Towards the Succinct Representation of m Out of n

Abstract

Combinations of n objects taken m at the time are ubiquitous in a wide range of combinatorial problems. In this paper, we introduce a novel approach to generate combinations from given integer numbers by using a gradient-based algorithm through plural number of CPU and GPU processors. The time complexity is bounded by O(m2) when using a single processor, and bounded by O(mlog m) when using at most O(m/ log m) processors. Relevant computational experiments confirmed the practical efficiency within computationally allowable limits. Our approach offers the building blocks to represent combinations with succinct encoding and complexity being independent of n, which is meritorious when n is very large, or when n is time-varying.