Analysis of fully distributed splitting and naming probabilistic procedures and applications

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)1−o(n−1) (resp. with probability 1−o(n−c)1−o(n−c) for any c≥1c≥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(log⁡n)O(log⁡n) with probability 1−o(n−1)1−o(n−1) and the expected value of the size of labels is also O(log⁡n)O(log⁡n). In the second case, the size of labels is O((log⁡n)(log⁎⁡n)2)O((log⁡n)(log⁎⁡n)2) with probability 1−o(n−c)1−o(n−c) for any c≥1c≥1; their expected size is O((log⁡n)(log⁎⁡n))O((log⁡n)(log⁎⁡n)).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(log⁡n)O(log⁡n) messages with probability 1−o(n−1)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)1−o(n−1) or with probability 1−o(n−c)1−o(n−c) for any c≥1c≥1. The size of messages has the same value as the size of labels. The number of messages is O(mlog⁡n)O(mlog⁡n) for electing and computing a spanning tree; it is O(nlog⁡n)O(nlog⁡n) 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 ϵvϵv; the error probability ϵvϵv 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(nlog⁡n)O(nlog⁡n) with probability 1−o(n−1)1−o(n−1).