Michalis Christou

Efficient seed computation revisited

The notion of the cover is a generalization of a period of a string, and there are linear time algorithms for finding the shortest cover. The seed is a more complicated generalization of periodicity, it is a cover of a superstring of a given string, and the shortest seed problem is of much higher algorithmic difficulty. The problem is not well understood, no linear time algorithm is known. In the paper we give linear time algorithms for some of its versions-computing shortest left-seed array, longest left-seed array and checking for seeds of a given length. The algorithm for the last problem is used to compute the seed array of a string (i.e., the shortest seeds for all the prefixes of the string) in O(n^2) time. We describe also a simpler alternative algorithm computing efficiently the shortest seeds. As a by-product we obtain an O(nlog(n/m)) time algorithm checking if the shortest seed has length at least m and finding the corresponding seed. We also correct some important details missing in the previously known shortest-seed algorithm Iliopoulos et al. (1996) [14].

IDENTIFYING ALL ABELIAN PERIODS OF A STRING IN QUADRATIC TIME AND RELEVANT PROBLEMS

Abelian periodicity of strings has been studied extensively over the last years. In 2006 Constantinescu and Ilie defined the abelian period of a string and several algorithms for the computation of all abelian periods of a string were given. In contrast to the classical period of a word, its abelian version is more flexible, factors of the word are considered the same under any internal permutation of their letters. We show two O(|y|2) algorithms for the computation of all abelian periods of a string y. The first one maps each letter to a suitable number such that each factor of the string can be identified by the unique sum of the numbers corresponding to its letters and hence abelian periods can be identified easily. The other one maps each letter to a prime number such that each factor of the string can be identified by the unique product of the numbers corresponding to its letters and so abelian periods can be identified easily. We also define weak abelian periods on strings and give an O(|y|log(|y|)) algorithm for their computation, together with some other algorithms for more basic problems.

Efficient seeds computation revisited

The notion of the cover is a generalization of a period of a string, and there are linear time algorithms for finding the shortest cover. The seed is a more complicated generalization of periodicity, it is a cover of a superstring of a given string, and the shortest seed problem is of much higher algorithmic difficulty. The problem is not well understood, no linear time algorithm is known. In the paper we give linear time algorithms for some of its versions -- computing shortest left-seed array, longest left-seed array and checking for seeds of a given length. The algorithm for the last problem is used to compute the seed array of a string (i.e., the shortest seeds for all the prefixes of the string) in O(n2) time. We describe also a simpler alternative algorithm computing efficiently the shortest seeds. As a by-product we obtain an O(n log (n/m)) time algorithm checking if the shortest seed has length at least m and finding the corresponding seed. We also correct some important details missing in the previously known shortest-seed algorithm (Iliopoulos et al., 1996).

Abelian borders in binary words

Abstract In this article we study the appearance of abelian borders in binary words, a notion closely related to the abelian period of a word. We show how many binary words have shortest border of a given length by identifying relations with Dyck words. Furthermore, we give some bounds on the number of abelian border-free words of a given length and on the number of abelian words of a given length that have at least one abelian border. Finally, using some techniques employed in a recent paper by Christodoulakis et al. (2013), we show that there exists an algorithm that finds the shortest abelian border of a binary word that is not abelian border-free in Θ ( n ) time on average.

Computing all subtree repeats in ordered trees

We consider the problem of computing all subtree repeats in a given labeled ordered tree. We first transform the tree to a string representing its postfix notation, and then present an algorithm based on the bottom-up technique to solve it. The proposed algorithm consists of two phases: the preprocessing phase and the phase where all subtree repeats are computed. The linear time and space complexity of the proposed algorithm are important parts of its quality.

Computing all subtree repeats in ordered ranked trees

We consider the problem of finding all subtree repeats in a given ordered ranked tree. Specifically, we transform the given tree to a string representing its postfix notation, and then propose an algorithm based on the bottom-up technique. The proposed algorithm is divided into two phases: the preprocessing phase, and the phase where all subtree repeats are computed. The linear runtime of the algorithm, as well as the use of linear auxiliary space, are important aspects of its quality.

On the right-seed array of a string

We consider the problem of finding the repetitive structure of a given fixed string y. A factor u of y is a cover of y, if every letter of y falls within some occurrence of u in y. A factor v of y is a seed of y, if it is a cover of a superstring of y. There exist linear-time algorithms for solving the minimal cover problem. The minimal seed problem is of much higher algorithmic difficulty, and no linear-time algorithm is known. In this article, we solve one of its variants - computing the minimal and maximal right-seed array of a given string. A right seed of y is the shortest suffix of y that it is a cover of a superstring of y. An integer array RS is the minimal right-seed (resp. maximal right-seed) array of y, if RS[i] is the minimal (resp. maximal) length of right seeds of y[0 . . i]. We present an O(n log n) time algorithm that computes the minimal right-seed array of a given string, and a linear-time solution to compute the maximal right-seed array by detecting border-free prefixes of the given string.

Bipartite Ramsey numbers involving stars, stripes and trees

The Ramsey number R(m, n) is the smallest integer p such that any blue-red colouring of the edges of the complete graph Kp forces the appearance of a blue Km or a red Kn. Bipartite Ramsey problems deal with the same questions but the graph explored is the complete bipartite graph instead of the complete graph. We consider special cases of the bipartite Ramsey problem. More specifically we investigate the appearance of simpler monochromatic graphs such as stripes, stars and trees under a 2-colouring of the edges of a bipartite graph. We give the Ramsey numbers Rb(mP2, nP2), Rb(Tm, Tn), Rb(Sm, nP2), Rb(Tm, nP2) and Rb(Sm, Tn).

On left and right seeds of a string

We consider the problem of finding the repetitive structure of a given string y of length n. A factor u of y is a cover of y, if every letter of y lies within some occurrence of u in y. A string v is a seed of y, if it is a cover of a superstring of y. A left seed of y is a prefix of y, that is a cover of a superstring of y. Similarly, a right seed of y is a suffix of y, that is a cover of a superstring of y. An integer array LS is the minimal left-seed (resp. maximal left-seed) array of y, if LS[i] is the minimal (resp. maximal) length of left seeds of y[0..i]. The minimal right-seed (resp. maximal right-seed) arrayRS of y is defined in a similar fashion. In this article, we present linear-time algorithms for computing all left and right seeds of y, a linear-time algorithm for computing the minimal left-seed array of y, a linear-time solution for computing the maximal left-seed array of y, an O(nlogn)-time algorithm for computing the minimal right-seed array of y, and a linear-time solution for computing the maximal right-seed array of y. All algorithms use linear auxiliary space.