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.