Position Heaps for Cartesian-Tree Matching on Strings and Tries

Abstract

The Cartesian-tree pattern matching is a recently introduced scheme of pattern matching that detects fragments in a sequential data stream which have a similar structure as a query pattern. Formally, Cartesian-tree pattern matching seeks all substrings $S'$ of the text string S such that the Cartesian tree of $S'$ and that of a query pattern P coincide. In this paper, we present a new indexing structure for this problem, called the Cartesian-tree Position Heap (CPH). Let n be the length of the input text string S, m the length of a query pattern P, and $\sigma$ the alphabet size. We show that the CPH of S, denoted $\mathsf {CPH}(S)$, supports pattern matching queries in $O(m (\sigma + \log (\min \{h, m\})) + occ )$ time with O(n) space, where h is the height of the CPH and $occ$ is the number of pattern occurrences. We show how to build $\mathsf {CPH}(S)$ in $O(n \log \sigma )$ time with O(n) working space. Further, we extend the problem to the case where the text is a labeled tree (i.e. a trie). Given a trie $\boldsymbol{T}$ with N nodes, we show that the CPH of $\boldsymbol{T}$, denoted $\mathsf {CPH}(\boldsymbol{T})$, supports pattern matching queries on the trie in $O(m (\sigma ^2 + \log (\min \{h, m\})) + occ )$ time with $O(N \sigma )$ space. We also show a construction algorithm for $\mathsf {CPH}(\boldsymbol{T})$ running in $O(N \sigma )$ time and $O(N \sigma )$ working space.