a b s t r a c tThe following problem looking as a high-school exercise hides an unexpected difficulty. Do the matrices A = 2 0 0 3 and B = 3 5 0 5 satisfy any nontrivial equation with the multiplication symbol only? This problem was mentioned as open in Cassaigne et al. [J. Cassaigne, T. Harju, J. Karhumäki, On the undecidability of freeness of matrix semigroups, Internat. J. Algebra Comput. 9 (3-4) (1999) 295-305] and in a book by Blondel et al. [V. Blondel, J. Cassaigne, J. Karhumäki, Problem 10.3: Freeness of multiplicative matrix semigroups, in: V. Blondel, A. Megretski (Eds.), Unsolved Problems in Mathematical Systems and Control Theory, Princeton University Press , 2004, pp. 309-314] as an intriguing instance of a natural computational problem of deciding whether a given finitely generated semigroup of 2 × 2 matrices is free. In this paper we present a new partial algorithm for the latter which, in particular, easily finds that the following equation holds for the matrices above. 1 Our algorithm turns out quite practical and allows us to settle also other related open questions posed in the mentioned article.

Abstract. Given an array A containing arbitrary (positive and negative) numbers, we consider the problem of supporting range maximumsum segment queries on A: i.e., given an arbitrary range [i, j]Chen and Chao [Disc. App. Math. 2007] presented a data structure for this problem that occupies Θ(n) words, can be constructed in Θ(n) time, and supports queries in Θ(1) time. Our first result is that if only the indices [i , j ] are desired (rather than the maximum sum achieved in that subrange), then it is possible to reduce the space to Θ(n) bits, regardless the numbers stored in A, while retaining the same construction and query time. Our second result is to improve the trivial space lower bound for any encoding data structure that supports range maximum-sum segment queries from n bits to 1.89113n − Θ(lg n), for sufficiently large values of n. Finally, we also provide a new application of this data structure which simplifies a previously known linear time algorithm for finding k-covers: given an array A of n numbers and a number k, find k disjoint subranges [i1, j1], ..., [i k , j k ], such that the total sum of all the numbers in the subranges is maximized. As observed by Csürös [IEEE/ACM TCBB 2004], k-covers can be used to identify regions in genomes.

Abstract. We study a generalization of the order-preserving pattern matching recently introduced by Kubica et al. (Inf. Process. Let., 2013) and Kim et al. (submitted to Theor. Comp. Sci.), where instead of looking for an exact copy of the pattern, we only require that the relative order between the elements is the same. In our variant, we additionally allow up to k mismatches between the pattern of length m and the text of length n, and the goal is to construct an efficient algorithm for small values of k. Our solution detects an order-preserving occurrence with up to k mismatches in O(n(log log m + k log log k)) time.

A minimax tree is similar to a Huffman tree except that, instead of minimizing the weighted average of the leaves' depths, it minimizes the maximum of any leaf's weight plus its depth. Golumbic (1976) introduced minimax trees and gave a Huffman-like, O(n log n)-time algorithm for building them. Drmota and Szpankowski (2002) gave another O(n log n)-time algorithm, which checks the Kraft Inequality in each step of a binary search. In this paper we show how Drmota and Szpankowski's algorithm can be made to run in linear time on a word RAM with Ω(log n)-bit words. We also discuss how our solution applies to problems in data compression, group testing and circuit design.