Ranking Bracelets in Polynomial Time

TL;DR

This paper introduces the first polynomial-time algorithm for ranking bracelets, a combinatorial object, by decomposing the problem into related ranking tasks over necklaces and palindromic structures.

Contribution

It presents a novel polynomial-time algorithm for ranking bracelets, including the first unranking algorithm, by leveraging new concepts of palindromic and enclosing apalindromic necklaces.

Findings

Algorithm runs in O(k^2 n^4) time

Introduces ranks over necklaces, palindromic necklaces, and apalindromic necklaces

Enables efficient ranking and unranking of bracelets

Abstract

The main result of the paper is the first polynomial-time algorithm for ranking bracelets. The time-complexity of the algorithm is O(k^2 n^4), where k is the size of the alphabet and n is the length of the considered bracelets. The key part of the algorithm is to compute the rank of any word with respect to the set of bracelets by finding three other ranks: the rank over all necklaces, the rank over palindromic necklaces, and the rank over enclosing apalindromic necklaces. The last two concepts are introduced in this paper. These ranks are key components to our algorithm in order to decompose the problem into parts. Additionally, this ranking procedure is used to build a polynomial-time unranking algorithm.