Optimal algorithms for broadcast and gossip in the edge-disjoint path modes

Abstract

The communication power of the one-way and two-way edgedisjoint path modes for broadcast and gossip is investigated. The complexity of communication algorithms is measured by the number of communication steps (rounds). The main results achieved are the following: 1. For each connected graph Gn of n nodes, the complexity of broadcast in Gn, Bmin(Gn), satisfies [log2 n] ≥ Bmin(Gn) ≥ [log2 n] + 1. The complete binary trees meet the upper bound, and all graphs containing a Hamiltonian path meet the lower bound. 2. For each connected graph Gn of n nodes, the one-way (two-way) gossip complexity R(Gn) (R2(Gn)) satisfies (Formula presented). All these lower and upper bounds are tight.3. All planar graphs of n nodes and degree h have a two-way gossip complexity of at least 1.5 1og2 n - log2 log2 n - 0.5 log2 h - 2, and the two-dimensional grid of n nodes has the gossip complexity 1.5 log2 n- log2 log2 n ± O(1), i.e. two-dimensional grids are optimal gossip structures among planar graphs. Similar results are obtained for one-way mode too. Moreover, several further upper and lower bounds on the gossip complexity of fundamental networks are presented.