Beyond the worst-case bisection bound: Fast sorting and ranking on meshes

Abstract

Sorting is an important subroutine in many parallel algorithms and has been studied extensively on meshes and related networks. If every processor of an n × n mesh is the source and destination of at most k elements, then sorting requires at least k k; n/2 steps in the worst-case, and simple algorithms have recently been proposed that nearly match this bound. However, this lower bound does not extend to non-worst-case inputs, or weaker definitions of sorting that are sufficient in many applications. In this paper, we give algorithms and lower bounds for several such problems. We first present a very simple scheme for k-k routing that performs optimally under both average-case and worst-case inputs. As an application of this scheme, we describe a simple k-k sorting algorithm based on sample sort that nearly matches this bound. The main part of the paper considers several 'sorting-like' problems. In the ranking problem, the ranks of all elements have to be determined, but there is no requirement about their final positions. We describe an algorithm running in time (1 +o(l)) k n/4 steps, which is nearly optimal under the considered model of the mesh. We show that integer versions of the sorting and ranking problems, where keys are drawn from {0,…, m — 1}, can be solved asymptotically faster than the general problems for small values of m. A related problem, the excess counting problem, can be solved in O(n) steps in many interesting cases.