Position Heaps for Cartesian-tree Matching on Strings and Tries

TL;DR

This paper introduces the Cartesian-tree Position Heap (CPH), an efficient indexing structure for pattern matching based on Cartesian trees in strings and tries, with optimized query and construction times.

Contribution

The paper presents the novel CPH data structure for Cartesian-tree pattern matching on strings and tries, improving query efficiency and providing construction algorithms.

Findings

Supports pattern matching in O(m (σ + log(min{h,m})) + occ) time for strings.

Supports pattern matching in O(m (σ^2 + log(min{h,m})) + occ) time for tries.

Constructs CPH in O(n log σ) time for strings and O(N σ) time for 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 $T$ with $N$ nodes, we show that the CPH of $T$, denoted $\mathsf{CPH}(T)$, 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 $\mathsf{CPH}(T)$ running in $O(N \sigma)$ time and $O(N \sigma)$ working space.