Probabilistic recurrence relations

This paper is concerned with recurrence relations that arise frequently in the analysis of divide-and-conquer algorithms. In order to solve a problem instance of size x, such an algorithm invests an amount of work a(x) to break the problem into subproblems of sizes h1(x), h2(x),..., hk(x), and then proceeds to solve the subproblems. Our particular interest is in the case where the sizes hi(x) are random variables; this may occur either because of randomization within the algorithm or because the instances to be solved are assumed to be drawn from a probability distribution. When the hi are random variables the running time of the algorithm on instances of size x is also a random variable T(x). We give several easy-to-apply methods for obtaining fairly tight bounds on the upper tails of the probability distribution of T(x), and present a number of typical applications of these bounds to the analysis of algorithms. The proofs of the bounds are based on an interesting analysis of optimal strategies in certain gambling games.