The Power of Collision: Randomized Parallel Algorithms for Chaining and Integer Sorting

We address the problem of sorting n integers each in the range {0, ..., m - 1} in parallel on the PRAM model of computation. We present a randomized algorithm that runs with very high probability in time O(lg n/lg lg n + lg lg m) with a processor-time product of O(n lg lg m) and O(n) space on the CRCW (Collision) PRAM [7]. The main features of this algorithm is that it matches the run-time and processor requirements of the algorithms in the existing literature [2, 10], while it assumes a weaker model of computation and uses a linear amount of space. The techniques used extend to improved randomized algorithms for the problem of chaining [11, 15], which is the following: given an array x1, ..., x n , such that m of the locations contain non-zero elements, to chain together all the non-zero elements into a linked list. We give randomized algorithms that run in O(1) time using n processors, whenever m is not too close to n. A byproduct of our research is the weakening of the model of computation required by some other sorting algorithms.