multiple message broadcasting in communication networks

Multiple Message Broadcasting in -

Communication Networks*

Oh-Heum Kwon and Kyung-Yong Chwa

Department of Computer Science, Korea Advanced Institute of Science and Technology Kusong-dong 373-1, Yusong-gu, Taejon 305-701, Republic of Korea

Broadcasting refers to the process of dissemination of a set of messages originating from one node to all other nodes in a communication network. We assume that, at any given time, a node can transmit a message along at most one incident link and simultaneously receive a message along at most one incident link. We first present an algorithm for determining the amount of time needed to broadcast k messages in an arbitrary tree. Second, we show that, for every n , there exists a graph with n nodes whose k-message broadcast time matches the trivial lower bound [log n l + k - 1 by designing a broadcast scheme for complete graphs. We call those graphs minimal broadcast graphs. Finally, we construct an n node minimal broadcast graph with fewer than ([log nl + 1 )2@”’-’ edges. Q 7995 John Wi/ey & Sons, Inc.

1. INTRODUCTION

Consider n processors connected by a point-to-point communication network. Broadcasting refers to the process of dissemination of a set of messages originating from one node to all other nodes in the network. This problem has been studied by many authors. For a comprehensive list of references, see the survey in [ 1 3 1.

Several works have treated the problem of broadcasting multiple messages [I , 2, 6, 7, 14, 151. If one needs to broadcast more than one message, e.g., large files of information, simple modifications of one-message broadcast schemes would be too inefficient. For example, neither repeating a one-message broadcast scheme k times nor having each call take k time units produces a very efficient k-message broadcast scheme. It is easy to confirm that there are more efficient broadcast schemes. In this paper, we consider various communication networks including trees, hypercubes, and complete graphs and design multiple message broadcast schemes with optimal completion time for each of these networks. Moreover, we propose a

* This work was partially supported by Electronics and Telecornmuni- cations Research Institute under Contract No. NN 15260.

NETWORKS, Vol. 26 (1 995) 253-261 CJ 1995 John Wiley & Sons, Inc.

new class of graphs in which multiple message broadcasting can be done efficiently.

The network is modeled by a graph G = (V, E), where the node set V represents the set of processors and the edge set E represents the bidirectional communication link between processors. We assume that the time required to cross any link is the same for all messages and is taken to be one unit. We also assume that messages can be simultaneously transmitted along a link in both directions with error-free transmission. Moreover, we assume that, at any given time, a node can transmit a message along at most one incident link and can simultaneously receive a message along at most one incident link. Several communication problems using this model have been considered elsewhere in [ I , 3, 14, 151.

The goal of the broadcasting process is to disseminate messages owned by an originator to all other members as quickly as possible. Let t k ( l ] , G ) denote the minimum time needed to broadcast k messages from a node u in the graph G. Define the k-message broadcast rime uf the graph G as zk(G) = max { tk( u, G ) I u E V ( G ) } and let fk( n) denote the minimal k-message broadcast time tk( G ) over all n node graphs G. Finally, an n node graph G is said to be a minimal broadcast graph if tk( G ) = tp( n ) for all positive integers k .

CCC 0028-3045/95/040253-09

253

254 KWON AND CHWA

0 1 2 3 4 5 6 7 8 9 1 0 1 1 1 2

D = {5.7.7.8.8.10.12)

Fig. 2.1. A schedule with latest starting time.

The problem of determining t l ( u , G) for any node u in an arbitrary graph G is NP-hard [ 101. It is easily seen that tk( n) 2 Flog n l + k - 1, since the last message can leave the originator after k - I steps and the number of nodes receiving the message can be at most doubled after each time unit. For a special case where n is a power of 2, it is known that ln( n) = log n + k - 1 [ 141. In [ 141, Ho presented a Multiple Spanning Binomial Tree algorithm which performs k-message broadcasting within m + k - 1 units of time for rn-dimensional hypercubes. For a general n, it is known that tx( n ) I [log n l + k [ 1 1 . In [ I ] , they present an algorithm which works in an n node complete graph and completes k-message broadcasting within [log n l + k units of time.

There have been some previous work on multiple message broadcasting under different communication models [ 5-7, 161. In [5-71, it was assumed that a processor can participate in at most one call, as either a sender or a receiver, at any given time, i.e.. it is forbidden for a node to send one message and receive another simultaneously. Farley [ 71 and Cockayne and Thomason [6] determined the minimum amount of time necessary to broadcast k messages in the complete graphs using this model. In [ 161. Karp et al. considered another model where message de- livery involves communication latency.

