Subset-lex: did we miss an order?

TL;DR

This paper extends a classic subset generation algorithm to various combinatorial objects, offering efficient, often loopless algorithms and new Gray codes for subset-lex order and compositions.

Contribution

It generalizes subset-lex algorithms to multisets, compositions, and partitions, introducing efficient, loopless algorithms and new Gray codes for these orders.

Findings

Algorithms are often loopless and require minimal extra variables.

Performance is highly competitive even when not loopless.

New Gray codes for subset-lex order and compositions are introduced.

Abstract

We generalize a well-known algorithm for the generation of all subsets of a set in lexicographic order with respect to the sets as lists of elements (subset-lex order). We obtain algorithms for various combinatorial objects such as the subsets of a multiset, compositions and partitions represented as lists of parts, and for certain restricted growth strings. The algorithms are often loopless and require at most one extra variable for the computation of the next object. The performance of the algorithms is very competitive even when not loopless. A Gray code corresponding to the subset-lex order and a Gray code for compositions that was found during this work are described.