The Parallel Simplicity of Compaction and Chaining

Given a set of values x 1, x 2,..., x n , of which k are nonzero, the compaction problem is the problem of moving the nonzero elements into the first k consecutive memory locations. The chaining problem asks that the nonzero elements be put into a linked list. One can in addition require that the elements remain in the same order, leading to the problems of ordered compaction and ordered chaining, respectively. This paper introduces a technique involving perfect hash functions that leads to a deterministic algorithm for ordered compaction running on a CRCW PRAM in time O(log k/log log n) using n processors. A matching lower bound for unordered compaction is given. The ordered chaining problem is shown to be solvable in time O(α( k)) with n processors (where α is a functional inverse of Ackermann′s function) and unordered chaining is shown to he solvable in constant time with n processors when k < n 1/4− ϵ.