Fast algorithms for identifying maximal common connected sets of interval graphs

Abstract

Given a family of interval graphs F = { G 1 = ( V , E 1 ) , … , G k = ( V , E k ) } on the same vertices V, a set S ⊂ V is a maximal common connected set of F if the subgraphs of G i , 1 ⩽ i ⩽ k , induced by S are connected in all G i and S is maximal for the inclusion order. The maximal general common connected set for interval graphs problem (gen-CCPI) consists in efficiently computing the partition of V in maximal common connected sets of F. This problem has many practical applications, notably in computational biology. Let n = | V | and m = ∑ i = 1 k | E i | . For k ⩾ 2 , an algorithm in O ( ( kn + m ) log n ) time is presented in Habib et al. [Maximal common connected sets of interval graphs, in: Combinatorial Pattern Matching (CPM), Lecture Notes in Computer Science, vol. 3109, Springer, Berlin, 2004, pp. 359–372]. In this paper, we improve this bound to O ( kn log n + m ) . Moreover, if the interval graphs are given as k sets of n intervals, which is often the case in bioinformatics, we present a simple O ( kn log 2 n ) time algorithm.