arXiv subjects

Braeden Sopp

Publications and source records attributed to Braeden Sopp.

5 recordsLinked to original sources

arXiv2026

Parametric Shortest Paths in a Linearly Interpolated Graph

We consider the parametric shortest paths problem in a linearly interpolated graph. Given two positively-weighted directed graphs $G_0=(V,E,\omega_0)$ and $G_1=(V,E,\omega_1),$ the linearly interpolated graph is the family of graphs $(1-\lambda)G_0+\lambda G_1$, parameterized by $\lambda\in [0,1]$. The problem is to compute all distinct parametric shortest paths. We compute a data structure in $\Theta(k|E|\log |V|)$ time, where~$k$ is the number of distinct parametric shortest paths over all~$\lambda\in [0,1]$ that exist for a nontrivial interval of parameters, each corresponding to a linear function in a maximal sub-interval of $[0,1]$. Using this data structure, a shortest path query takes~$\Theta(\log k)$ time.

cs.CG↗

arXiv2026

Computing and Enumerating Minimal Common Supersequences Between Two Strings

Given $k$ strings each of length at most $n$, computing the shortest common supersequence of them is a well-known NP-hard problem (when $k$ is unbounded). On the other hand, when $k=2$, such a shortest common supersequence can be computed in $O(n^2)$ time using dynamic programming as a textbook example. In this paper, we consider the problem of computing a \emph{minimal} common supersequence and enumerating all minimal common supersequences for $k=2$ input strings. Our results are summarized as follows. A minimal common supersequence of $k=2$ input strings can be computed in $O(n)$ time. (The method also works when $k$ is a constant). All minimal common supersequences between two input strings can be enumerated with a data structure of $O(n^2)$ space and an $O(n)$ time delay, and the data structure can be constructed in $O(n^3)$ time.

cs.DS↗

arXiv2026

Computing Maximal Repeating Subsequences in a String

In this paper we initiate the study of computing a maximal (not necessarily maximum) repeating pattern in a single input string, where the corresponding problems have been studied (e.g., a maximal common subsequence) only in two or more input strings by Hirota and Sakai starting 2019. Given an input string $S$ of length $n$, we can compute a maximal square subsequence of $S$ in $O(n\log n)$ time, greatly improving the $O(n^2)$ bound for computing the longest square subsequence of $S$. For a maximal $k$-repeating subsequence, our bound is $O(f(k)n\log n)$, where $f(k)$ is a computable function such that $f(k) < k\cdot 4^k$. This greatly improves the $O(n^{2k-1})$ bound for computing a longest $k$-repeating subsequence of $S$, for $k\geq 3$. Both results hold for the constrained case, i.e., when the solution must contain a subsequence $X$ of $S$, though with higher running times.

cs.DS↗

arXiv2025

Fast Order Statistics with Group Inequality Testing

Suppose that a group test operation is available for checking order relations in a set, can this speed up problems like finding the minimum/maximum element, determining the rank of element, and computing order statistics? We consider a one-sided group inequality test to be available, where queries are of the form $u \le_Q V$ or $V \le_Q u$, and the answer is `yes' if and only if there is some $v \in V$ such that $u \le v$ or $v \le u$, respectively. We restrict attention to total orders and focus on query-complexity; for min or max finding, we give a Las Vegas algorithm that makes $\mathcal{O}(\log^2 n)$ expected queries. We observe that rank determination can be solved with existing ``defect-counting'' algorithms, but also give a simple Monte Carlo approximation algorithm with expected query complexity $\tilde{\mathcal{O}}(\frac{1}{\delta^2} \log \frac{1}{\epsilon})$, where $1-\epsilon$ is the probability that the algorithm succeeds and we allow a relative error of $1 \pm \delta$ for $\delta > 0$ in the estimated rank. We then give a Monte Carlo algorithm for approximate selection that has expected query complexity $\tilde{\mathcal{O}}(\frac{1}{\delta^4}\log \frac{1}{\epsilon \delta^2} )$; it has probability at least $\frac{1}{2}$ to output an element $x$, and if so, $x$ has the desired approximate rank with probability $1-\epsilon$. Keywords: Order statistics, Group inequality testing, Randomized algorithms

cs.DS↗

arXiv2025

On the difficulty of order constrained pattern matching with applications to feature matching based malware detection

We formulate low-level malware detection using algorithms based on feature matching as Order-based Malware Detection with Critical Instructions (General-OMDCI): given a pattern in the form of a sequence $M$ of colored blocks, where each block contains a critical character (representing a unique sequence of critical instructions potentially associated with malware but without certainty), and a program $A$, represented as a sequence of $n$ colored blocks with critical characters, the goal is to find two subsequences, $M'$ of $M$ and $A'$ of $A$, with blocks matching in color and whose critical characters form a permutation of each other. When $M$ is a permutation in both colors and critical characters the problem is called OMDCI. If we additionally require $M'=M$, then the problem is called OMDCI+; if in this case $d=|M|$ is used as a parameter, then the OMDCI+ problem is easily shown to be FPT. Our main (negative) results are on the cases when $|M|$ is arbitrary and are summarized as follows: OMDCI+ is NP-complete, which implies OMDCI is also NP-complete. For the special case of OMDCI, deciding if the optimal solution has length $0$ (i.e., deciding if no part of $M$ appears in $A$) is co-NP-hard. As a result, the OMDCI problem does not admit an FPT algorithm unless P=co-NP. In summary, our results imply that using algorithms based on feature matching to identify malware or determine the absence of malware in a given low-level program are both hard.

cs.CC↗