Improved Algorithms for the Boxed-Mesh Permutation Pattern Matching Problem

Abstract

Recently, various types of permutation patterns such as mesh patterns, boxed-mesh patterns, and consecutive patterns, have been studied where relative order between characters is considered rather than characters themselves. Among these, we focus on boxed-mesh patterns and study the problem of finding all boxed-subsequences of a text $T$ of length $n$ whose relative order between characters is the same as that of a pattern $P$ of length $m$. Recently, it is known that this problem can be solved in $O(n^3)$ time. In this paper, we first propose an $O(n^2 m)$-time algorithm for the problem based on interesting properties of boxed subsequences. Then, we give a further improved algorithm which runs in $O(n^2 \log m)$ time using preprocessed information on $P$ and order-statistics trees.