Pattern Matching for Separable Permutations

Abstract

Given a permutation $\pi$ (called the text) of size n and another permutation $\sigma$ (called the pattern) of size k, the NP-complete permutation pattern matching problem asks whether $\sigma$ occurs in $\pi$ 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 (Finding pattern matchings for permutations, Information Processing Letters 61 (1997), no. 6) 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 $O(n^2 k)$ time and O(nk) space algorithm. Furthermore, we show how to use this approach to decide in $O(n k^3 \ell ^2)$ time whether a separable permutation of size n is a disjoint union of two given permutations of size k and $\ell$.

• Given a permutation of size n and a separable permutation of size k, we propose an $O(n^ 6k)$ time and $O(n^4 \log k)$ space algorithm to compute the largest common separable permutation that occurs in the two input permutations. This improves upon the existing $O(n^8)$ time algorithm by Rossin and Bouvel (The longest common pattern problem for two permutations, Pure Mathematics and Applications 17 (2006)).

• Finally, we give a $O(n^6 k)$ time and 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).