Tight Bounds for Broadcasting in the Linear Cost Model

This work considers broadcast protocols made of successive communication rounds in the linear cost model: the time needed to send a message of length L is deened as + LL. In this model, the communication time of any algorithm A is expressed as the sum R A + T A , where R A is the number of rounds and T A the transmission cost of the algorithm. In order to design an eecient algorithm realizing a given communication pattern, it appears that minimizing R A and T A are antinomic goals. We study this trade-oo issue for broadcast protocols. Surprisingly, such a general theoretical study has almost never been done. In the literature, only the two opposite issues are actually considered: minimizing the number of rounds in the case of short messages, or minimizing the transmission cost in the case of large messages. Our results concern the fully-connected N-nodes network K N , with N = (k + 1) T , in the bidirectional k?ports mode. We derive tight bounds on the communication time for broadcasting in T + r rounds, our lower bounds holding for any network. An extended abstract of this paper written by the second author has been accepted in the 5th International Colloquium on Structural Information and Communication Complexity (SIROCCO'98). SLOOP is a joint project with the CNRS and the University of Nice-Sophia Antipolis (I3S laboratory). RRsumm : Ce travail considdre des protocoles de diiusion se ddroulant en tapes successives sous le moddle linnaire du coot de communication : le temps nncessaire pour envoyer un message de longueur L est dddni par + LL. Dans ce moddle, le temps de communication d'un algorithme s'exprime par la somme R A + T A , oo R A est le nombre d''tapes et T A le coot de transmission de l'algorithme. En vue de concevoir un algorithme eecace rralisant un schhma de communication donnn, il apparaat que minimiser R A et T A sont des objectifs antinomiques. Nous tudions le compromis nncessaire entre ces deux coots dans le cadre de la diiusion. Il s'avre qu'une telle tude thhorique ggnnrale n'a pratiquement jamais tt entreprise. Dans la litttrature, seulement les deux cas extrrmes sont considdrrs : minimisation du nombre d''tapes dans le cas de messages courts, ou minimisation du coot de transmission dans le cas de messages longs. Nos rrsultats concernent le rrseau complet K N , avec …