Computing Common Intervals of K Permutations, with Applications to Modular Decomposition of Graphs

Abstract

Abstract. We introduce a new approach to compute the common intervals of K permutations based on a very simple and general notion of generators of common intervals. This formalism leads to simple and efficient algorithms to compute the set of all common intervals of K permutations that can contain a quadratic number of intervals, as well as a linear space basis of this set of common intervals. Finally, we show how our results on permutations can be used for computing the modular decomposition of graphs.

A more efficient algorithm for perfect sorting by reversals

“…The strong interval tree of P can be computed in linear time and space (see [2,3] for example). We call the tree T S (P ) the strong interval tree of P , and we identify a vertex of T S (P ) with the strong interval it represents.…”

IEEE/ACM Transactions on Computational Biology and Bioinformatics

“…The combinatorial structures we consider here are common intervals of signed permutations [7], [22], [41]. Roughly speaking, a common interval of two signed permutations is a set of genes that forms an interval in both permutations, or in other words, that is conserved in the two permutations up to local rearrangements.…”

“…They proposed an O(kn) time algorithm to compute the set of irreducible common intervals of k permutations of n elements. A simpler algorithm is given in [7] (see also [12] for a related work). Note that, if I is a common interval of P and J is an interval of P that does not commute with I, then reversing J in P leads to a permutation P such that I is not a common interval of P .…”

“…The vertex I, or equivalently the strong interval I of P , is either: This representation for strong intervals was first given implicitly in [22], and explicitly in [25], where it was shown that T s (P ) can be related to a data structure widely used in graph theory, called PQ-tree. It can be computed in O(n) time using algorithms similar to the ones given in [7], [22], [25]. A formal link between P Q-trees and conserved structures in signed permutations was first proposed in [5], in the context of conserved intervals, a subset of common intervals.…”

“…The crux is the use of strong interval tree as a guide (we assume it is given, and refer to [7], [12] for simple algorithms building this tree.) Indeed, we obtain a characterization of perfect scenarios of a signed permutation P in terms of T s (P ):…”