Ranking Binary Unlabelled Necklaces in Polynomial Time

TL;DR

This paper presents the first polynomial-time algorithm for ranking unlabelled binary necklaces, a complex equivalence class of cyclic words under rotation and relabelling, with a detailed ranking method and new concepts.

Contribution

It introduces a polynomial-time algorithm for ranking unlabelled binary necklaces and defines new concepts like symmetric unlabelled necklaces and necklaces with enclosing labelling.

Findings

Algorithm runs in $O(n^6 \, \log^2 n)$ time

Provides a method to compute ranks using three different rank calculations

First polynomial-time solution for this ranking problem in binary necklaces

Abstract

Unlabelled Necklaces are an equivalence class of cyclic words under both the rotation (cyclic shift) and the relabelling operations. The relabelling of a word is a bijective mapping from the alphabet to itself. The main result of the paper is the first polynomial-time algorithm for ranking unlabelled necklaces of a binary alphabet. The time-complexity of the algorithm is $O(n^6 \log^2 n)$, where $n$ is the length of the considered necklaces. The key part of the algorithm is to compute the rank of any word with respect to the set of unlabelled necklaces by finding three other ranks: the rank over all necklaces, the rank over symmetric unlabelled necklaces, and the rank over necklaces with an enclosing labelling. The last two concepts are introduced in this paper.