(c n ) but enabling also to nd the optimal algorithm. Then, in Section 3, we shall use the optimal algorithm for gossiping on C n to reach some new, eective algorithms for gossiping in cube connected cycles (CCC n ) and buttery networks (BF n ). Namely, we show that r(ccc k ) r(c k ) + 3k? 1 d7k=2e + d2 p dk=2ee? 2 and r(bf k ) r(c k ) + 2k d5k=2e + d2 p dk=2ee? 1 Since b5k=2c? 1 b(ccc k ) d5k=2e? 1 [LP88], k + bk=2c b(bf k ) 2k + 5 [St89], and r(g) b(g) for any graph G, we conjecture that the obtained gossip algorithms for CCC k and BF k are not very far from the optimal algorithms. 4

5 1 Broadcasting in Shue-Exchange Networks In this section we shall give an eective algorithm for broadcasting in SE-networks which works in 2k rounds. Let SE k denote the SE-network of dimension k(k 1), i.e. SE k = (V k ; E k ), where V k = f0; 1g k and E k consists of shue edges fa; ag and exchange edges fa; ag, 2 f0; 1g k?1 ; a; a 2 f0; 1g; a 6= a. There exist nodes in SE k with the distance 2k? 1 (namely 00 : : :0 and 11 : : :1) and this implies b(se k ) 2k? 1. Theorem 1.1 b(se k ) 2k for k 1. Proof. Let for each word w = a 1 a 2 : : : a k 2 f0; 1g k w 1 = a 1 and w t = a t+1 a t+2 : : : a k for t k. If w = then w 1 =. Now, we shall write the broadcasting algorithm for an arbitrary source node in SE k. Algorithm BSE for t = 0 to k? 1 do for all 2 f0; 1g t do in parallel if t 62 f 1 g then begin t sends to t+1 t 1 (shue round) end; end; t+1 t 1 sends to t+1 t 1 (exchange round) Now, we need to prove the following two facts: (1) there is no conict in any of 2k rounds, i.e. algorithm BSE works in the one-way mode (if a node is active in a round then it is active only via one edge in one direction). (2) after 2r rounds (r executions of the loop) all nodes r ; 2 f0; 1g r, have learned the piece of information of. 5

6 (1) There is no conict in any exchange round because each sender has the last bit t 1 and each receiver has the last bit t 1. Let there be a conict in a shue round, i.e., t = t+1 t 1 for some ; 2 f0; 1g +. It implies t 1 t+1 = t+1 1 ) t 1 = t+1 1 = : : : = k?1 1 = 1 ) t 2 f 1 g. But this is a contradiction because we do not use Shue-operation for t 2 f 1 g. (2) This will be proved by induction according to r = t + 1. It has to be shown that the nodes t+1 t 1 ; t 2 1 (which do not receive the information in the r-th execution of the loop), have got the information already in previous rounds. Clearly, our induction hypothesis [that all nodes r, for each 2 f0; 1g r have learned the piece of information of after r executions of the loop] is fullled after the rst execution of the loop. Now, let us consider the situation after r executions of the loop. Clearly, if r?1 62 ( 1 ) then all r for 2 f0; 1g r know the piece of information of. If r?1 2 ( 1 ) then r r 1 = r?1 1 r 1 which knows already the piece of information of according to the induction hypothesis. 2 2 Optimal Gossiping in Cycles In this section we shall give an optimal method for gossiping in cycles. The best bounds known for this problem were n=2 r(c n ) n=2 + O( p n) achieved in [BHMS90]. So, we shall start rst with the hardest task { to prove a lower bound on r(c n ). Later we present the algorithm for gossiping in cycles and some new, derived algorithms for gossiping in CCC-networks and BF-networks. 6

