Randomized parallel list ranking for distributed memory multiprocessors

Abstract

We present a randomized parallel list ranking algorithm for distributed memory multiprocessors, using a BSP type model. We first describe a simple version which requires, with high probability, log(3p)+log ln(n)=O(logp+log logn) communication rounds (h-relations withh=O(n/p)) andO(n/p)) local computation. We then outline an improved version that requires high probability, onlyr⩽(4k+6) log(2/3p)+8=O(k logp) communication rounds wherek=min{i⩾0 |ln(i+1)n⩽(2/3p)2i+1}. Notek<ln*(n) is an extremely small number. Forn andp⩾4, the value ofk is at most 2. Hence, for a given number of processors,p, the number of communication rounds required is, for all practical purposes, independent ofn. Forn⩽1, 500,000 and 4⩽p⩽2048, the number of communication rounds in our algorithm is bounded, with high probability, by 78, but the actual number of communication rounds observed so far is 25 in the worst case. Forn⩽10010100 and 4⩽p⩽2048, the number of communication rounds in our algorithm is bounded, with high probability, by 118; and we conjecture that the actual number of communication rounds required will not exceed 50. Our algorithm has a considerably smaller member of communication rounds than the list ranking algorithm used in Reid-Miller’s empirical study of parallel list ranking on the Cray C-90.(1) To our knowledge, Reid-Miller’s algorithm(1) was the fastest list ranking implementation so far. Therefore, we expect that our result will have considerable practical relevance.