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 Gn of n nodes, the complexity of broadcast in Gn, 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 graph Gn of n nodes, the one-way (two-way) gossip complexity R(Gn) (R2(Gn)) satisfies $$\begin{gathered}\left\lceil {\log _2 n} \right\rceil \leqslant R^2 (G_n ) \leqslant 2 \cdot \left\lceil {\log _2 n} \right\rceil + 1, \hfill \\1.44...\log _2 n \leqslant R(G_n ) \leqslant 2 \cdot \left\lceil {\log _2 n} \right\rceil + 2. \hfill \\\end{gathered}$$ . 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 1.5log2n−log2log2n−0.5log2h−2, and the two-dimensional grid of n nodes has the gossip complexity 1.5log2n−log2log2n±O(1), i.e. two-dimensional grids are optimal gossip structures among planar graphs. Similar results are obtained for one-way mode too.