O(1) Time Generation of Adjacent Multiset Combinations

TL;DR

This paper presents an O(1) time algorithm for generating adjacent multiset combinations, ensuring minimal change between successive combinations, based on a novel twisted lexicographic tree approach.

Contribution

It introduces a new O(1) time algorithm for adjacent multiset combinations using a twisted lexicographic tree, differing from previous non-adjacent algorithms.

Findings

Achieves constant-time generation of adjacent multiset combinations.

Uses a twisted lexicographic tree to ensure minimal changes between outputs.

Provides an iterative traversal method for efficient generation.

Abstract

We solve the problem of designing an O(1) time algorithm for generating adjacent multiset combinations in a different approach from Walsh. By the word adjacent, we mean that two adjacent multiset combinations are different at two places by one in their vector forms. Previous O(1) time algorithms for multiset combinations generated non-adjacent multiset combinations. Our algorithm in this paper can be derived from a general framework of combinatorial Gray code, which we characterise to suit our need for combinations and multiset combinations. The central idea is a twisted lexico tree, which is obtained from the lexicographic tree for the given set of combinatorial objects by twisting branches depending on the parity of each node. An iterative algorithm which traverses this tree will generate the given set of combinatorial objects in constant time as well as with a fixed number of changes from the present combinatorial object to the next.