(Probabilistic) recurrence relations revisited

Abstract

The performance attributes of a broad class of randomised algorithms can be described by a recurrence relation of the form T(x)=a(x)+T(H(x)), where a is a function and H(x) is a random variable. For instance, T(x) may describe the running time of such an algorithm on a problem of size x. Then T(x) is a random variable, whose distribution depends on the distribution of H(x). To give high probability guarantees on the performance of such randomised algorithms, it suffices to obtain bounds on the tail of the distribution of T(x). Karp derived tight bounds on this tail distribution, when the distribution of H(x) satisfies certain restrictions. However, his proof is quite difficult to understand. In this paper, we derive bounds similar to Karp's using standard tools from elementary probability theory, such as Markov's inequality, stochastic dominance and a variant of Chernoff bounds applicable to unbounded variables. Further, we extend the results, showing that similar bounds hold under weaker restrictions on H(x). As an application, we derive performance bounds for an interesting class of algorithms that was outside the scope of the previous results.

Supported by the ESPRIT Basic Research Actions Program of the EC under contract No. 7141 (project ALCOM II).

Work done while at the Max-Planck-Institute für Informatik supported by the ESPRIT Basic Research Actions Program of the EC under contract No. 7141 (project ALCOM II).