Computing a Longest Common Palindromic Subsequence

Abstract

The longest common subsequence (LCS) problem is a classic and well-studied problem in computer science. Palindrome is a word which reads the same forward as it does backward. The longest common palindromic subsequence (LCPS) problem is a variant of the classic LCS problem which finds a longest common subsequence between two given strings such that the computed subsequence is also a palindrome. In this paper, we study the LCPS problem and give two novel algorithms to solve it. To the best of our knowledge, this is the first attempt to study and solve this problem.

A hardness result and new algorithm for the longest common palindromic subsequence problem

“…Secondly, we propose a new algorithm for the 2-LCPS problem which runs in O(σM 2 + n) time and uses O(M 2 + n) space, where σ denotes the number of distinct characters occurring in both A and B. We remark that our new algorithm is faster than Chowdhury et al's sparse algorithm with O(M 2 log 2 n log log n + n) running time [5] when σ = o(log 2 n log log n).…”

“…Hence our method runs in O(σM 2 + n) = O(n 4 /σ) expected time. On the other hand, the conventional dynamic programming algorithm of Chowdhury et al [5] takes Θ(n 4 ) time for any input strings of length n each. Thus, our method achieves a σ-factor speed-up in expectation.…”

“…[4,3,12,15,6,7,20,8,21,22]). Along this line of research, Chowdhury et al [5] introduced the longest common palindromic subsequence problem for two strings (2-LCPS problem for short), which asks one to compute (the length of) a longest common subsequence between strings A and B with the additional constraint that the subsequence must be a palindrome. The problem is equivalent to finding (the length of) a longest palindrome that appears as a subsequence in both strings A and B, and is motivated for biological sequence comparison [5].…”

“…For simplicity, we assume that input strings are of equal length n. However, all algorithms mentioned and proposed in this paper are applicable for strings of different lengths 2. The original time bound claimed in[5] is O(M 2 log 2 n log log n), since they assume that the matching position pairs are already computed. For given strings A and B of length n each over an integer alphabet of polynomial size in n, we can compute all matching position pairs of A and B in O(M + n) time.…”

Longest property-preserved common factor: A new string-processing framework

“…We are given two or more strings and our goal is to compute a factor common to all strings that preserves a specific property and has maximal length. An analogous line of research was introduced in [12]. The goal is to compute a subsequence (rather than a factor) common to all strings that preserves a specific property and has maximal length.…”