Optimal Algorithms for Broadcast and Gossip in the Edge-Disjoint Path Modes (Extended Abstract)

The communication power of the one-way and two-way edge-disjoint 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 G n of n nodes, the complexity of broadcast in G n , B min (G n), satisses dlog 2 ne B min (G n) dlog 2 ne +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 G n of n nodes, the one-way (two-way) gossip complexity R(G n) (R 2 (G n)) satisses dlog 2 ne R 2 (G n) 2 dlog 2 ne + 1, 1:44 : : : log 2 n R(G n) 2 dlog 2 ne + 2. 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 log 2 n ?log 2 log 2 n ?0:5 log 2 h ?2, and the two-dimensional grid of n nodes has the gossip complexity 1:5 log 2 n ?log 2 log 2 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.