In this paper, we first consider a tree network. The problem of broadcasting in trees was studied in [ 17-20]. Slater et al. [ 201 designed a linear time algorithm for finding the broadcast center ofany tree T, i.e., the set of nodes u in T for which t i ( u , T) is minimum over all nodes. Their results include a linear time algorithm for determining one-message broadcast time t l ( u , T) for any node u in a tree T. In Section 2. we generalize their results and present an algorithm for finding the k-message broadcast time lk( u. T) for an arbitrary integer k in any tree T.

In Section 3, we present a broadcast scheme for a complete graph with n nodes which complete k-message broadcasting in [log nl + k - 1 time units.* Note that this bound matches the lower bound stated above; thus. tk (n) = [log n l + k - 1 for all positive integers n .

Finally, we characterize a new class of minimal broadcast graphs with a relatively small number of edges. There has been much research on graph G with f , (C) = rlog nl having a relatively small (or minimum) number of edges [ 4, 8, I 1 ,

AAer submission of this paper. the same result was reported in [Z].

12, 2 I ] . It is well known that an rndimensional hypercube is a graph with minimum number of edges over all 2"' node graphs G with t l ( G ) = m [ 131. Hence, a hypercube is also a minimal broadcast graph with minimum number of edges. In the case where n is not a power of 2, the best-known upper bound on the number of edges of graph G with t l (G) = t 1 ( n ) is approximately rd log nl/2 - O( n) (for the exact values of the bound, see [12, 211). In Section 4, we characterize a new class of minimal broadcast graphs with fewer than (Flog nl+ 1)2"Opn1-I edges.

2. BROADCASTING IN TREES

A tree T = (1,'. E ) is a connected graph with the set of n nodes V and n - I edges E. Often, a tree is represented by a rooted tree T,, obtained by specifying a node r E V as its root. For any node u # r , the subtree TY of tree T, is rooted at u and consists of u and all its descendants.

Broadcasting in a rooted tree T, will denote a process of disseminating k messages M I , M 2 , . . . , Mk from the root r to all other nodes of the tree. In this section, we first consider the following decision problem: Given a rooted trw T, and t ~ * o positive integers t and k, is it possible to hroudcusr k messugcs,from r within t tinits qf time?The algorithm proposed to solve this problem will be extended to determine the minimum broadcast time at the end of this section.

A broadcast scheme S in T, is completely specified by a set of cull . s c ~ q i i c m ~ ~ matrices S, = ( sU), 1 I i I I, 1 I j I k , for all internal nodes u of T,, where I indicates the number of child nodes of u and sli indicates the time instance at which u starts to transmit a message Mj to its i-th child node. A broadcast scheme is said to be monotone if each node receives messages in the increasing order of their indices. It is easy to show that there is an optimal time monotone broadcast scheme in a tree network. Hence. we will consider only monotone broadcast schemes in this section.

One simple observation applies throughout this section: Consider a collection of p jobs, J = { J I , J2, . . . . J p } . Assume that each job Ji is associated with a due date di and the processing time of each job is one unit. A schedule of jobs on a single machine is said to be feasible if no job violates its due date. The starting time of a feasible schedule is the starting time of the job processed first in the schedule. We want to find the schedule with the latest starting time among all feasible schedules. This can be done by selecting the jobs in the nonincreasing order of their due dates and assigning the selected job into the latest idle time slot without exceeding the due date. For example, let ( 5 , 7, 7, 8 ,8 , 10, 12) be the due dates ofjobs JI , J 2 , . . . , 5,. A schedule with the latest starting time is depicted in Figure 2. I .

MULTIPLE MESSAGE BROADCASTING 255

In general, the latest starting time can be represented by the following recursive function f on the multiset D = { d , , d2, . . . , d p } of due dates:

i f D = { d l } ; J ( D ) = [ d l - I

min { d,,,;,, f ( D - { d,;,} ) } - 1 otherwise,

where d,,, is the smallest element in D. It is obvious that if.f( D ) -= 0, no feasible schedule exists: otherwise, f ( D) is the starting time of the schedule with the latest starting time among all feasible schedules.

Now, return to the original problem: Let J:', 1 I i I k , denote the task of sending message M, to node ti

from the parent of 11. For root node r, we imagine a virtual source of messages outside the network and define J: as the task of sending message M, to the root node from a virtual source. Define the due date a:'( t ) of the job J:' as the maximum time instance at which job J:' is completed over all monotone broadcast schemes with completion time no more than t , if such a scheme exists, and negative infinite otherwise. For any broadcast scheme, we assume that the job J: is completed just before the first time instance at which r starts to transmit message M , to any child node. According to this assumption. the due dates 6: ( t ) is well defined for all nodes u of the tree.

