Finding all common intervals of k permutations

Abstract

Given k permutations of n elements, a k-tuple of intervals of these permutations consisting of the same set of elements is called a common interval. We present an algorithm that finds in a family of k permutations of n elements all K common intervals in optimal O(nk+K) time and O(n) additional space. This extends a result by Uno and Yagiura (Algorithmica 26, 290-309, 2000) who present an algorithm to find all K common intervals of k = 2 permutations in optimal O(n+K) time and O(n) space. To achieve our result, we introduce the set of irreducible intervals, a generating subset of the set of all common intervals of k permutations.