Average case analysis for a simple compression algorithm

In this paper we treat the static dictionary problem, very well known in computer science. It consists in storing a set S of m elements in the range [1 ... n] so that membership queries on S's elements can be handled in O(1) time. It can be approached as a table compression problem in which a size n table has m ones and the other elements are zeros. We focus our attention on sparse case (m much less than n). We use a simple algorithm to solve the problem and make an average-case analysis of the total space required when the input derives from uniform probability distribution. We also find some conditions able to minimize storage requirements. We then propose and analyze a new algorithm able to reduce storage requirements drastically to O(m(4/3)).