In this paper, we do not actually calculate the due dates of the tasks, while we calculate an upper bound of the due dates. This upper bound of 6: ( 1 ) will be defined later and denoted by A: ( 1 ) . Assume that we have calculated A: ( t ) for every node u and message M, . It is obvious that if, for any 11 and i, A:'(t) is negative, then no feasible monotone broadcast scheme with completion time of no more than t exists. However, the inverse of the statement is not trivial. In Lemma 2.2 and Corollary 2.1, we prove that the inverse also holds, i.e., if A:'([) 2 0 for every node zi and message M, , there exists a monotone broadcast scheme with completion time of no more than t , giving the necessary and sufficient conditions for the existence of a monotone broadcast scheme with completion time of no more than t .

In Definition 2.1, we will define A:'(f) formally. The basic idea behind the definition is as follows: For leaf node u , the due date 6 : ' ( t ) , 1 I i I k , is obviously no more than t . So, let A:' ( t ) = t , 1 I i I k , for every leaf node 11. Suppose that u I , u2, . . . , u1 are child nodes of node v and that A?([), I I i I k , 1 I j I I , which is assumed to be an upper bound of 6 g ( t ) , is given. How can we obtain a meaningful upper bound of the due date I S ; ( [ ) of job Jy , 1 I i 4 k? Note that job Jy must precede all jobs 53, I I j 5 1. By the monotonicity of the broadcast scheme, job J? must precede ( k - i) jobs J ] , where i + 1 I h 4 k ; thereby, job JP must precede all jobs J], i I h I k , 1 ~j 5 1. Hence, the due date 6 y ( t ) ofjob Jy is no more than the latest starting time of the schedule

ofthejobs {J]li I h I k , 1 s j I I ) , taking A](t)as their due dates. Definition 2.1 is just a summary of the discussion above.

Definition 2.1. Given a positive integer t and a node u in Tr, Ay(t) , 1 I i I k, is defined as follows:

ifu is a leaf node

D i ) otherwise, AY(t) =

whereD;= { A Y / ( t ) l i i h i k , I ~ j ~ I } a n d u ~ , u ~ , . . . . u1 are child nodes of u. Let A"(t ) denote a sequence (AY(t), A i ( t ) , . . . . APft)).

For instance, suppose that A'"'(t) = (10, 12, 15, 16), AIV2(f) = (7, 10, 14, 15), AH'(t) = (3, 7, 13, 17), and A " b ( t ) = (4 ,7 ,9, 16) for the child nodes w I , w2, w3, and w4 of a node M'. Then, A "'( t ) is equal to (2, 5. 8. 13).

Lemma 2.1. I f A i ( t ) is negative, there is no monotone broadcast scheme with completion time less than or equal to 1.

Pruuf.' Obvious from the fact that A { ( t ) is an upper bound of a { ( [ ) . To examine the inverse of Lemma 2.1, assume that A{( I ) is nonnegative. If each task JY is completed within time A: ( t ) by any monotone broadcast scheme S , then the completion time of S is no more than t , because AY(t) is f for all leaf nodes w . Now, we will prove the inverse of Lemma 2.1 by giving a mechanism for constructing such a broadcast scheme S. Since all messages are initially available for the root node r , it can be thought that tasks J:, I I i s k , are completed within time A;( 1 ) . For any node u and its child nodes u, , u2, . . . , uI, suppose that each of the tasks Jy , 1 I i I k , is completed within time Ay(t) . It is sufficient to construct a call sequence matrix S, = (so), 1 I i I I, 1 ~j I k , which satisfies the following three conditions:

Condition 1. All elements of S, are distinct; Condition 2. sij < AY(t) . for all i , j , 1 I i I I , 1 s.j

Condition3.min{suII 1 i 1 1 } z A Y ( t ) , 1 ~ j ~ k . I k;

The first condition represents the restriction that no node can send two or more messages simultaneously, and the last condition represents the restriction that node u can send messages that the node has received. The second condition is needed to apply the construction mechanism to the child nodes of u recursively.

We claim that Algorithm 2.1 produces such a call sequence matrix S,. Algorithm 2.1 constructs a matrix S, in a column-by-column manner from the k-th column down to the first column. For the j-th column, the jobs

256 KWON AND CHWA

{ J J l , Jy, . . . , J y } are considered in the nonincreasing order of the values of A?( t ) s . When J y is considered, so

is determined to the latest idle time such that sjj < A?( t ) . Figure 2.2 is an example of a call sequence matrix such as obtained by Algorithm 2.1.

/* Let q , v z , . . .,v1 be t he child nodes of a node v i n T, */ /* 2 denotes a set of non-negative integers */

Input :

Output :

a set of AY(t)s, l < i < f . 1 < j 5 k. a call sequence matrix S, = (sjj), 1 i 5 1 , 1 5 j 5 k.

