Pattern Matching for Separable Permutations

Abstract

Abstract. Given a permutation π (called the text) of size n and another permutation σ (called the pattern) of size k, the NP-complete pattern containment problem asks whether σ is contained in π as an order-isomorphic subsequence. In this paper, we focus on separable permutations (those permutations that avoid both 2413 and 3142, or, equivalently, that admit a separating tree). The main contributions presented in this paper are as follows.-We simplify the algorithm of Ibarra to detect an occurrence of a separable permutation in a permutation and show how to reduce the space complexity from O(n 3 k) to O(n 3 log k). -In case both the text and the pattern are separable permutations, we give a more practicable alternative O(n 2 k) time and O(nk) space algorithm. Furtheremore, we show how to use this approach to decide in O(nk 3 2 ) time whether a separable permutation is a disjoint union of two given permutations of size k and -We give a O(n 6 k) time and O(n 4 log k) space algorithm to compute the longest common pattern of two permutations of size at most n (provided that at least one of these permutations is separable). This improves upon the existing O(n 8 ) time algorithm. -Finally, we give a O(n 6 k) time and O(kn 4 ) space algorithm to detect an occurrence of a bivincular separable permutation in a permutation. (Bivincular patterns generalize classical permutations by requiring that positions and values involved in an occurrence may be forced to be adjacent).

Unshuffling Permutations: Trivial Bijections and Compositions

“…Given (3142, 2413)-avoiding (a.k.a. separable) permutations π, σ 1 and σ 2 , it can be decided in polynomial time whether π ∈ σ 1 ¡ σ 2 [8].…”