Randomized parallel list ranking for distributed memory multiprocesors

Abstract

We present a randomized parallel list ranking algorithm for distributed memory multiprocessors. A simple version requires, with high probability, log(3p)+log ln(n)=O(log p+log log n) communication rounds (h-relations with h=O(n/p)) and O(n/p) local computation. An improved version requires, with high probability, only r ≤ (4k+6) log (2/3p)+8=O(k log p) communication rounds where k= min{i ≥ 0¦ln(i+1)n ≤ (2/3p)2i+1}. Note that k < ln*(n) is an extremely small number. For \(n \leqslant 10^{10^{100} }\)and p ≥ 4, the value of k is at most 2. For a given number of processors, p, the number of communication rounds required is, for all practical purposes, independent of n. For \(n \leqslant 10^{10^{100} }\)and 4 ≤ p ≤ 2048, the number of communication rounds in our algorithm is bounded, with high probability, by 118. We conjecture that the actual number of communications rounds will not exceed 50.