7 Theorem 2.1 r(c n ) n=2 + d p 2ne? 1 for n even, and r(c n ) dn=2e + d p 2n? 1=2e? 1 for n odd. Proof. Since the optimal algorithm for gossiping in an n-node cycle in two-way mode uses at least dn=2e rounds [FP80] we may assume that r(c n ) = dn=2e+f(n) for some function f from positive integers to nonnegative integers. We shall show by a detailed analysis that f(n) p 2n? 3=2. Let A be an arbitrary optimal algorithm for gossiping in C n in one-way mode. This algorithm A can be specied as a sequence of sets A 1 ; A 2 ; : : :; A t(a), where A i is the set of communications (directed edges) in the i-th round and t(a) = dn=2e + f(n) is the number of rounds in A. To prove our lower bound we use the fact that there are exactly two simple paths between any two nodes v and u in C n. We shall denote by L(v; u) (R(v; u)) the path from v (u) to u (v) in the clockwise direction of the cycle (The clockwise direction in the cycle is the direction leading from the rith-most node to the left-most node through the lower half of the cycle.). We dene the left distance d l (v; u) from v to u as the length of the path L(v; u) and the right distance d r (v; u) from v to u as the length of the path R(v; u). Obviously, d l (v; u) + d r (v; u) = n. The rough idea of the proof is based on the following observation (see Fig. 1). Observation For each node v of C n there exists an edge (l v ; r v ) such that (a) the distances d l (l v ; v); d r (r v ; v) 2 [dn=2e? f(n)? 1; dn=2e + f(n)] (b) for each node u on the path L v = L(l v ; v) which does not go through the edge (l v ; r v ), the piece of information I(u) (originally distributed in u) is submitted to v through the path which does not involve the edge (l v ; r v ). [note, that the submission of I(u) also through (l v ; r v ) is not forbidden] (c) for each node x on the path R v = R(r v ; v) which does not go through the edge (l v ; r v ), 7

8 the piece of information I(x) is submitted to v through the path which does not involve the edge (l v ; r v ). Figure 1 To specify the idea of the proof more precise we need the following denition: Let X = x 1 ; : : :; x m be a simple path in C n and let t 1 ; t 2 ; : : :; t m?1 be an increasing sequence of numbers in f1; 2; : : :; t(a)g. We say that X[t 1 ; : : :; t m?1 ] is a time-way of A when for each i 2 f1; : : :; m? 1g x i sends a message to x i+1 in the t i -th round. Obviously, there can exist several time-ways of A for some simple path in C n. Let X[T ], for some X = x 1 ; : : :; x m ; T = t 1 ; : : :; t m?1 ; be a time-way of A. If t i+1?t i?1 = k i 0 for an i 2 f1; : : :; m? 2g then we say that X[T ] has the k i -delay at the node x i+1. The global delay of X[T ] is d(x[t ]) = t 1? 1 + P m?2 i=1 k i. The global time of X[T ] is t(x[t ]) = t m?1 = m? 1 + d(x[t ]). Now, let us give a more precise description of the proof idea. We shall divide the nodes in C n into two disjoint subsets B A (including all nodes which have learned the cummulative message in one round from a neighbouring node which knowed already the whole cummulative message) and C A (the complement of B A ) according to the algorithm A. Let, for each node v of C n t A v be the minimal number of rounds after that v has learned the cumulative message in the algorithm A. We dene C A := fvj no message submitted to v in any round t t A v was the cumulative messageg, and B A := fvj the message submitted to v in the round t A v is the cumulative message g. So, we can view the algorithm A as a process in which rst the nodes in C A concentrate the whole cumulative message of C n and then the nodes in C A distribute the cumulative message to the nodes in B A. (Note that these two parts of the process may be non-disjoint 8