Begin

N =(!I;

f o r j = k down t o 1 do

Let (CI, c2,. . . ,CI) be a sequence of indices such t h a t

a?(t) 2 arc.+l(t), 15 a 5 - 1; --- (1)

f o r i = 1 t o I do

Sclj = max{d E Zld < AT(2) and d 4 N}; --- (2)

N = N u {Sc, j } ; --- (3)

endf o

endf o

End.

Algorithm 2.1. Constructing call sequence matrix.

Lemma 2.2. For any node v and its child nodes v l , v2. . . . , vl in T,, Algorithm 2.1 correctlv produces a call sequence matrix S, satisfiing Conditions 1. 2, and 3.

Proof. Conditions 1 and 2 obviously hold. Assume that Condition 3 does not hold, i.e., assume that there exists an integerj' such that min { so' I 1 I i I I } < A>( t ) . Let a = min{sotI 1 I i I I } , and let (s,,,~,, smzr . . . , s,,+,~) be a maximal sequence such that sPlq, = a, sm2 = a + 1, ..., s p A x = a + x - I , 1 ~ p ~ s l , j ' ~ q ~ ~ k , I I C I X . It must be that A%( t ) I a + x , I I c I x ; otherwise. one of sPdcs, 1 I c I x, must be at least a + x by the nature of the algorithm. Hence, f( { A z l ( t ) , A p ( t ) , . . . . A 2 ( t ) 1 ) 2 f( D j t ) is at most a, which contradicts to the assumption.

Corollary 2.1. There exists a monotone broadcast scheme with completion time of no more than t if A{( t ) is nonnegative.

Theorem 2.1. For any tree T, and a node v , the k-message broadcast time tk(v, T ) is no more than t if and onlv if A;( t ) is nonnegative.

Now, consider the problem of finding the minimum broadcasting time fk( r , Tr) . I t is obvious that Ay( t ) + c = A: ( I + c ) for any positive integer c. So, we can obtain the minimum integer t such that A ; ( t ) is nonnegative by calculating A;(to) for any fixed value lo. Moreover, we can construct the call sequence matrix for each node ex- plicitly by Algorithm 2.1.

Theorem 2.2. We can,find a broadcast scheme with min- irniim completion time,for an arbitrary tree T, in O(k2n

Matrix of A','s S,

Fig. 2.2. Example of call sequence matrix.


+ kn log n ) steps. where n is the number of nodes in T, and k is the number of messages.

Proof: Consider the time required to execute Algo- rithm 2. l for any node u which has l child nodes. State- ment ( 1 ) requires O( I log I) steps for each execution. Statements (2 ) and (3) can be implemented to require O( kl) steps for I inner loop executions. Summing up, the time required over all nodes is O(k2n + kn log n).

3. BROADCASTING IN HYPERCUBES AND COMPLETE GRAPHS

In this section, we present two k-message broadcast schemes, one for an m-dimensional hypercube Qm and another for an n node complete graph K,,. The algorithm for hypercubes is just another representation of the one presented in [14]. The completion time of these two schemes match the known lower bounds m + k - 1 and Hog n l + k - 1, respectively.

3.1. Broadcasting in Hypercubes

Each node of an mdimensional hypercube Qm is given a distinct rn-bit label I = 1112 0 I,,,, such that any two nodes connected by a link have labels that differ in exactly by one bit. The number of bits in which labels x and y differ is denoted by hd(x, y); this is the Hamming distance between the nodes. A link connecting two nodes has dimension i if the labels of the nodes differ only in the i-th bit. Let u’ denote the neighbor of the node u connected

by a link of dimension i and V denote the node where hd( u, 3 ) = m. A k-dimensional subcube (k-subcube) of Qm, where k I m, is a subgraph of Qm which is a k- dimensional hypercube. Especially, define Q’ as an (m - 1 )-subcube of Q? induced by all nodes of which the i- th bit is zero and Q’ as the (m - 1 )-subcube induced by all nodes of which the i-th bit is one.

Since the hypercube is a node symmetric graph, it is sufficient to describe a broadcast scheme for a specific node as an originator. In our presentation, node r = 00 * - - 0 is taken as an originator.

