Approximate Cartesian Tree Matching with Substitutions
Panagiotis Charalampopoulos, Jonas Ellert, Manal Mohamed
https://arxiv.org/abs/2602.08570 https://arxiv.org/pdf/2602.08570 https://arxiv.org/html/2602.08570
arXiv:2602.08570v1 Announce Type: new
Abstract: The Cartesian tree of a sequence captures the relative order of the sequence's elements. In recent years, Cartesian tree matching has attracted considerable attention, particularly due to its applications in time series analysis. Consider a text $T$ of length $n$ and a pattern $P$ of length $m$. In the exact Cartesian tree matching problem, the task is to find all length-$m$ fragments of $T$ whose Cartesian tree coincides with the Cartesian tree $CT(P)$ of the pattern. Although the exact version of the problem can be solved in linear time [Park et al., TCS 2020], it remains rather restrictive; for example, it is not robust to outliers in the pattern.
To overcome this limitation, we consider the approximate setting, where the goal is to identify all fragments of $T$ that are close to some string whose Cartesian tree matches $CT(P)$. In this work, we quantify closeness via the widely used Hamming distance metric. For a given integer parameter $k>0$, we present an algorithm that computes all fragments of $T$ that are at Hamming distance at most $k$ from a string whose Cartesian tree matches $CT(P)$. Our algorithm runs in time $\mathcal O(n \sqrt{m} \cdot k^{2.5})$ for $k \leq m^{1/5}$ and in time $\mathcal O(nk^5)$ for $k \geq m^{1/5}$, thereby improving upon the state-of-the-art $\mathcal O(nmk)$-time algorithm of Kim and Han [TCS 2025] in the regime $k = o(m^{1/4})$.
On the way to our solution, we develop a toolbox of independent interest. First, we introduce a new notion of periodicity in Cartesian trees. Then, we lift multiple well-known combinatorial and algorithmic results for string matching and periodicity in strings to Cartesian tree matching and periodicity in Cartesian trees.
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Revised January 8, 2026: Simulation of prosthetic vision with the PRIMA system and enhancement of face representation https://arxiv.org/abs/2503.11677 retinal implant
Another evening of filter refactoring and optimization, a few more nice performance jumps.
I've now done a first pass (remove deprecated method signatures, add explicit input location, add NVTX trace data, do easy GPU optimization if I see an obvious low effort win) on all filters A-F alphabetically plus a few later on that were priorities for one reason or other.
90 down (of which 23 were optimized and the rest just refactored), 115 to go.
Some of the remaining ones sh…
The Associated Press has uncovered a pattern where Bangladeshi workers are enticed to Russia under false pretences of civilian employment, subsequently coerced into combat roles.
Many of these individuals faced threats of violence, imprisonment, or even death if they refused.
h…
D05 - Animals
CAPTAIN: One of our pursuit units engaged an unidentified planet hopper. They refused to obey a stand and be searched order and the pursuit commander opened fire.
SERVALAN: Why all the interest? What's one civilian cargo ship more or less?
https://blake.torpidity.net/m/405/97
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