9 in time, i.e. in the same round one node in B A can receive the cummulative message from a neighbour, and one node in C A can receive a partial information in order to complete its knowledge about the cummulative message.) Let l A denote the cardinality of C A, and let C A = fu 1 ; u 2 ; : : :; u la g. For each i 2 f1; : : :; l A g we dene B(u i ) := fu i g [ fv 2 B A j there is a time-way u i ; : : :; v[t 1 ; : : :; t A v ] from u i to v for some t 1 > t A u i g. Clearly S l A i=1 B(u i ) includes all nodes of the cycle C n i A is a gossip algorithm for C n. Obviously jb(u i )j 2(t(A)? t A u i ) = 2(dn=2e + f(n)? t A u i ). So, if A is a gossip algorithm the following inequality must be true (1) n P l A i=1 jb(u i )j 2 P l A i=1 (dn=2e + f(n)? t A u i ) = 2 l A (dn=2e + f(n))? 2 P l A i=1 t A u i. To use (1) for obtaining a lower bound for f(n) we need to obtain a lower bound on P la i=1 t A u i. To do it we need the idea of collisions and delays. We shall investigate the collisions among the paths bringing the pieces of information to the nodes in C A in what follows. We consider only the paths leading to the nodes in C A from that reason that we are able to secure in this way that no two distinct collisions take part simultaneously at the same place and in the same round. So, all collisions among the paths leading to nodes in C A will be "disjoint" in this sense, and we shall have no troubles to add them. Now, let us give the formal denition of collisions between two time-ways going in opposite directions. Let X[T ] = x 1 ; : : :; x m [t 1 ; : : :; t m?1 ] and Y [T 0 ] = y 1 ; : : :; y k [t 0 1 ; : : :; t0 k?1 ] be two time-ways of A going in opposite directions, and let x m 6= y k. We say that X[T ] and Y [T 0 ] have a collision in a node v i (x 1 = y 1 = v) or (1 o ) 9i 2 f1; : : :; m? 1g and 9j 2 f1; : : :; k? 1g (i; j) 6= (1; 1), such that x i = y j = v and (2 o ) (t i?1 < t 0 j?1 < t i) or (t 0 j?1 < t i?1 < t 0 j ) 9

10 Following this denition and Fig. 2.a we see that each collision according (1 o ) and (2 o ) causes at least two "delays" (either one 2-delay on one of the two time-ways or one 1-delay on each of the two time-ways in collision). Later we shall show that the fact x 1 = y 1 causes at least 3 delays in x 1 for some special time-ways. Figure 2.a Let X[T ] = x 1 ; : : :; x m [t 1 ; : : :; t m?1 ] and Y [T 0 ] = y 1 ; : : :; y k [t 0 1 ; : : :; t0 k?1 ] be two time-ways of A going in opposite directions. We say that X[T ] and Y [T 0 ] have a relative collision in a node v i (i) x m = y k = v or (ii) 9i 2 f1; : : :; m? 1g such that x i = y k = v and t 0 k?1 < t i (see Fig. 1.2.b) or (iii) 9j 2 f1; : : :; k? 1g such that x m = y j = v and t m?1 < t 0 j Figure 2.b We note that the relative collision dened above is called relative because one time-way has already nished its work in the node v when the second time-way still ows via the node v. On the other hand we are speaking about a collision also in this case because the information cumulated in v cannot be distributed in the rounds in which the second time-way reaches and leaves the node v. In what follows we shall say that two time-ways X[T ] and Y [T 0 ] have a conict if there is either a collision or a relative collision between X[T ] and Y [T 0 ]. Let X[T ] = x 1 ; : : :; x m [t 1 ; : : :; t m?1 ] and Y [T 0 ] = y 1 ; : : :; y k [t 0 1 ; : : :; t0 k?1 ] be two time-ways of A going in the same direction. We say that X[T ] and Y [T 0 ] overlap in the node u when one of the following conditions holds: 10