It is well known that t l ( r , Qm) = m. The broadcast scheme performing this consists of communication over the edges of dimension i in each of the steps i = 1,2, . . . , m. This scheme establishes a binomial spanning tree

The broadcast scheme MCAST for broadcasting k messages M I , M 2 , . . . , Mk from the node r in Qm is sum- marized in Scheme 3.1. In our presentation, step i actually refers to the time interval [ i - 1, i]. In each time step i, 1 I i 5 m + k - 1. of MCAST, the messages are exchanged bidirectionally between pairs of nodes connected through the edges of dimension ( i - 1 ) mod m + I . In each communication step, every node but r transmits the message with the highest index among the messages that they have. The originator r injects a message M , at step i, I I i I k, and a message Mk at every steps j , k s j I m + k - 1. The process of MCAST can be represented by k edge labeled trees as depicted in Figure 3.1, each of which represents the paths traversed by a message. The edge labels indicate the time at which the communication occurs.

of Q,,,.

/* To broadcast k messages M I , Mz,. . . , Mk from T = 00.. .O in Qm */

Begin

for t = l t o m + k - 1 do

d = ( t - 1) mod m + 1;

for every node v having any message do

Select a message Mi such that, if v = T , then i = min{t, k}, otherwise, i = max{jI node v has the message Mj};

Send M; t o vd;

endf o

endf o

End.

Scheme 3.1 MCAST

258 KWON AND CHWA

01 I

T3

Fig. 3.1. Tree representation of Scheme 3.1

Lemma 3.1. The scheme MCAST broadcasts k me.wage.7 M I MZ. . . . , MA ,from a node r to all w her nodes correctlv.

Prmt Focus on message M I , I I i I k - 1, and let i' = ( i - 1 ) mod m + 1. At step t = i, the node r transmits message MI to a node r". During the steps between f = i + 1 and t = i + m - I , no node in the subcube Qi' may have a message with an index larger than i, since r is in Q" and no message can cross the edges of dimension i'. Hence, after step f = i + m - 1. M I reaches all nodes in Qj'and, also, at step t = i + m. all nodes in Qnr. Message MA is emitted from r at step k and transmitted through the edges of dimension ( j - I)mod m + 1 in each of steps j = k , k + I , . . . , k + m - 1. This process finds a binomial spanning tree rooted at r .

In the next subsection, we construct a broadcast scheme for a complete graph based on the scheme MCAST. Before proceeding, let us review some properties of the scheme MCAST. Let Y i ( I ) denote the set of nodes of Qm that receives message Mi at step t by MCAST, and let L, denote the set of nodes that receives message M , at step in

+ i when i # k (at step m + k - 1 when i = k). i.e.. L, = Y i ( m + i), 1 I i I k - 1 and Lk = Y,(m + k - 1 ) . We mean by f; the leaves of the i-th tree in Figure 3.1. In Lemma 3.1, we actually proved the following three primitive facts which are to be used in the next subsection:

1. Each message M i , 1 I i I k , is emitted from node r

2. Each message M , , 1 5 i I k, is received by all nodes at step i;

by step m + i (by step m + k - 1 when i = k);

3. Any node which receives message Mi at step m + i (at step m + k - 1 when i = k) is in Q(i-')modmfl (Q((P-2)mddnl+l when i = k ) .

3.2. Broadcasting in Complete Graphs Suppose that there is a broadcast scheme S for any graph G. Scheme S can be viewed as a set of communication tasks. Each task T E S is specified as a quadruple ( u , w , i, t ) which represents the task of sending message Mi from node u to ,vat time step t . Let Hbe another graph. Assume that the number of nodes of H is less than that of G. We are concerned with the way of constructing a broadcast scheme for H from the scheme S.

Let 4 be a surjection from the nodes of G onto the nodes of H which preserves adjacency, i.e., if u and u are adjacent in G, then either C#J( u) = 4( u ) or 4( u ) and C#J( u ) are also adjacent in H. Let S, denote the transformed scheme { ( 4 ( u ) , 4(w), i, t ) l ( u , w, i, t ) E S and 4 ( u ) # $I( M')} . We say that is out-conflict free for S , if there are no pairs of communication tasks ( u l , 1 1 , i, 1 ) and ( u2, ,r . ,j ,t) inSsuchthat6(ul)=9(u,)and {q5(uI)) r l {&(u), $( \I*) } = 0. Similarly. we say that 4 is in-conflict free for S . if there are no pairs of communication tasks ( 2 4 , u l , i, I ) and ( w, u2J, t ) in Ssuch that 4( u l ) = 4( u 2 ) and { 4( u , ) } n { 4( 1 1 ) . q5( w)} = 0. If, for any broadcast scheme S in G, there is an adjacency preserving and in/out-conflict free surjection 4 from the nodes of G to the nodes of H , then S, becomes a broadcast scheme which works in H and has the same completion time with S. In our concern, G is an m-dimensional hypercube and H is a complete graph with n nodes where 2"-' < n < 2". Since His a complete graph, we can neglect the adjacency preservation of the surjection.

