Longest Common Subsequences in Permutations and Maximum Cliques in Circle Graphs
For two strings a, b, the longest common subsequence (LCS) problem consists in comparing a and b by computing the length of their LCS . In a previous paper, we defined a generalisation, called “the all semi-local LCS problem”, for which we proposed an efficient output representation and an efficient algorithm. In this paper, we consider a restriction of this problem to strings that are permutations of a given set. The resulting problem is equivalent to the all local longest increasing subsequences (LIS) problem. We propose an algorithm for this problem, running in time O(n 1.5) on an input of size n. As an interesting application of our method, we propose a new algorithm for finding a maximum clique in a circle graph on n nodes, running in the same asymptotic time O(n 1.5). Compared to a number of previous algorithms for this problem, our approach presents a substantial improvement in worst-case running time.
KeywordsMaximum Clique Circle Graph Longe Common Subsequence Interval Model Maximum Clique Problem
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