Optimal and sublogarithmic time randomized parallel sorting algorithms

Reviewer: Pradip K. Srimani

This paper describes the first optimal randomized parallel algorithm for the integer sorting problem, that is, sorting n integers in the range [1, n ]. The authors use a CRCW PRAM model of computation and prove that the product of the time and processor bounds of their algorithm is bounded above by a linear function of the input size. In addition, the authors present a deterministic sublogarithmic parallel algorithm for prefix sum computation and a randomized sublogarithmic parallel algorithm for random permutation generation. They also provide sublogarithmic time randomized algorithms for general as well as integer sorting. Although they prove that their algorithm for general sorting is optimal, it is not yet known whether their integer sorting algorithm is optimal. Note that all the sorting algorithms in this paper are unstable; stable versions of these algorithms are still open problems. To quote the authors, “it seems that parallel stable sort is inherently more complex than parallel unstable sort.” The paper is lucidly written and very readable; it contains ample pointers to the existing literature on relevant results in the theory of sorting. The idea of the assignment problem and the concept of coarse and fine sorting seem especially interesting. Anybody interested in the theory of parallel sorting would benefit from reading this paper.