11 (i) there exists a node u = x i = y j for some i 2 f1; : : :; m? 1g; j 2 f1; : : :; k? 1g such that maxft i?1 ; t 0 j?1 g < minft i; t 0 j g (we set t o = t 0 o = 0) (ii) x m = y j = u for some j 2 f1; 2; : : :; k? 1g and t 0 j?1 < t m?1 < t 0 j (iii) y k = x i = u for some i 2 f1; 2; : : :; m? 1g and t i?1 < t 0 k?1 < t i We shall need still one denotation. Let X and Y be two simple paths in the cycle C n, and let Y be contained in X. We shall denote this fact by Y X. Now, for each node v in C A we shall x some time-ways bringing the cumulative message to v. Let v 2 C A, and let (l v ; r v ) be an edge satisfying (a), (b), (c) of Observation Let L v = L(l v ; v) (R v ) denote the path from l v (r v ) to v which does not go through the edge (l v ; r v ). We shall call the edge (l v ; r v ) suitable for v when the following property (d) holds. (d) There exists two time-ways L v [z 1 ; : : :; z m ] and R v [k 1 ; : : :; k b ] such that z m t A v and k b t A v. Now, let us x one suitable edge (l v ; r v ) for each node v in C A. The time-way L v [T A ] for T A = t A 1 ; : : :; ta m will be called the left time-way of A for v if, for any time-way L v[z 1 ; : : :; z m ] of A, t A i z i for each i 2 f1; : : :; mg. Analogously, R v [T A ] is called the right time-way of A for v if, for any time-way R v [a 1 ; : : :; a m ] of A, t A i a i for each i 2 f1; : : :; mg. Let L v [T A ](R v [T A ]) be the left (right) time-way of A for some v 2 C A. The number t A v? t(l v [T A ]) = k (t A v? t(r v [T A ])) is called k-relative delay of L v (R v ) and denoted by rd(l v [T A ]) (rd(r v [T A ])). Clearly, the relative delays can be caused by relative collisions. Now, let us formulate an important property of nodes in C A. Lemma For each two nodes x and y in C A the left (right) time-ways L x [T A ] and L y [T A ] (R x [T A ] and R y [T A ]) do not overlap. 11

12 Proof. The proof is done by contradiction. Let L x [T A ] and L y [T A ] overlap in a node v (see Fig. 3.a and Fig. 4). W.l.o.g. we can assume that x is laid on the path L y (if x does not lay on the path L y then y is laid on L x and the proof can be realized in the same way be exchanging the roles of x and y). Obviously, in the node v each of the two time-ways L x [T A ] and L y [T A ] has learned all pieces of information distributed in the nodes on the paths from l x to v and from l y to v. Thus there must exist a node on the path between x and y which has learned the cumulative message at least in the round minft A x ; ta y g. It can be shown that the time-way L y [T A ] (or R x [T A ]) disseminates the cumulative message to y (to x) exactly in the round t A y (t A x ). Clearly, this is a contradiction with the fact that y 2 C A (x 2 C A ): If R x [T A ] and R y [T A ] overlap the contradiction can be achieved in the same way. 2 The proof of Theorem 2.1 continued Now, let us give a lemma which helps us to add the number of delays caused by collisions among the left and right time-ways of nodes in C A. Lemma For each two nodes x; y 2 C A there are two conicts (between L x [T A ] and L y [T A ] on one side and R y [T A ] and R x [T A ] on the other side) causing at least 4 delays and relative delays. Proof. We shall consider two possibilities according to the positions of x; y; l x ; l y ; r x ; r y. I (L x 6 L y ) ^ (L y 6 L x ) ^ (R x 6 R y ) ^ (R y 6 R x ) [see Fig. 3.a, Fig. 3.b] II (L x L y ) _ (L y L x ) _ (R x R y ) _ (R y R x ) [see Fig. 4] I. Case I is still divided into two subcases I.1 (see Fig. 3.a) and I.2 (see Fig. 3.b) according to the positions of x; y; l x ; l y ; r x ; r y. I.1 (I) ^ (L x 6 R y ) ^ (R x 6 L y ) ^ (R y 6 L x ) ^ (L y 6 R x ) I.2 (I) ^ ((L x R y ) _ (R x L y ) _ (R y L x ) _ (L y R x )) I.1 Let us rst prove that there are two collisions in case (I.1). We give the proof only for the case described at Fig. 3.a; all other cases of type (I.1) are symmetrical. 12