In the following, we first define a surjection +x: V ( Q n l ) + V ( K,,) which maps each node of Q, onto a node of K,. But, unfortunately, the function 4x is not conflict free for MCAST. Hence, we will modify the scheme MCAST slightly. The modified scheme does no longer work for hypercubes. Indeed, the modified scheme works for the complete graph with 2" nodes. Finally, we will prove that the function 4~ is in/out-conflict free for the modified scheme. Before defining function &, observe a simple fact on the scheme MCAST, which gives some insight into the definition:

Fact 1. A node u of Qn, is in Li .for any i. 1 I i I k, if and onlv if V is not in Li .

Proqf.' Immediate from the fact that if u E Q', then 3 E Q', and vice versa.

Fact I means that, in MCAST, no pairs of node u and 6 either send or receive any message Mi simultaneously. We can prevent some cases of in/out-conflict by mapping nodes u and V onto a node of K,,.


Definition 3.1. Let X be a set 0f2m - n nodes of Qm with the property that i f u E X; then, V 4 X . Thejiinction dx

as follows: The nodes in V ( Qm) - X are mapped onto the nodes of K, by an arbitrary one-to-one mapping and the node x E X is mapped onto the node of K,, onto which

Fact 3. Suppose that there are two tasks (w, u, i, t ) and ( u , V , j , t ) in MCAST. Then, u E Lj or V E Lj.

- u(,- I )rnodm+ I and -

contradicts Fact 2.

from the A'ode set of Qm onto the node set 0 f K n is defined proo, suppose that 4 f i and 3 4 L~. since = V(,- I I , then w = U, which

rn node X is mapped.

Since 2m-1 < n < 2", we can always find such a set X. It is not difficult to confirm that dX is not conflict free for the scheme MCAST. Let us see Fig. 3.1. Suppose that 01 1 E X , i.e., nodes 01 1 and 100 are mapped onto a node of K,, by dx. There are two tasks, (01 1, 010, 1, 4 ) and ( 100, 101,3,4), which constitute an out-conflict. The following three facts give useful properties of the conflicts that may occur:

Fact 4. Suppose that there is a task (u , w, i, t ) such lhat w E L; in MCAST. Then, there is another task (W, U , i, I ) .

ProqK Immediate from the Fact 1 and the fact that w = u " a n d V = ( G ) " , w h e r e t ' = ( t - l ) m o d m + 1. rn

Suppose that two nodes u and ij are mapped onto a node of K,, by dx, i.e., either u E X or V E X, and that there occurs an out-conflict by two tasks (u , u, i, t ) and (V, w, j , t ) . From Fact 2, we can assume that. without loss of generality, zi E f i . We know that, from the Fact 4, there is another task (6 , 3, i, t ) , where 3 is also in L j .

Fact 2. Suppose that there arc' two tasks ( u , M', i , t ) and (3. 1 1 , j , I ) in MCAST. Then, either M I E Li or ti E Lj.

Prmf.' From Fact 1, i # j . Suppose, without loss of generality, that i < j and that u = uIu2. . .unl E Ti( 1) and V = E l i & * * -V,,, E Tj(I'). Then, I + 1 I t I m + i - I and 1'+ 1 I t , i.e., Is m + i - 2 and I' I m + i - 2. As the node r(i-1 )modJ?l+ I , which received the message Mi from r at step i , is in Q("2)rnO'"''+1 and no edges of dimension ( i - 2)mod m + 1 are used during the steps between i + 1 and i + m - 2, it must be that u

= 1. This implies that any edge of dimension ( i - 2)mod m + 1 is used for transmission of message Mj during the steps between j and I' I m + i - 2, which contradicts to

E ~ ( i - Z ) r n o d m + I * Thus, u( i -~)rnodni+l = 0 and i(i-2)rnm.1m+1

the preceding observation. rn

We modify the scheme MCAST by substituting two tasks (u , ti,i,~)and(ii,V,i,t)bythetasks(u,i~,i,t)and(zi, ti, i , t ) . From the definitions of the conflict, we can see that the new tasks (u, V, i , t ) and ( U , zi, i , t ) do not con- tribute any types of conflict at all. In-conflict cases can be done in similar ways.

Now, let us formally describe the modified scheme. The scheme MCAST2 consists of all task 7 = ( u , u, i, t ) E MCAST such that u 4 Li and task T' = ( u , U , i, t ) such that ( 1 1 , M', i , t ) E MCAST and w E f;. Figure 3.2 depicts an example of the tree representation of the modified scheme. The modified broadcast scheme is sum- marized in Scheme 3.2 in algorithmic form.

