Computing the Number of Longest Common Subsequences

This note provides very simple, efficient algorithms for computing the number of distinct longest common subsequences of two input strings and for computing the number of LCS embeddings.Comment: 3 pages, LaTe

DLCSS: Dynamic Longest Common Subsequences

Autonomous driving is a key technology towards a brighter, more sustainable future. To enable such a future, it is necessary to utilize autonomous vehicles in shared mobility models. However, to evaluate, whether two or more route requests have the potential for a shared ride, is a compute-intensive task, if done by rerouting. In this work, we propose the Dynamic Longest Common Subsequences algorithm for fast and cost-efficient comparison of two routes for their compatibility, dynamically only incorporating parts of the routes which are suited for a shared trip. Based on this, one can also estimate, how many autonomous vehicles might be necessary to fulfill the local mobility demands. This can help providers to estimate the necessary fleet sizes, policymakers to better understand mobility patterns and cities to scale necessary infrastructure

Bounds on the Number of Longest Common Subsequences

This paper performs the analysis necessary to bound the running time of known, efficient algorithms for generating all longest common subsequences. That is, we bound the running time as a function of input size for algorithms with time essentially proportional to the output size. This paper considers both the case of computing all distinct LCSs and the case of computing all LCS embeddings. Also included is an analysis of how much better the efficient algorithms are than the standard method of generating LCS embeddings. A full analysis is carried out with running times measured as a function of the total number of input characters, and much of the analysis is also provided for cases in which the two input sequences are of the same specified length or of two independently specified lengths.Comment: 13 pages. Corrected typos, corrected operation of hyperlinks, improved presentatio

Computing longest common square subsequences

A square is a non-empty string of form YY. The longest common square subsequence (LCSqS) problem is to compute a longest square occurring as a subsequence in two given strings A and B. We show that the problem can easily be solved in O(n^6) time or O(|M|n^4) time with O(n^4) space, where n is the length of the strings and M is the set of matching points between A and B. Then, we show that the problem can also be solved in O(sigma |M|^3 + n) time and O(|M|^2 + n) space, or in O(|M|^3 log^2 n log log n + n) time with O(|M|^3 + n) space, where sigma is the number of distinct characters occurring in A and B. We also study lower bounds for the LCSqS problem for two or more strings

A Central Limit Theorem for the Length of the Longest Common Subsequences in Random Words

Let $(X_i)_{i \geq 1}$ and $(Y_i)_{i\geq1}$ be two independent sequences of independent identically distributed random variables taking their values in a common finite alphabet and having the same law. Let $LC_n$ be the length of the longest common subsequences of the two random words $X_1\cdots X_n$ and $Y_1\cdots Y_n$. Under a lower bound assumption on the order of its variance, $LC_n$ is shown to satisfy a central limit theorem. This is in contrast to the limiting distribution of the length of the longest common subsequences in two independent uniform random permutations of $\{1, \dots, n\}$, which is shown to be the Tracy-Widom distribution.Comment: Some corrections, typos corrected and improvement

Longest common parameterized subsequences with fixed common substring

In this paper we consider the problem of the longest common parameterized subsequence with fixed common substring (STR-IC-LCPS). in particular, we show that STR-IC-LCPS is NP-complete. We describe an approach to solve STR-IC-LCPS. This approach is based on an explicit reduction from the problem to the satisfiability problem

The Longest Common Exemplar Subsequence Problem

In this paper, we propose to find order conserved subsequences of genomes by finding longest common exemplar subsequences of the genomes. The longest common exemplar subsequence problem is given by two genomes, asks to find a common exemplar subsequence of them, such that the exemplar subsequence length is maximized. We focus on genomes whose genes of the same gene family are in at most s spans. We propose a dynamic programming algorithm with time complexity O(s4 s mn) to find a longest common exemplar subsequence of two genomes with one genome admitting s span genes of the same gene family, where m, n stand for the gene numbers of those two given genomes. Our algorithm can be extended to find longest common exemplar subsequences of more than one genomes