TIGHT BOUNDS FOR BROADCASTING IN THE LINEAR COST MODEL

Abstract

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 defined as α + Lτ, where α stands for a start-up time while Lτ represents the cost of sending L bits of data in a channel with bandwidth 1/τ. In this model, the communication time of any algorithm [Formula: see text] is expressed as the sum [Formula: see text], where [Formula: see text] is the number of rounds and [Formula: see text] the transmission cost of the algorithm. In order to design an efficient algorithm realizing a given communication pattern, it appears that minimizing [Formula: see text] and [Formula: see text] are antinomic goals. We study this trade-off 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 KN, 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.