13 First, let us show that there is a collision between L x [T A ] and R y [T A ] which causes at least two delays. Clearly, if r y 6= l x then there is a collision between L x [T A ] and R y [T A ] because L x [T A ] starts from l x in the direction to r y, R y [T A ] has to go through l x in order to reach y. We shall still handle the special case when r y = l x. According to the denition of the collision this is also a collision but we shall prove that this special collision also causes at least two delays. We claim that none of R y [T A ] and L x [T A ] can start in the rst round. Let us assume that R y [T A ] does (the case for L x [T A ] is analogous). Then R x [T A ] cannot start in the rst round because r x receives the message from r y in this round. But this is a contradiction with Lemma because the time-ways R y [T A ] and R x [T A ] overlap in the node r x. Figure 3.a Now, we have two possibilities. If there is a collision between L y [T A ] and R x [T A ] then the proof for case (I.1) is done. Let there be no collision between L y [T A ] and R x [T A ]. This implies that either L y [T A ] reaches y before R x [T A ] goes through y or that R x [T A ] reaches x before L y [T A ] goes through x. Let us deal only with the case that L y [T A ] = L y [t 1 ; : : :; t m ] reaches y before R x [T A ] goes through y because the second case can be solved analogously. We see that the node y is reached by R x [T A ] after the round t m. Since R x [T A ] and R y [T A ] do not overlap according to the result of Lemma we obtain that R y [T A ] cannot reach y before R x [T A ] leaves y. So, this relative collision of L y [T A ] and R x [T A ] causes a 2-relative delay for L y [T A ]. I.2 Let us now prove Lemma for the case L x R y and L y R x described at Fig. 1.3.b. All other cases of the type (I.2) are symmetrical, i.e., R x L y and R y L x (see Fig. 1.3.c for the border case with x = l y ). Figure 3.b 13

14 Thus we have L x R y and L y R x. Now, let us show that there is a conict between L x [T A ] and R y [T A ]. If R y [T A ] leaves x before L x [T A ] reaches x then R y [T A ] and L x [T A ] have a collision in a node leading on the way L x. If L x [T A ] reaches x before R y [T A ] leaves x then we have a relative collision in the node x which causes either two relative delays (L x [T A ] reaches x before R y [T A ] does) or one relative delay and one delay (R y [T A ] reaches x as rst). To show that there is a conict between L y [T A ] and R x [T A ] one can use the same consideration as described above for the conict between L x [T A ] and R y [T A ]. Figure 3.c II We show that this case (see Fig. 4) cannot appear. We shall distinguish the following three possibilities: (i) R x [T A ] reaches x before L y [T A ] does (ii) L y [T A ] reaches y before R x [T A ] does (iii) R x [T A ] and L y [T A ] have a conict in a node between x and y. Figure 4 Let us consider the case (i); the cases (ii) and (iii) can be solved analogously. Following Lemma and Fig. 1.4 we see that also L x [T A ] reaches x before L y [T A ] reaches it. So, x has learned the cumulative message before L y [T A ] has reached x. This implies that L y [T A ] brings the cumulative message from x to y which is a contradiction with the fact that y 2 C A. 2 The proof of Theorem 2.1 continued Now, we are already able to formulate our lower bound for P l A i=1 t A u i. Lemma (2) P la i=1 t A u i l A (n? 1)=2 + l A (l A? 1) + l A =2. 14

15 Proof. P la i=1 t A u i = P l A i=1 t(l ui [T A ])+rd(l ui [T A ])+t(r ui [T A ])+rd(r ui [T A ]) 2 = P l A d l (l ui ;u i )+d r (r ui ;u i ) i=1 + P l A d(l ui [T A ])+rd(l ui [T A ])+d(r ui [T A ])+rd(r ui [T A ]) 2 i=1 2 l A (n?1) ( 4l A(l A?1) 2 + l A ) = l A (n? 1)=2 + l A (l A? 1) + l A =2. (*) The inequality (*) follows from the following facts: (i) the paths l v ; : : :; v and v; : : :; r v include together all edges of C n except (l v ; r v ), i.e., d l (l v ; v) + d r (r v ; v) = n? 1 (ii) Following Lemma we see any pair of nodes x; y from C A brings at least 4 delays to d(l x [T A ]) + rd(l x [T A ]) + d(r x [T A ]) + rd(r x [T A ]) + d(l y [T A ]) + rd(l y [T A ]) + d(r y [T A ]) + rd(r y [T A ]), and following Lemma we see that these delays are disjoint from any delays caused by another pair of nodes from C A. Clearly, the number of distinct pairs of nodes in C A is l A (l A? 1)=2. (iii) For each node x 2 C A there is one relative delay following from the relative collision between L x [T A ] and R x [T A ]. 2 The Proof of Theorem 2.1 continued Applying the inequality (2) of lemma in the inequality (1) we obtain n 2l A (dn=2e + f(n))? 2[l A (n? 1)=2 + l A (l A? 1) + l A 2 ] (3) which directly implies f(n) n 2l A? dn=2e + 1 l A (l A n?1 2 + l A (l A? 1) + l A 2 ) Thus, we obtain n 2l A + l A + n?1 2? dn=2e?

