Optimal Algorithms for Broadcast and Gossip in the Edge-Disjoint Path Modes

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 graphGnofnnodes, the complexity of broadcast inGn,Bmin(Gn), satisfies ?log2n??Bmin(Gn)??log2n?+1. The complete binary trees meet the upper bound, and all graphs containing a Hamiltonian path meet the lower bound. 2. For each connected graphGnofnnodes, the one-way (two-way) gossip complexityR(Gn) (R2(Gn)) satisfies?log2n??R2(Gn)?2·?log2n?+1,1.44...log2n?R(Gn)?2·?log2n?+2.All these lower and upper bounds are shown to be sharp up to 1. 3. All planar graphs ofnnodes and degreehhave a two-way gossip complexity of at least 1.5log2n?log2log2n?0.5log2h?8, and the two-dimensional grid ofnnodes has the gossip complexity 1.5log2n?log2log2n±O(1); i.e., two-dimensional grids are optimal gossip structures among planar graphs of bounded degree. Some upper bounds are also obtained for the one-way mode. 4. Thed-dimensional grid,d?3, ofnnodes has the two-way gossip complexity (1+1/d)·log2n?log2nlog2n±O(d).

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