Probabilistic Recurrence Relations for Work and Span of Parallel Algorithms

Abstract: In this paper we present a method for obtaining tail-bounds for random variables satisfying certain probabilistic recurrences that arise in the analysis of randomized parallel divide and conquer algorithms. In such algorithms, some computation is initially done to process an input x, which is then randomly split into subproblems $h_1(x), ..., h_n(x)$, and the algorithm proceeds recursively in parallel on each subproblem. The total work on input x, W(x), then satisfies a probabilistic recurrence of the form $W(x) = a(x) + \sum_{i=1}^n W (h_i(x))$, and the span (the longest chain of sequential dependencies), satisfies $S(x) = b(x) + \max_{i=1}^n S(h_i(x))$, where a(x) and b(x) are the work and span to split x and combine the results of the recursive calls. Karp has previously presented methods for obtaining tail-bounds in the case when n = 1, and under certain stronger assumptions for the work-recurrence when n > 1, but left open the question of the span-recurrence. We first show how to extend his technique to handle the span-recurrence. We then show that in some cases, the work-recurrence can be bounded under simpler assumptions than Karp's by transforming it into a related span-recurrence and applying our first result. We demonstrate our results by deriving tail bounds for the work and span of quicksort and the height of a randomly generated binary search tree.