Combinatorial Algorithms for Multidimensional Necklaces

Abstract

A necklace is an equivalence class of words of length $n$ over an alphabet under the cyclic shift (rotation) operation. As a classical object, there have been many algorithmic results for key operations on necklaces, including counting, generating, ranking, and unranking. This paper generalises the concept of necklaces to the multidimensional setting. We define multidimensional necklaces as an equivalence classes over multidimensional words under the multidimensional cyclic shift operation. Alongside this definition, we generalise several problems from the one dimensional setting to the multidimensional setting for multidimensional necklaces with size $(n_1,n_2,...,n_d)$ over an alphabet of size $q$ including: providing closed form equations for counting the number of necklaces; an $O(n_1 \cdot n_2 \cdot ... \cdot n_d)$ time algorithm for transforming some necklace $w$ to the next necklace in the ordering; an $O((n_1 \cdot n_2 \cdot ... \cdot n_d)^5)$ time algorithm to rank necklaces (determine the number of necklaces smaller than $w$ in the set of necklaces); an $O((n_1\cdot n_2 \cdot ... \cdot n_d)^{6(d + 1)} \cdot \log^d(q))$ time algorithm to unrank multidimensional necklace (determine the $i^{th}$ necklace in the set of necklaces). Our results on counting, ranking, and unranking are further extended to the fixed content setting, where every necklace has the same Parikh vector, in other words every necklace shares the same number of occurrences of each symbol. Finally, we study the $k$-centre problem for necklaces both in the single and multidimensional settings. We provide strong approximation algorithms for solving this problem in both the one dimensional and multidimensional settings.