Subsequence Matching and LCS under Cartesian-Tree Equivalence

Two strings of the same length are said to Cartesian-tree match (CT-match) if their Cartesian-trees are isomorphic [Park et al., TCS 2020]. Cartesian-tree matching is a natural model that allows for capturing similarities of numerical sequences. Oizumi et al. [CPM 2022] showed that subsequence pattern matching under CT-matching model (CT-MSeq) can be solved in $O(nm \log \log n)$ time, where $n$ and $m$ are text and pattern lengths, respectively. This current article follows this line of research, and gives the following new results: (1) An $O(nm)$-time CT-MSeq algorithm for binary alphabets; (2) An $O((nm)^{1-ε})$-time conditional lower bound for the CT-MSeq problem on alphabets of size 4, for any constant $ε> 0$, under the Orthogonal Vector Hypothesis (OVH). Further, we introduce the new problem of longest common subsequence under CT-matching (CT-LCS) for two given strings $S$ and $T$ of length $n$, and present the following results: (3) An $O(n^6)$-time CT-LCS algorithm for general ordered alphabets; (4) An $O(n^2 / \log n)$-time CT-LCS algorithm for binary alphabets; (5) An $O(n^{2-ε})$-time conditional lower bound for the CT-LCS problem on alphabets of size 5, for any constant $ε> 0$, under OVH.