On the Common Factors in a Set of Linear Orders

We give an O(n) algorithm for finding the PQ tree of a consecutive-ones matrix if a consecutive-ones ordering is given, where n is the number of columns and an O(n)-space representation of the matrix is given. We use this to obtain the modular decomposition of permutation graphs and two-dimensional partial orders inO(n) time when their compact representation with two linear orders is given. More generally, given a set of k linear orders on a set V , we find a decomposition tree that gives a representation of all sets that form consecutive intervals in all of the linear orders. There is a natural associative, commutative intersection operator on such decomposition trees and show how to evaluate it in O(|V |) time. We use these results to obtain a linear time bound for modular decomposition of 2-structures.