The longest common subsequence (LCS) problem is a fundamental algorithmic problem. Given a pair of strings, the problem asks for the length of the longest string that is a subsequence in both input strings. Among the many relatives of this problem, there is its natural version where one or both of input strings have periodic structure. The case where only one of the input strings is periodic has been considered before; in this work, we develop an efficient algorithm for the more difficult case where both input strings are periodic. The algorithm is based on the existing algebraic framework for the LCS problem, developed by the third author; in particular, we extend this framework to dealing with affine (i.e. doubly-infinite periodic) permutations instead of finite ones. Given input strings that are a k-repeat of a period of length m and an 𝓁-repeat of a period of length n, the resulting algorithm runs in time O(mn+n log n log k), which is a substantial improvement over existing approaches. The algorithm has been implemented by the first author; by running his code, one can process pairs of periodic input strings with lengths far beyond the reach of all known alternative algorithms.

The longest common subsequence (LCS) problem on a pair of strings is a classical problem in string algorithms. Its extension, the semi-local LCS problem, provides a more detailed comparison of the input strings, without any increase in asymptotic running time. Several semi-local LCS algorithms have been proposed previously; however, to the best of our knowledge, none have yet been implemented. In this paper, we explore a new hybrid approach to the semi-local LCS problem. We also propose a novel bit-parallel LCS algorithm. In the experimental part of the paper, we present an implementation of several existing and new parallel LCS algorithms and evaluate their performance.

The longest common subsequence (LCS) problem is fundamental in computer science and its many applications. Parallel algorithms for this problems have been studied previously, in particular in the bulk-synchronous parallelism (BSP) model, which treats local computation, communication and synchronisation as independent scarce resources of a computing system. We consider primarily BSP algorithms running in either constant or polylogarithmic synchronisation. Based on our previous results on the algebraic structure and efficient algorithms for semi-local LCS, we present BSP algorithms for parallel semi-local LCS, improving on the existing upper bounds on communication and synchronisation; in particular we present the first constant-sync work-optimal LCS algorithm.

Given a fixed alignment scoring scheme, the bounded length (respectively, bounded total length) Smith-Waterman alignment problem on a pair of strings of lengths m, n, asks for the maximum alignment score across all substring pairs, such that the first substring’s length (respectively, the sum of the two substrings' lengths) is above the given threshold w. The latter problem was introduced by Arslan and Eğecioğlu under the name "local alignment with length threshold". They proposed a dynamic programming algorithm solving the problem in time O(mn^2), and also an approximation algorithm running in time O(rmn), where r is a parameter controlling the accuracy of approximation. We show that both these problems can be solved exactly in time O(mn), assuming a rational scoring scheme; furthermore, this solution can be used to obtain an exact algorithm for the normalised bounded total length Smith - Waterman alignment problem, running in time O(mn log n). Our algorithms rely on the techniques of fast window-substring alignment and implicit unit-Monge matrix searching, developed previously by the author and others.

Monge matrices play a fundamental role in optimisation theory, graph and string algorithms. Distance multiplication of two Monge matrices of size n can be performed in time O(n 2). Motivated by applications to string algorithms, we introduced in previous works a subclass of Monge matrices, that we call simple unit-Monge matrices. We also gave a distance multiplication algorithm for such matrices, running in time O(n 1.5). Landau asked whether this problem can be solved in linear time. In the current work, we give an algorithm running in time O(nlogn), thus approaching an answer to Landau’s question within a logarithmic factor. The new algorithm implies immediate improvements in running time for a number of algorithms on strings and graphs. In particular, we obtain an algorithm for finding a maximum clique in a circle graph in time O(nlog2 n), and a surprisingly efficient algorithm for comparing compressed strings. We also point to potential applications in group theory, by making a connection between unit-Monge matrices and Coxeter monoids. We conclude that unit-Monge matrices are a fascinating object and a powerful tool, that deserve further study from both the mathematical and the algorithmic viewpoints.

Monge matrices play a fundamental role in optimisation theory, graph and string algorithms. Distance multiplication of two Monge matrices of size n can be performed in time O(n2). Motivated by applications to string algorithms, we introduced in previous works a subclass of Monge matrices, that we call simple unit-Monge matrices. We also gave a distance multiplication algorithm for such matrices, running in time O(n1.5). Landau asked whether this problem can be solved in linear time. In the current work, we give an algorithm running in time O(n log n), thus approaching an answer to Landau's question within a logarithmic factor. The new algorithm implies immediate improvements in running time for a number of algorithms on strings and graphs. In particular, we obtain an algorithm for finding a maximum clique in a circle graph in time O(n log2 n), and a surprisingly efficient algorithm for comparing compressed strings. We also point to potential applications in group theory, by making a connection between unit-Monge matrices and Coxeter monoids. We conclude that unit-Monge matrices are a fascinating object and a powerful tool, that deserves further study from both the mathematical and the algorithmic viewpoints.

In this paper, we show new parallel algorithms for a set of classical string comparison problems: computation of string alignments, longest common subsequences (LCS) or edit distances, and longest increasing subsequence computation. These problems have a wide range of applications, in particular in computational biology and signal processing. We discuss the scalability of our new parallel algorithms in computation time, in memory, and in communication. Our new algorithms are based on an efficient parallel method for (min,+)-multiplication of distance matrices. The core result of this paper is a scalable parallel algorithm for multiplying implicit simple unit-Monge matrices of size n x n on p processors using time O( n log n ‾ p). communication O(n log p) ‾ p) and O(log p) supersteps. This algorithm allows us to implement scalable LCS computation for two strings of length n using time O(n2 ‾ p) and communication O(n ‾ √ p), requiring local memory of size O(n ‾ √ p) on each processor. Furthermore, our algorithm can be used to obtain the first generally work-scalable algorithm for computing the longest increasing subsequence (LIS). Our algorithm for LIS computation requires computation O(n log2 n ‾ p), communication O(n log p)/ p), and O(log2 p) supersteps for computing the LIS of a sequence of length n. This is within a log n factor of work-optimality for the LIS problem, which can be solved sequentially in time O(n log n) in the comparison-based model. Our LIS algorithm is also within a log p-factor of achieving perfectly scalable communication and furthermore has perfectly scalable memory size requirements of O(n ‾ p) per processor.