A Practical O(R\log\log n+n) time Algorithm for Computing the Longest Common Subsequence

TL;DR

This paper introduces three new algorithms for the Longest Common Subsequence problem, improving efficiency by reformulating the problem and utilizing advanced data structures, achieving near-linear time complexity in certain cases.

Contribution

The paper presents three novel algorithms for LCS with improved time complexities, reformulating the problem through an abstract data type and leveraging data structures like van Emde Boas trees.

Findings

First algorithm: O(R log log n + n) time, O(R) space

Second algorithm: O(R log L + n) time, O(R) space

Third algorithm: O(nL) time, O(R) space

Abstract

In this paper, we revisit the much studied LCS problem for two given sequences. Based on the algorithm of Iliopoulos and Rahman for solving the LCS problem, we have suggested 3 new improved algorithms. We first reformulate the problem in a very succinct form. The problem LCS is abstracted to an abstract data type DS on an ordered positive integer set with a special operation Update(S,x). For the two input sequences X and Y of equal length n, the first improved algorithm uses a van Emde Boas tree for DS and its time and space complexities are O(R\log\log n+n) and O(R), where R is the number of matched pairs of the two input sequences. The second algorithm uses a balanced binary search tree for DS and its time and space complexities are O(R\log L+n) and O(R), where L is the length of the longest common subsequence of X and Y. The third algorithm uses an ordered vector for DS and its time and space complexities are O(nL) and O(R).