/* To broadcast k messages M I , Mz, . . ., Mk from T */ Begin

For t = l t o m + k - 1 do

d = ( t - 1) mod m t 1 ;

For every node v having any message do

Select a message M; such that i = min{t,k} i f v = T ,

i = max{il node v have message M i } , otherwise;

If t = m + i and i # k or t = m + k - 1 and i = k

Send message Mj to V ; else

Send Mi t o v d ; endf o

endf o End.

Scheme 3.2 MCASTP

260 KWON AND CHWA

Ooo

Ooo TI T2

Ooo Ooo

101 110 01 I 011

6 111

6 010 001

d Ooo

T3 =4

Fig. 3.2. Tree representation of modified scheme.

Lemma 3.2. The jimction @x is in/ozit-con/licts .bee for the scheme MCASTL .

ProoL Suppose that the occurrence of an out-conflict. i.e., there are two tasks ( u, 2 1 , i, t ) and (C, w, j . 1 ) for the nodes u, b such that { u, b n X f 0. From Fact 1. it is obvious that i # j. From Fact 2, either ZI E -Ci or M' E -Cj. Without loss of generality, assume that ZI E L j , which means that zi = b. It is contradictory to the definition of out-conflict. Moreover, from Fact 3 and a similar argument, we can prove that no in-conflict occurs. H

Theorem 3.1. For every positive integer n. tk( n ) = flog n l + k - 1.

4. NEW MINIMAL BROADCAST GRAPHS

In this section, we investigate minimal broadcast graphs with a relatively small number of edges. It is well known that the mdimensional hypercube is a graph with the minimum number of edges among all graphs with 2"' nodes in which a message can be broadcasted within m units of time [ 131. Hence, an m-dimensional hypercube is also a minimal broadcast graph with the minimum number of edges among all graphs with 2 "' nodes.

Now, consider the case 2"-' c n < 2"'. In broadcast scheme MCAST2, it is not that all edges of a complete graph are used. By classifying the edges used, we can construct a new class of minimal broadcast graphs. Consider the scheme MCASTZ. Two types of edges are used. The

first is the edges of hypercube Q,, and the other is the edges between nodes of Hamming distance m.

Now, define H 2 m ( la) as a graph with 2"' nodes, each of which is given an m bits binary label, and edges between two nodes 2 1 , u such that hd( 21, u ) = I or hd(u, u ) = m. The scheme MCAST2 works for the graph Hzm( 0). Let Xo be an arbitrary 2"' - n nodes set of H z m ( 0) such that r = 00- * - 0 4 Xo, and if u E Xo, then b 4 X, . Another graph H,( X o ) with n nodes is defined by contracting each pairs of nodes u E Xo and b of H 2 4 0) and labeling the contracted node with 3.

In the following, we will prove that the graph H,(Xo) is a minimal broadcast graph. Since the graph H,,(X,) is no longer node symmetric, we need to show that, for every node u of H n ( X o ) , there is an adjacency preserving and in/out conflict free surjection 4: V ( H z m ( 0 ) ) +

L'( H,(Xo) ) such that +(OO. - - 0 ) = u. Let 0 denote the binary exclusive-or operator between

two m-bits binary strings and. for any set of strings Y and a binary string u, let Y 0 u denote the set of strings { y 0 u l y E Y ;. Consider any node u of H,(Xo) . Let X , denote Xu 0 u. It is obvious that 00 - - 0 4 X , (since u 4 X o ) . Moreover, for any y. if J' E Xu then

Let 4irU: V ( Ifzrn( 0) ) + V ( H,(Xo)) be a function such that 4:vb( 1 1 ) = ZI 0 u if ZI 0 u 4 Xo and 4$,(u) = ZI 0 u otherwise. It is obvious that $5." is an adjacency preserving surjection.

4 X, .

Lemma 4.1. The jmction 4i,v is in/out-conflict .fie(> .fin- MCAST2.

Prool.' Suppose that the occurrence of an out-conflict, i.e., there are two tasks ( 2 1 , HI, i , t ) and (x, y, j , t ) in MCAST2 such that @i,v(z~) = 4$,"(x) and 4 i ( z i ) # $J:~~( w) and 45r( 2 1 ) # r#~i.,,(y). Suppose, without loss of generality, that 4$"( 1 1 ) = ZI 0 u and 4$"(x) = x 0 u = X 0 u. This implies that ZI = 2. A similar argument as in Lemma 3.2 leads us to a contradiction. In-conflict cases can be done similarly.

Lemma 4.2. A graph H,(Xo) is a minimal broad- casf graph.

Theorem 4.1. For any positive integer n, fhere is an n node minimal broudcast graph with no more than ([log nl + I )2"%"'/2 - 2""gn1 + n edges.