16 f(n) n 2l A + l A? 1 for n even, and (4) n 2l A + l A? 3=2 for n odd. Since (4) holds for any unknown l A (note that (4) holds for any gossip algorithm A) we take f(n) minf n +l A?aja = 1 for n even; a = 3=2 for n odd; 1 l A ng = minfg n (l A )j1 l A ng l A 2l A l A Since the function g n (l A ) has the global minimum l A = p n=2 on the real interval [1; n] and f is the function into the set of positive integers we obtain f(n) d2 p n=2e? 1 for n even, and f(n) d2 p n=2? 3=2e for n odd. Since r(c n ) dn=2e + f(n) Theorem 2.1 is proved 2 Corollary 2.2 If A is an optimal gossip algorithm for C n that meets the lower bound of Theorem 2.1 then A must have the number of l A of concentrators between b p n=2c and d p n=2e. Next we use the observation of Corollary 2.2 to nd the optimal gossip algorithm for C n, n even. Theorem 2.3 r(c n ) dn=2e + d2 p dn=2ee? 1 Proof. To explain the idea we give rst the algorithm for n = 2l 2, l even. Then we extend the algorithm for any positive integer n. Let us divide the cycle C n into l disjoint paths of lengths 2l, the i-th path starting with v i and ending with u i ; as depicted at Fig. 5. Let vi 0(u0 i ) be a node between v i and u i with the distance l? 1 from v i (u i ). Now, the algorithm works in two phases. 1. Phase For each i 2 f1; : : :; lg: there is a time-way of the length n=2 from v i to v (i+l=2?1)modl+1 going through u i, and there is a time-way of the length n=2? 1 from u i?1 to v (i+l=2?1)modl+1 going through v i?1. (Clearly, the time-ways starting in v i 's go in opposite direction as the time-ways starting in u i 's.) 16

17 Note that after the 1. phase all nodes v i 's know already the cumulative message because for each v i there are two time-paths; one from v (i+l=2?1)modl+1 to v i and the second one from u (i+l=2?1?1)modl+1 to v i. 2. Phase For each i 2 f1; : : :lg: v i sends the cumulative message to u i?1 v i sends the cumulative message to v 0 i, u i sends the cumulative message to u 0 i, Figure 5 Now, let us add the number of rounds. Each time-way starting in a v i in the 1. phase has length n=2 and it has exactly (n=2l)? 1 = l? 1 collisions. When the collision of two time-ways is solved in such a way that the collision causes 1-delay for each time-way then the 1. phase uses n=2 + l? 1 rounds. Since the distance between v i and u i is 2l? 1 the 2. phase uses l rounds. One can simply see that n=2 + 2l? 1 = n=2 + d p 2ne? 1. Now, let us give an algorithm for even n's. For each even, positive integer n > 3 there is a positive integer l such that 2l 2 n < 2(l + 1) 2 = 2l 2 + 4l + 2 Thus n = 2l 2 + 2i for some i 2 f0; 1; : : :; 2lg. If 1 i l then we divide the cycle into such l parts P 1 ; : : :; P l that i parts have the length 2(l + 1) and l? i parts have the length 2l. For each i the part P i starts with the node v i and ends with the node u i (similarly as at Fig. 5, only the distances between v i and u i may be dierent for distinct i's). Then the generalized algorithm realizes the time-ways of the length n=2 from all nodes v i 's, and the time-ways of the length n=2? 1 from all nodes u i 's. Clearly, this 1. phase can be realized in n 2 + l? 1 rounds. After 17

