The k-Centre Selection Problem for Multidimensional Necklaces

This paper introduces the natural generalisation of necklaces to the multidimensional setting – multidimensional necklaces. One-dimensional necklaces are known as cyclic words, two-dimensional necklaces correspond to toroidal codes, and necklaces of dimension three can represent periodic motives in crystals. Our central results are two approximation algorithms for the k-Centre selection problem, where the task is to find k uniformly spaced objects within a set of necklaces. We show that it is NP-hard to verify a solution to this problem even in the one dimensional setting. This strong negative result is complimented with two polynomialtime approximation algorithms. In one dimension we provide a 1 + f(k,N) approximation algorithm where f(k,N) = logq (kN) N−logq (kN) − log 2 q(kN) 2N(N−logq (kN)) . For two dimensions we give a 1 + g(k,N) approximation algorithm where g(k,N) = logq (kN) N−logq (kN) − log 2 q(k) 2N(N−logq (kN)) . In both cases N is the size of the necklaces and q the size of the alphabet. Alongside our results for these new problems, we also provide the first polynomial time algorithms for counting, generating, ranking and unranking multidimensional necklaces.

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