The K-Centre Problem for Necklaces

In graph theory, the objective of the k-centre problem is to find a set of $k$ vertices for which the largest distance of any vertex to its closest vertex in the $k$-set is minimised. In this paper, we introduce the $k$-centre problem for sets of necklaces, i.e. the equivalence classes of words under the cyclic shift. This can be seen as the k-centre problem on the complete weighted graph where every necklace is represented by a vertex, and each edge has a weight given by the overlap distance between any pair of necklaces. Similar to the graph case, the goal is to choose $k$ necklaces such that the distance from any word in the language and its nearest centre is minimised. However, in a case of k-centre problem for languages the size of associated graph maybe exponential in relation to the description of the language, i.e., the length of the words l and the size of the alphabet q. We derive several approximation algorithms for the $k$-centre problem on necklaces, with logarithmic approximation factor in the context of l and k, and within a constant factor for a more restricted case

The k-center Problem for Classes of Cyclic Words

The problem of finding k uniformly spaced points (centres) within a metric space is well known as the k-centre selection problem. In this paper, we introduce the challenge of k-centre selection on a class of objects of exponential size and study it for the class of combinatorial necklaces, known as cyclic words. The interest in words under translational symmetry is motivated by various applications in algebra, coding theory, crystal structures and other physical models with periodic boundary conditions. We provide solutions for the centre selection problem for both one-dimensional necklaces and largely unexplored objects in combinatorics on words - multidimensional combinatorial necklaces. The problem is highly non-trivial as even verifying a solution to the k-centre problem for necklaces can not be done in polynomial time relative to the length of the cyclic words and the alphabet size unless P= NP. Despite this challenge, we develop a technique of centre selection for a class of necklaces based on de-Bruijn Sequences and provide the first polynomial O(k· n) time approximation algorithm for selecting k centres in the set of 1D necklaces of length n over an alphabet of size q with an approximation factor of O(1+logq(k·n)n-logq(k·n)). For the set of multidimensional necklaces of size n1× n2× … × nd we develop an O(k· N2) time algorithm with an approximation factor of O(1+logq(k·N)N-logq(k·N)) in O(k· N2) time, where N= n1· n2· … · nd by approximating de Bruijn hypertori technique

Combinatorial Algorithms for Multidimensional Necklaces

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

The k-Centre Problem for Classes of Cyclic Words

The problem of finding k uniformly spaced points (centres) within a metric space is well known as the k-centre selection problem. In this paper, we introduce the challenge of k-centre selection on a class of objects of exponential size and study it for the class of combinatorial necklaces, known as cyclic words. The interest in words under translational symmetry is motivated by various applications in algebra, coding theory, crystal structures and other physical models with periodic boundary conditions. We provide solutions for the centre selection problem for both one-dimensional necklaces and largely unexplored objects in combinatorics on words - multidimensional combinatorial necklaces. The problem is highly non-trivial as even verifying a solution to the k-centre problem for necklaces can not be done in polynomial time relative to the length of the cyclic words and the alphabet size unless P= NP. Despite this challenge, we develop a technique of centre selection for a class of necklaces based on de-Bruijn Sequences and provide the first polynomial O(k· n) time approximation algorithm for selecting k centres in the set of 1D necklaces of length n over an alphabet of size q with an approximation factor of O(1+logq(k·n)n-logq(k·n)). For the set of multidimensional necklaces of size n1× n2× … × nd we develop an O(k· N2) time algorithm with an approximation factor of O(1+logq(k·N)N-logq(k·N)) in O(k· N2) time, where N= n1· n2· … · nd by approximating de Bruijn hypertori technique

The k-centre problem for classes of cyclic words

The problem of finding k uniformly spaced points (centres) within a metric space is well known as the k-centre selection problem. In this paper, we introduce the challenge of k-centre selection on a class of objects of exponential size and study it for the class of combinatorial necklaces, known as cyclic words. The interest in words under translational symmetry is motivated by various applications in algebra, coding theory, crystal structures and other physical models with periodic boundary conditions. We provide solutions for the centre selection problem for both one-dimensional necklaces and largely unexplored objects in combinatorics on words - multidimensional combinatorial necklaces. The problem is highly non-trivial as even verifying a solution to the k-centre problem for necklaces can not be done in polynomial time relative to the length of the cyclic words and the alphabet size unless P= NP. Despite this challenge, we develop a technique of centre selection for a class of necklaces based on de-Bruijn Sequences and provide the first polynomial O(k· n) time approximation algorithm for selecting k centres in the set of 1D necklaces of length n over an alphabet of size q with an approximation factor of O(1+logq(k·n)/n-logq(k·n)). For the set of multidimensional necklaces of size n1× n2× … × nd we develop an O(k· N2) time algorithm with an approximation factor of O(1+logq(k·N)/N-logq(k·N)) in O(k· N2) time, where N= n1· n2· … · nd by approximating de Bruijn hypertori technique.</p

The k-Centre Problem for Classes of Cyclic Words

The problem of finding k uniformly spaced points (centres) within a metric space is well known as the k-centre selection problem. In this paper, we introduce the challenge of k-centre selection on a class of objects of exponential size and study it for the class of combinatorial necklaces, known as cyclic words. The interest in words under translational symmetry is motivated by various applications in algebra, coding theory, crystal structures and other physical models with periodic boundary conditions. We provide solutions for the centre selection problem for both one-dimensional necklaces and largely unexplored objects in combinatorics on words - multidimensional combinatorial necklaces. The problem is highly non-trivial as even verifying a solution to the k-centre problem for necklaces can not be done in polynomial time relative to the length of the cyclic words and the alphabet size unless P= NP. Despite this challenge, we develop a technique of centre selection for a class of necklaces based on de-Bruijn Sequences and provide the first polynomial O(k· n) time approximation algorithm for selecting k centres in the set of 1D necklaces of length n over an alphabet of size q with an approximation factor of O(1+logq(k·n)n-logq(k·n)). For the set of multidimensional necklaces of size n1× n2× … × nd we develop an O(k· N2) time algorithm with an approximation factor of O(1+logq(k·N)N-logq(k·N)) in O(k· N2) time, where N= n1· n2· … · nd by approximating de Bruijn hypertori technique.</p