5. CONCLUSION

In this paper, we have considered the problem of broadcasting multiple messages in trees, hypercubes, and complete graphs under a reasonable assumption about the communication capability of networks and characterized a class of minimal broadcast graphs with no more than (Hog nl + 1 ) 2' log nl / 2 edges.


It is unlikely that the upper bound on the number of edges of the minimal broadcast graph presented in Section 4 is very tight. In general, we expect that the number of edges of the minimal broadcast graph with n vertices, where n is between 2 mp ' and 2" and is very close to 2 "- I , is much smaller than the number of edges of (rn - 1 )- dimensional hypercubes. But, we have no idea for im- proving the upper bound. It would be interesting to find minimal broadcast graphs with fewer edges than the graphs proposed.

REFERENCES

A. Bar-Noy and S. Kipnis, Broadcasting multiple messages in simultaneous sendlreceive systems. Proceedings qf the 5th IEEE Symposizim on Parallel and Distrihzited Pro- cessing ( 1993) 344-347. A. Bar-Noy. S. Kipnis. and B. Schieber. Optimal multiple message broadcasting in telephone-like communication systems. Proceedings of the 61h IEEE Sjwposizrm on Purallel and Distributed Procewing ( 1994) 2 16-223. D. P. Bertsekas and J. N. Tsitsiklis, Parallel and Distrib- uted Cornpiitation: Nzimericul M(>thods. Prentice-Hall. Englewood Cliffs, NJ (1989). S. Chau and A. L. Liestman, Constructing minimal broadcast networks. J. Comhin. In/?). S.19.st. Sci. 10 ( 1985) 110-122. P. Chinn, S. Hedetniemi, and S. Mitchell. Multiple-message broadcasting in complete graphs. Procecding.s of the 10th SE Conlerence on Comhinatorics. Graph Theory and Coinpu~ing. Utilitas Mathematica. Winnipeg ( 1979) 25 I - 260. E. J. Cockayne and A. Thomason, Optimal multi-message broadcasting in complete graphs. Proceedings uf the 1 1 f h SE Conf2renee on Combinatorics. Graph Theory and Compziting. Utilitas Mathematica. Winnipeg ( 1980) 18 I - 199.

A. Farley, Broadcast time in communication networks. SIAM J. Appl. Math. 39 ( 1980) 385-390. A. Farley, Minimal broadcast networks. Nctwrks 9 (1979) 313-332.

A. Farley, S. T. Hedetniemi. A. Proskurowski. and S. Mitchell, Minimum broadcast graphs. Discr. Math. 25

M. Carey and D. Johnson, Compziters and Intractability: A Guide 10 the Theory of NP-Complc~tene.~.s. Freeman, San Francisco ( 1979). L. Gargano and U. Vaccaro. On the construction of minimal broadcast networks. Neiic~orks 19 ( 1989) 673-689. M. Crigni and D. Peleg, Tight bounds on minimum broadcast networks. SIAM J. Disc. Math. 4 ( I99 I ) 207- 222. S. M. Hedetniemi. S. T. Hedetniemi. and A. L. Liestman. A survey ofgossiping and broadcasting in communication networks. Netwrks 18 ( 1988) 3 19-349. C. T. Ho, Optimal broadcasting on SlMD hypercubes without indirect addressing capability. J. Parulld Disirih. Comput. 13 (1991 ) 246-255.

S. L. Johnsson and C. T. Ho. Optimum broadcasting and personalized communication in hypercubes. IEEE Tr. Comput. C-38(9) (1989) 1249-1268. R. M. Karp, A. Sahay, E. E. Santos. and K. E. Schauser, Optimal broadcast and summation in the LogP model. Proceedings oft he 5th '4 CM S!vnposiiim on Parallel Al- gorithms and Archirc~ctzir~~.s ( 1993) 142-153. J. M. Koh and D. W. Tcha, Information dissemination in trees with nonuniform edge transmission times. IEEE Tr. Coinpiit. C-40 ( I99 I ) 1 174- 1 177.

J. M. Koh and D. W. Tcha, Local broadcasting in a tree under minsum criterion. Nmwrks 23 ( 1993) 7 1-79. A. Proskurowski. Minimum broadcast trees. IEEE Tr. Cornpiit. C-30 ( 198 I ) 363-366. P. J. Slater. E. Cockayne. and S. T. Hedetniemi. Infor- mation dissemination in trees. SL4M J. Coinpiit. 10

J. A. Ventura and X. Weng. A new method for constructing minimal broadcast networks. Networks 23 ( 1993) 481-497.

(1979) 189-193.

( 1981 ) 692-701.

Received February 22, 1994 Accepted July 3, 1995

multiple message broadcasting in communication networks

Documents