Analysis of Fully Distributed Splitting and Naming Probabilistic Procedures and Applications

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 1 − o(n − 1) (resp. with probability 1 − o(n − c) for any c ≥ 1) 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 1 − o(n − 1) and the expected value of the size of labels is also O(logn). In the second case, the size of labels is \(O\left((\log n)(\log^* n)^2 \right)\) with probability 1 − o(n − c) for any c ≥ 1; their expected size is \(O\left((\log n)(\log^* n) \right)\).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 1 − o(n − 1).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 1 − o(n − 1) or with probability 1 − o(n − c) for any c ≥ 1. 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.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 1 − o(n − 1). (Proofs are omitted for lack of space).