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

An explicit universal cycle for the (n-1)-permutations of an n-set

We show how to construct an explicit Hamilton cycle in the directed Cayley graph Cay({\sigma_n, sigma_{n-1}} : \mathbb{S}_n), where \sigma_k = (1 2 >... k). The existence of such cycles was shown by Jackson (Discrete Mathematics, 149 (1996) 123-129) but the proof only shows that a certain directed graph is Eulerian, and Knuth (Volume 4 Fascicle 2, Generating All Tuples and Permutations (2005)) asks for an explicit construction. We show that a simple recursion describes our Hamilton cycle and that the cycle can be generated by an iterative algorithm that uses O(n) space. Moreover, the algorithm produces each successive edge of the cycle in constant time; such algorithms are said to be loopless

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

Ranking Bracelets in Polynomial Time

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

k-Universality of Regular Languages

A subsequence of a word w is a word u such that u = w[i1]w[i2] . . . w[ik], for some set of indices 1 ≤ i1 < i2 < · · · < ik ≤ |w|. A word w is k-subsequence universal over an alphabet Σ if every word in Σk appears in w as a subsequence. In this paper, we study the intersection between the set of k-subsequence universal words over some alphabet Σ and regular languages over Σ. We call a regular language L k-∃-subsequence universal if there exists a k-subsequence universal word in L, and k-∀-subsequence universal if every word of L is k-subsequence universal. We give algorithms solving the problems of deciding if a given regular language, represented by a finite automaton recognising it, is k-∃-subsequence universal and, respectively, if it is k-∀-subsequence universal, for a given k. The algorithms are FPT w.r.t. the size of the input alphabet, and their run-time does not depend on k; they run in polynomial time in the number n of states of the input automaton when the size of the input alphabet is O(log n). Moreover, we show that the problem of deciding if a given regular language is k-∃-subsequence universal is NP-complete, when the language is over a large alphabet. Further, we provide algorithms for counting the number of k-subsequence universal words (paths) accepted by a given deterministic (respectively, nondeterministic) finite automaton, and ranking an input word (path) within the set of k-subsequence universal words accepted by a given finite automaton

DEVELOPMENT OF A HIGH-THROUGHPUT ASSAY TO MEASURE DNA MISMATCH REPAIR EFFICIENCY IN VIVO

Whether due to mutagens or DNA replication errors, mismatches arise spontaneously in vivo. If left unrepaired, accumulation of mutations at a high rate can be detrimental to the survival of the organism. Cells recognize the mismatches and repair them via a dedicated mismatch repair system. Although its efficiency has been shown to depend on the type and the sequence context of the mismatch, only a small subset of possible mismatched sequences could be examined thus far. In this work, I describe a high-throughput sequencing based approach that can assess the repair efficiency of many different mismatches in parallel, enabling a systematic analysis of the sequence effect on mismatch repair. In this scheme, an in vitro synthesized plasmid containing a single mismatch is introduced to an E. coli cell. If the mismatch is repaired prior to replication, all of the descendants will share the same sequence. If, however, replication precedes mismatch repair, the descendants have a mixture of two different sequences, and therefore the event frequencies of these two types provide information on the repair efficiency. Use of DNA barcodes enables obtaining single-molecule level information regarding the fate of each mismatch carrying molecules, through which the repair of 4434 different mismatches and 1300 insertion loops were monitored in vivo under various conditions. The results showed that CC mismatches are always poorly repaired whereas local sequence context is a strong determinant of the highly heterogeneous repair efficiency of TT, AG and CT mismatches. In contrast, most of the insertion loops were repaired with a high efficiency without an appreciable sequence context dependence. The measurement of the repair efficiency in mutant cell strains of different mismatch repair pathway mutants also showed a residual repair capability, potentially an indication of side-processes that lead to an apparent repair of mismatch bearing plasmids