Cartesian Tree Subsequence Matching

Park et al. [TCS 2020] observed that the similarity between two (numerical) strings can be captured by the Cartesian trees: The Cartesian tree of a string is a binary tree recursively constructed by picking up the smallest value of the string as the root of the tree. Two strings of equal length are said to Cartesian-tree match if their Cartesian trees are isomorphic. Park et al. [TCS 2020] introduced the following Cartesian tree substring matching (CTMStr) problem: Given a text string $T$ of length $n$ and a pattern string of length $m$, find every consecutive substring $S = T[i..j]$ of a text string $T$ such that $S$ and $P$ Cartesian-tree match. They showed how to solve this problem in $\tilde{O}(n+m)$ time. In this paper, we introduce the Cartesian tree subsequence matching (CTMSeq) problem, that asks to find every minimal substring $S = T[i..j]$ of $T$ such that $S$ contains a subsequence $S'$ which Cartesian-tree matches $P$. We prove that the CTMSeq problem can be solved efficiently, in $O(m n p(n))$ time, where $p(n)$ denotes the update/query time for dynamic predecessor queries. By using a suitable dynamic predecessor data structure, we obtain $O(mn \log \log n)$-time and $O(n \log m)$-space solution for CTMSeq. This contrasts CTMSeq with closely related order-preserving subsequence matching (OPMSeq) which was shown to be NP-hard by Bose et al. [IPL 1998].