18 the 1. phase we have exactly l cumulative points in C A (note that these points may be dierent from v i 's and u i 's in this case), and the distance between two neighbouring cumulative points is at most 2(l + 1). So, the 2. phase of the distribution of the cumulative message works in l + 1 rounds. The total number of rounds is n=2 + 2l = n 2 + (2l + 1)? 1 = n 2 + d2 p dn=2ee? 1 because 2 p n=2 = 2 p l 2 + i < 2(l + 1=2) for 1 i l. If l < i 2l then the cycle is divided into l parts, where i? l parts have the length 2(l + 2) and 2l? i = l? (i? l) parts have the length 2(l + 1). The algorithm for gossiping works exactly in the way described above, the only dierence is that the 2. phase uses l + 2 rounds instead of l + 1 rounds. Thus the total number of rounds of the algorithm is n=2 + 2l + 1. Since 2 p n=2 = 2 p l 2 + i > 2(l+1=2) for l < i 2l we have n=2+2l+1 = n=2+(2l+2)?1 = n=2+d2 p n=2e?1. In the case that n > 1 is an odd positive integer one can use the algorithm for C n+1 to design the algorithm for gossiping in C n. Clearly, the number of rounds of such algorithm is at most r(c n+1 ) = (n + 1)=2 + d2 p (n + 1)=2e? 1 = dn=2e + d2 p dn=2ee? 1. 2 Combining the results of Theorem 2.1 and Theorem 2.3 we obtain the main result of our paper. Theorem 2.4: r(c n ) = n=2 + d p 2ne? 1 for each even n > 3, and dn=2e + d p 2n? 1=2e? 1 r(c n ) dn=2e + d2 p dn=2ee? 1 for each odd n 3. 3 Utility of Optimal Gossiping in Cycles for other Networks In this section we show how to construct eective algorithms for gossiping in CCC k -networks and BF k -networks using the optimal algorithm for gossiping in C k (k denotes the dimension of networks here). 18

19 Both CCC k and BF k consists of 2 k cycles of k nodes denoted by C (k) for 2 f0; 1g k. Let (i; ) denote a node in the i-th level of CCC k or BF k for some 2 f0; 1g k. Now, we dene set to set broadcasting as in [RL88]. Let A and B be two sets of nodes. The set to set broadcasting from A to B is a communication process in which each node in B has learned all pieces of information distributed in A. In what follows we give algorithms for set to set broadcasting from the i-th level to the i-th level in BF k, and for set to set broadcasting from the i-th level to the ((i? 1)modk)-th level in CCC k. SET CCC k for j = 0 to k? 1 do for all 2 f0; 1g k do in parallel begin exchange information between ((i + j)mod k; ) and ((i + j)mod k; ((i + j)mod k)) f * needs two rounds * g; if j < k? 1 then ((i + j)mod k; ) sends to ((i + j + 1)mod k; ) f * needs 1 round * g end; SET BF k for j = 0 to k? 1 do for all 2 f0; 1g k do in parallel begin ((i + j)mod k; ) sends to ((i + j + 1)mod k; ((i + j)mod k)) ((i + j)mod k; ) sends to ((i + j + 1)mod k; ) end; Now, gossiping in CCC k and BF k can be done as follows. Algorithm GOSSIP-BF K 19

