On the Optimality of General Lower Bounds for Broadcasting and Gossiping

Abstract

In this paper we show that many general lower bounds on the broadcasting and gossiping time are optimal. In particular, let b(G) be the broadcasting time of a network G under the basic one-port model. The only lower bound on b(G) holding for every n vertices graph G is max$(\log_2 n, Diam(G))$, but the $\log_2 n$ factor cannot be achieved in bounded degree networks. In fact, let the parameter d be defined in undirected graphs as the maximum degree minus one and for directed graphs as the maximum out-degree. Then, in [ SIAM J. Discrete Math., 1 (1998), pp. 531--540; SIAM J. Discrete Math., 5 (1992), pp. 10--24] it has been proved that, for any graph G of parameter d, b(G) \geq \frac{\log_2 n}{\log_2 \xi}$, where $\xi$ is the largest real number such that $\xi^{d} -\xi^{d-1} - \xi^{d-2} -\cdots - \xi-1=0$. Since then many papers have proposed constructions of bounded degree networks having a small broadcast time [Proceedings of the 2nd International Euro-Par Conference (EUROPAR), Lecture Notes in Comput. Sci. 1123, Springer-Verlag, New York, 1996, pp. 313--324; IEEE Trans. Comput., 33 (1984), pp. 190--194], but so far the optimality of [SIAM J. Discrete Math., 1 (1998), pp. 531--540; {SIAM J. Discrete Math.}, 5 (1992), pp. 10--24] was still an open question. In this paper we prove that the above lower bound is tight, improving all the existing upper bounds by means of probabilistic methods. Namely, we show that for n arbitrarily large there exist families of n vertices graphs in which a uniformly drawn graph has broadcasting time as predicted by [SIAM J. Discrete Math., 1 (1998), pp. 531--540; SIAM J. Discrete Math., 5 (1992), pp. 10--24] with probability converging to 1. Moreover, we show that [SIAM J. Discrete Math., 1 (1998), pp. 531--540; SIAM J. Discrete Math., 5 (1992), pp. 10--24] is attained even in the case of gossiping and systolic gossiping in the full-duplex mode. Finally, new upper bounds on bounded-degree and systolic gossiping are also determined in the directed and half-duplex modes. While the systolic construction is tight and matches the lower bound of [Inform and Comput., to appear], we strongly conjecture that the bounded-degree result is optimal and that a corresponding matching lower bound is still to be proven.