Longest Common Subsequence in k-length substrings

TL;DR

This paper introduces a new computational biology problem, LCSk, generalizing the classic LCS, and provides efficient algorithms for both LCSk and a related distance measure, EDk, with practical time and space complexities.

Contribution

It defines the LCSk problem, extending LCS to k-length substrings, and presents algorithms with optimal time complexity for solving LCSk and EDk distance.

Findings

LCSk can be computed in O(n^2) time for strings of length n.

EDk distance measure can be computed in O(nm) time and O(km) space.

The algorithms generalize classical LCS solutions to k-length substrings.

Abstract

In this paper we define a new problem, motivated by computational biology, $LCSk$ aiming at finding the maximal number of $k$ length $substrings$, matching in both input strings while preserving their order of appearance. The traditional LCS definition is a special case of our problem, where $k = 1$. We provide an algorithm, solving the general case in $O(n^2)$ time, where $n$ is the length of the input strings, equaling the time required for the special case of $k=1$. The space requirement of the algorithm is $O(kn)$. %, however, in order to enable %backtracking of the solution, $O(n^2)$ space is needed. We also define a complementary $EDk$ distance measure and show that $EDk(A,B)$ can be computed in $O(nm)$ time and $O(km)$ space, where $m$, $n$ are the lengths of the input sequences $A$ and $B$ respectively.