Computing SEQ-IC-LCS of Labeled Graphs

TL;DR

This paper introduces the SEQ-IC-LCS problem for labeled graphs, proposing algorithms to compute the longest common subsequence constrained by a third graph, with solutions for acyclic and cyclic cases.

Contribution

The paper defines the SEQ-IC-LCS problem on labeled graphs and provides algorithms with complexity analysis for both acyclic and cyclic graph cases.

Findings

Algorithms for acyclic graphs run in $O(|E_1||E_2||E_3|)$ time.

Algorithms for cyclic graphs run in $O(|E_1||E_2||E_3| + |V_1||V_2||V_3| ext{log}| extSigma|)$ time.

Space complexity is $O(|V_1||V_2||V_3|)$ for both cases.

Abstract

We consider labeled directed graphs where each vertex is labeled with a non-empty string. Such labeled graphs are also known as non-linear texts in the literature. In this paper, we introduce a new problem of comparing two given labeled graphs, called the SEQ-IC-LCS problem on labeled graphs. The goal of SEQ-IC-LCS is to compute the the length of the longest common subsequence (LCS) $Z$ of two target labeled graphs $G_1 = (V_1, E_1)$ and $G_2 = (V_2, E_2)$ that includes some string in the constraint labeled graph $G_3 = (V_3, E_3)$ as its subsequence. Firstly, we consider the case where $G_1$, $G_2$ and $G_3$ are all acyclic, and present algorithms for computing their SEQ-IC-LCS in $O(|E_1||E_2||E_3|)$ time and $O(|V_1||V_2||V_3|)$ space. Secondly, we consider the case where $G_1$ and $G_2$ can be cyclic and $G_3$ is acyclic, and present algorithms for computing their SEQ-IC-LCS in $O(|E_1||E_2||E_3| + |V_1||V_2||V_3|\log|\Sigma|)$ time and $O(|V_1||V_2||V_3|)$ space, where $\Sigma$ is the alphabet.