Elsevier

Theoretical Computer Science

Volume 584, 13 June 2015, Pages 115-130
Theoretical Computer Science

Analysis of fully distributed splitting and naming probabilistic procedures and applications

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Abstract

This paper proposes and analyses two fully distributed probabilistic splitting and naming procedures which assign a label to each vertex of a given anonymous graph G without any initial knowledge. We prove, in particular, that with probability 1o(n1) (resp. with probability 1o(nc) for any c1) there is a unique vertex with the maximal label in the graph G having n vertices. In the first case, the size of labels is O(logn) with probability 1o(n1) and the expected value of the size of labels is also O(logn). In the second case, the size of labels is O((logn)(logn)2) with probability 1o(nc) for any c1; their expected size is O((logn)(logn)).

We analyse a basic simple maximum broadcasting algorithm and prove that if vertices of a graph G use the same probabilistic distribution to choose a label then, for broadcasting the maximal label over the labelled graph, each vertex sends O(logn) messages with probability 1o(n1).

From these probabilistic procedures we deduce Monte Carlo algorithms for electing or computing a spanning tree in anonymous graphs without any initial knowledge and for counting vertices of an anonymous ring; these algorithms are correct with probability 1o(n1) or with probability 1o(nc) for any c1. The size of messages has the same value as the size of labels. The number of messages is O(mlogn) for electing and computing a spanning tree; it is O(nlogn) for counting the vertices of a ring. These algorithms can be easily extended to also ensure for each vertex v an error probability bounded by ϵv; the error probability ϵv is decided by v in a totally decentralised way.

We illustrate the power of the splitting procedure by giving a probabilistic election algorithm for rings having n vertices with identities which is correct and always terminates; its message complexity is equal to O(nlogn) with probability 1o(n1).

Keywords

Monte Carlo algorithm
Spanning tree computation
Counting
Election algorithm
Probabilistic analysis
Splitting and naming

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