Sorting suffixes of two-pattern strings

Recently, several authors presented linear recursive algorithms for sorting suffixes of a string. All these algorithms employ a similar three-step approach, based on an initial division of the suffixes of x into two sets: in step 1 sort the first set using recursive reduction of the problem, in step 2 determine the order of the suffixes in the second set based on the order of the suffixes in the first set, and in step 3 merge the two sets together. To optimize such an algorithm either for space or time, it may not be sufficient to optimize one of the three steps, since in doing so, one might increase the resources required for the others to an unacceptable extent. Franek, Lu, and Smyth introduced two-pattern strings as a generalization of Sturmian strings. Like Sturmian strings, two-pattern strings are generated by iterated morphisms, but they exhibit a much richer structure. In this paper we show that the suffixes of two-pattern strings can be sorted in linear time using a variant of the three step approach outlined above. It turns out that, given the order of the suffixes in a two-pattern string, one can almost directly list in linear time all the suffixes of its expansion under a two-pattern morphism

Another algorithm for reducing bandwidth and profile of a sparse matrix

The paper describes a new bandwidth reduction method for sparse matrices which promises to be both fast and effective in comparison with known methods. The algorithm operates on the undirected graph corresponding to the incidence matrix induced by the original sparse matrix, and separates into three distinct phases: (1) determination of a spanning tree of maximum length, (2) modification of the spanning tree into a free level structure of small width, (3) level-by-level numbering of the level structure. The final numbering produced corresponds to a renumbering of the rows and columns of a sparse matrix so as to concentrate non-zero elements of the matrix in a band about the main diagonal

On baier's sort of maximal Lyndon substrings

We describe and analyze in terms of Lyndon words an elementary sort of maximal Lyndon factors of a string and prove formally its correctness. Since the sort is based on the first phase of Baier’s algorithm for sorting of the suffixes of a string, we refer to it as Baier’s sort

Graphs with small generalized chromatic number

Let G = (V,E) denote a finite simple undirected connected graph of order n = [V] and diameter D. For any integer k..

Computing periodicities in strings: A new approach

The most efficient methods currently available for the computation of repetitions or repeats in a string x = x[1..n] all depend on the prior computation of a suffix tree/array STx/SAx. Although these data structures can be computed in asymptotic Θ(n) time, nevertheless in practice they involve significant overhead, both in time and space. Since the number of repetitions/repeats in x can be reported in a way that is at most linear in string length, it therefore seems that it should be possible to devise less roundabout means of computing repetitions/repeats that take advantage of their infrequent occurrence. This survey paper provides background for these ideas and explores the possibilities for more efficient computation of periodicities in strings

The sum number of the cocktail party graph

A graph G is called a sum graph if there exists a labelling of the vertices of G by distinct positive integers such that the vertices labelled u and v are adjacent if and only if there exists a vertex labelled u + v. If G is not a sum graph, adding a finite number of isolated vertices to it will always yield a sum graph, and the sum number oe(G) of G is the smallest number of isolated vertices that will achieve this result. A labelling that realizes G + K oe(G) as a sum graph is said to be optimal. In this paper we consider G = H m;n , the complete n-partite graph on n 2 sets of m 2 nonadjacent vertices. We give an optimal labelling to show that oe(H 2;n ) = 4n \Gamma 5, and in the general case we give constructive proofs that oe(H m;n ) 2 \Omega\Gamma mn) and oe(H m;n ) 2 O(mn 2 ). We conjecture that oe(H m;n ) is asymptotically greater than mn, the cardinality of the vertex set; if so, then H m;n is the first known graph with this property. We also provide for the first time an optimal labelling of the complete bipatite graph Kmn whose smallest label is 1

Sum graphs of small sum number

See attache

Palindromes in starlike trees

In this note, we obtain an upper bound on the maximum number of distinct non-empty palindromes in starlike trees. This bound implies, in particular, that there are at most 4 n distinct non-empty palindromes in a starlike tree with three branches each of length n. For such starlike trees labelled with a binary alphabet, we sharpen the upper bound to 4 n − 1 and conjecture that the actual maximum is 4 n − 2. It is intriguing that this simple conjecture seems difficult to prove, in contrast to the proof of the bound

Approximate periodicity in strings

In many application areas (for instance, DNA sequence analysis) it becomes important to compute various kinds of “approximate period” of a given string y. Here we discuss three such approximate periods and the algorithms which compute them: an Abelian generator, a cover, and a seed. Let u be a substring of y. Then u is an Abelian generator of y iff y is a concatenation of substrings which are permutations of u: u is a cover of y iff every letter of y is contained in an occurrence of u in y and u is a seed of y iff y is a substring of a string y with cover u. Observe that, according to these definitions, y is an Abelian generator, a cover, and a seed of itself

Computing regularities in strings: A survey

The aim of this survey is to provide insight into the sequential algorithms that have been proposed to compute exact “regularities” in strings; that is, covers (or quasiperiods), seeds, repetitions, runs (or maximal periodicities), and repeats. After outlining and evaluating the algorithms that have been proposed for their computation, I suggest possibly productive future directions of research