A Versatile Algorithm to Generate Various Combinatorial Structures

TL;DR

This paper reviews and extends an algorithm for generating various combinatorial structures, demonstrating its versatility and efficiency in producing permutations, combinations, subsets, and other structures with minimal modifications.

Contribution

It introduces a unified approach to generate multiple combinatorial structures efficiently by modifying a single foundational algorithm.

Findings

The algorithm correctly generates all unique permutations with repetitions.

Minor modifications enable generation of combinations, subsets, and other structures.

The approach is efficient and versatile for various combinatorial generation tasks.

Abstract

Algorithms to generate various combinatorial structures find tremendous importance in computer science. In this paper, we begin by reviewing an algorithm proposed by Rohl that generates all unique permutations of a list of elements which possibly contains repetitions, taking some or all of the elements at a time, in any imposed order. The algorithm uses an auxiliary array that maintains the number of occurrences of each unique element in the input list. We provide a proof of correctness of the algorithm. We then show how one can efficiently generate other combinatorial structures like combinations, subsets, n-Parenthesizations, derangements and integer partitions & compositions with minor changes to the same algorithm.