Lower Bounds on Systolic Gossip

Gossiping is an information dissemination process in which each processor has a distinct item of information and has to collect all the items possessed by the other processors. In this paper we derive lower bounds on the gossiping time of systolic protocols, i.e. constituted by a periodic repetition of simple communication steps. In particular, if we denote by n the number of processors in the network, then for directed networks and for undirected networks in the half-duplex mode any s-systolic gossip protocol takes at least g(s) log2 n time steps, where g(4) = 1:8133, g(6) = 1:5310 and g(8) = 1:4721. For the case s = 4 this result is improved to 2:0218 log2 n for directed butterflies of degree 2 and we show that the 2:0218 log2 n and 1:8133 log2 n lower bounds hold also respectively for undirected Butterfly and de Bruijn networks of degree 2 in the full-duplex case. Our results are obtained by means of new technique relying on two novel concepts in the field: the notion of delay digraph of a systolic protocol and the use of matrix norm methods.