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

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), satisfies [log₂ n]≤B _min(G _n)≤[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 G _n of n nodes, the one-way (two-way) gossip complexity R(G _n) (R ²(G _n)) 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 1.5log₂ n−log₂log₂ n−0.5log₂ h−2, and the two-dimensional grid of n nodes has the gossip complexity 1.5log₂ n−log₂log₂ 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.

This work was partially supported by grants Mo 285/4-1, Mo 285/9-1 and Me 872/6-1 (Leibniz Award) of the German Research Association (DFG), and by the ESPRIT Basic Research Action No. 7141 (ALCOM II).

This author was partially supported by SAV Grant No. 88 and by EC Cooperation Action IC 1000 Algorithms for Future Technologies.

This author was supported by the Ministerium für Wissenschaft und Forschung des Landes Nordrhein-Westfalen.