20 1. Use in parallel for all the 1. phase of the optimal algorithm for gossiping in C k concentrating the cumulative message of C k in l = b p dk=2ec "regularly distributed" nodes in C k to concentrate the cumulative message of C (k) of BF k in l nodes (v i ; ) for 1 i l. 2. For all i 2 fv j j1 j lg do in parallel set to set broadcasting from the i-th level to the i-th level on BF k. 3. Use in parallel for all the 2. phase of the optimal algorithm for gossiping in the cycle to broadcast the cumulative message of BF k contained in the nodes (v i ; ); 1 i l; to the other nodes in the cycle C (k). Algorithm GOSSIP-CCC k 1 Use in parallel for all 2 f0; 1g 2k the 1. phase of the optimal algorithm for gossiping in C k to concentrate the cummulative message of C (k) of CCC k in l = b p dk=2ec nodes (v i ; ) for 1 i l. 2 For all i 2 fv j j1 j lg do in parallel set to set broadcasting from the i-th level of CCC k to the ((i? 1)mod k)-th level of CCC k. 3 Use in parallel for all the 2. phase of the optimal algorithm for gossiping in the cycle to broadcast the cummulative message of CCC k contained in the nodes ((v i? 1)mod k; ) of CCC k ; 1 i l, to the other nodes in the cycle C (k). Analyzing the complexity of the above stated procedures we obtain: Theorem 3.1 r(ccc k ) r(c k ) + 3k? 1 d7k=2e + d2 p dk=2ee? 2, and r(bf k ) r(c k ) + 2k d5k=2e + d2 p dk=2ee? 1. Concluding this section we show that this technique can be used also for gossiping in two-way mode. 20

21 Theorem 3.2 r 0 (CCC k ) k=2 + 2k = 5 dk=2e for k even, and r 0 (CCC k ) dk=2e + 2k + 2 = 5 dk=2e for k odd. Proof. To do gossiping in CCC k the following algorithm working in three phases can be used. 1. Use the optimal algorithm for gossiping in C k in two-way mode [FP80] to do gossiping in parallel on all cycles C (k) of CCC k. 2. For all odd i k? 1 do in parallel set to set broadcasting from the i-th level to the ((i? 1)modk)-th level on CCC k. 3. For all odd j k? 1 do in parallel: the j-th levels learns in parallel from the (j? 1)-th level [the (k? 1)-th level learns in parallel in one special round when k is odd]. The result of Theorem 3.2 follows directly from the fact r 0 (C k ) = k=2 for k even and r 0 (C k ) = dk=2e + 1 for k odd proved in [FP80] and from the fact that the information exchange in the algorithm SET CCC k performed in the two-way mode runs in one round. 2 References BHMS90 A. Bagchi S.L. Hakimi J. Mitchem E. Schmeichel: Parallel algorithms for gossiping by mail. Inform. Proces. Letters 34 (1990), No. 4, BP88 J.-C. Bermond C. Peyrat: Broadcasting in DeBruijn networks. In: Proc. 19th Southeastern Conference on Combinatorics, Graph Theory and Computing. Congressus Numerantium 66 (1988), BS72 B. Baker R. Shostak: Gossips and Telephones. Discr. Mathem. 2 (1972),

22 EM89 S. Even B. Monien: On the number of rounds necessary to disseminate information. Proc. 1st ACM Symp. on Parallel Algorithms and Architectures, Santa Fe, June ES79 R.C. Entringer P.J. Slater: Gossips and telegraphs. J. Franklin Institute 307 (1979), FP80 A.M. Farley A. Proskurowski: Gossiping in grid graphs. J. Combin. Inform. System Sci. 5 (1980), HHL88 S.M. Hedetniemi S.T. Hedetniemi A.L. Liestman: A survey of gossiping and broadcasting in communication networks. Networks 18 (1988), HMS72 A. Hajnal E.C. Milner E. Szemeredi: A cure for the telephone disease. Can. Math. Bull. 15 (1972), Kn75 W. Knodel: New gossips and telephones. Discrete Math. 13 (1975), 95. LP88 A.L. Liestman { J.G. Peters: Broadcast networks of bounded degree. SIAM J. Disc. Math. 1 (1988), RL88 D. Richards A.L. Liestman: Generalization of broadcasting and gossiping. Network 18 (1988), St89 E. Stohr: Broadcasting in the buttery graph, manuscript