Fast parallel generation of random permutations

Abstract

We study the problem of generating random permutations, i.e., of drawing random elements from the uniform distribution over the set of all permutations of a finite set. Our results are the following, where space bounds of O(n) are omitted.

(A) Random permutations of {1,...,n} can be computed on a CRCW PRAM either in constant expected time using O(nlog^(k) n) processors and O(nlog^(k) n) space, where k ε ℕ is arbitrary but fixed, or optimally in O(log*n) expected time using O(n/log*n) processors;

(B) Disregarding a small failure probability, the computation of random permutations reduces to integer sorting. In particular, random permutations of {1,...,n} can be computed on an EREW PRAM either in O(log n) expected time using n processors, or in O((log n)^5/3(log log n)^1/3) expected time using O(n/log n) processors;

(C) Random permutations of {1,..., n} can be computed either on an EREW PRAM in O(log n) time using n processors and O(n ²) space, on an EREW PRAM in O((log n)²) time using O(n/log n) processors, on a CREW PRAM in O(log n log log n) time using O(n/log log n) processors,or on a CREW PRAM in O(log n log log log n) time using O(n/log log log n) processors and O(n ^1+ε) space, for arbitrary fixed ε>0.

Supported in part by the Deutsche Forschungsgemeinschaft, SFB 124, TP B2, VLSI Entwurfsmethoden und Parallelität, and in part by the ESPRIT II Basic Research Actions Program of the EC under contract No. 3075 (project ALCOM).