The parallel simplicity of compaction and chaining

Given a set of values x1, x2, ... xn, 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. Parallel algorithms for these problems were considered by Hagerup and Nowak (ICALP 1989). This paper improves their results by introducing a technique involving perfect hash functions. This 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. In addition, 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 can be solved in constant time with n processors when k