A generic algorithm to find all common intervals of two permutations

Let K he the set of {1,2,....,m}, [x, y] denote the set of [x,x+1,...,y], where 1≤x,y≤m. Given two permutations σ_A and σ_B of a set ℵ, A 2-tuple of intervals ([x₁, y₁], [x₂, y₂]) is called common intervals if σ_A([x₁, y₁])=([x₂, y₂]). In this paper, we propose a sufficient and necessary condition for a 2-tuple of intervals to be common intervals. Based on these conditions, we present a generic algorithm that finds all common intervals of these two permutations.