Gossip algorithms for aggregation have recently received significant attention for sensor network applications because of their simplicity and robustness in noisy and uncertain environments. However, gossip algorithms can waste significant energy by essentially passing around redundant information multiple times. For realistic sensor network model topologies like grids and random geometric graphs, the inefficiency of gossip schemes is caused by slow mixing times of random walks on those graphs. We propose and analyze an alternative gossiping scheme that exploits geographic information. By utilizing a simple resampling method, we can demonstrate substantial gains over previously proposed gossip protocols. In particular, for random geometric graphs, our algorithm computes the true average to accuracy @math using @math radio transmissions, which reduces the energy consumption by a @math factor over standard gossip algorithms.
In @cite_18 the authors consider the related problem of computing the average of a network in . They propose a distributed algorithm to solve this problem and show how it can be related to cover times of random walks on graphs.
@cite_18
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We pose a new and intriguing question motivated by distributed computing regarding random walks on graphs: How long does it take for several independent random walks, starting from the same vertex, to cover an entire graph? We study the cover time - the expected time required to visit every node in a graph at least once - and we show that for a large collection of interesting graphs, running many random walks in parallel yields a speed-up in the cover time that is linear in the number of parallel walks. We demonstrate that an exponential speed-up is sometimes possible, but that some natural graphs allow only a logarithmic speed-up. A problem related to ours (in which the walks start from some probablistic distribution on vertices) was previously studied in the context of space efficient algorithms for undirected s-t-connectivity and our results yield, in certain cases, an improvement upon some of the earlier bounds.