SWAT 1994: Algorithm Theory — SWAT '94 pp 219-230

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

Extended abstract
• Juraj Hromkovič
• Ralf Klasing
• Walter Unger
• Hubert Wagener
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 824)

## 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. 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. 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. 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.

Moreover, several further upper and lower bounds on the gossip complexity of fundamental networks are presented.

## Keywords

communication algorithms parallel computations

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## Authors and Affiliations

• Juraj Hromkovič
• 1
• Ralf Klasing
• 1
• Walter Unger
• 1
• Hubert Wagener
• 1