A Randomized BSP/CGM Algorithm for the Maximal Independent Set Problem

This paper presents a randomized parallel algorithm for the Maximal Independent Set problem. Our algorithm uses a BSP-like computer with p processors and requires that n+m p = (p) for a graph with n vertices and m edges. Under this scalability assumption, and after a prepro-cessing phase, it computes a maximal independent set after O(log p) communication rounds, with high probability, each round requiring linear computation time O(n+m p). The preprocessing phase is deterministic and important in order to ensure that degree computations can be implemented eeciently. For this, we give an optimal parallel BSP/CGM algorithm to the p-quantiles search problem, which runs in O(m logp p) time and a constant number of communication rounds, and could be of interest in its own right, as shown in the text. Nous pr esentons dans ce papier un algorithme randomis e parall ele pour la recherche d'un stable maximal d'un graphe Maximal Independent Set, MIS]. Notre algorithme est ecrit pour des machines parall eles de type BSP (gros grain) a p processeurs, telles que n+m p = (p) pour un graphe a n sommets et m ar^ etes. Sous cette contrainte de granula-rit e, il calcule un stable maximal en O(log p) rounds de communications, avec forte probabilit e, chaque round n ecessitant un temps de calcul de O(n+m p). L' etape initiale de placement des donn ees, n ecessaire pour un calcul eecace du degr e des nnuds du graphe, est d eterministe. Cette etape, qui pourrait s'av erer utile en soi pour d'autres algorithmes, est bas ee sur une recherche des p-quantiles, dont nous proposons une pa-rall elisation optimale : son temps d'ex ecution est O(mlog p p) avec O(1) phases de communications. Ce papier a et e soumis pour pr esentation a ESA'98. Abstract This paper presents a randomized parallel algorithm for the Maximal Independent Set problem. Our algorithm uses a BSP-like computer with p processors and requires that n+m p = (p) for a graph with n vertices and m edges. Under this scalability assumption, and after a pre-processing phase, it computes a maximal independent set after O(logp) communication rounds, with high probability, each round requiring linear computation time O(n+m p). The preprocessing phase is deterministic and important in order to ensure that degree computations can be implemented eeciently. For this, we give an optimal parallel BSP/CGM algorithm to the p-quantiles search problem, which runs in O(m log …