arXiv subjects

Adiesha Liyanage

Publications and source records attributed to Adiesha Liyanage.

6 recordsLinked to original sources

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.

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arXiv2026

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 $\tilde{\mathcal{O}}(\frac{1}{δ^2} \log \frac{1}ε)$, where $1-ε$ is the probability that the algorithm succeeds and we allow a relative error of $1 \pm δ$ for $δ> 0$ in the estimated rank. We then give a Monte Carlo algorithm for approximate selection that has expected query complexity $\tilde{\mathcal{O}}(\frac{1}{δ^4}\log \frac{1}{εδ^2} )$; it has probability at least $\frac{1}{2}$ to output an element $x$, and if so, $x$ has the desired approximate rank with probability $1-ε$. Keywords: Order statistics, Group inequality testing, Randomized algorithms

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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.

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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.

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arXiv2023

The Longest Subsequence-Repeated Subsequence Problem

Motivated by computing duplication patterns in sequences, a new fundamental problem called the longest subsequence-repeated subsequence (LSRS) is proposed. Given a sequence $S$ of length $n$, a letter-repeated subsequence is a subsequence of $S$ in the form of $x_1^{d_1}x_2^{d_2}\cdots x_k^{d_k}$ with $x_i$ a subsequence of $S$, $x_j\neq x_{j+1}$ and $d_i\geq 2$ for all $i$ in $[k]$ and $j$ in $[k-1]$. We first present an $O(n^6)$ time algorithm to compute the longest cubic subsequences of all the $O(n^2)$ substrings of $S$, improving the trivial $O(n^7)$ bound. Then, an $O(n^6)$ time algorithm for computing the longest subsequence-repeated subsequence (LSRS) of $S$ is obtained. Finally we focus on two variants of this problem. We first consider the constrained version when $Σ$ is unbounded, each letter appears in $S$ at most $d$ times and all the letters in $Σ$ must appear in the solution. We show that the problem is NP-hard for $d=4$, via a reduction from a special version of SAT (which is obtained from 3-COLORING). We then show that when each letter appears in $S$ at most $d=3$ times, then the problem is solvable in $O(n^5)$ time.

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arXiv2022

Beyond the Longest Letter-duplicated Subsequence Problem

Given a sequence $S$ of length $n$, a letter-duplicated subsequence is a subsequence of $S$ in the form of $x_1^{d_1}x_2^{d_2}\cdots x_k^{d_k}$ with $x_i\inΣ$, $x_j\neq x_{j+1}$ and $d_i\geq 2$ for all $i$ in $[k]$ and $j$ in $[k-1]$. A linear time algorithm for computing the longest letter-duplicated subsequence (LLDS) of $S$ can be easily obtained. In this paper, we focus on two variants of this problem. We first consider the constrained version when $Σ$ is unbounded, each letter appears in $S$ at least 6 times and all the letters in $Σ$ must appear in the solution. We show that the problem is NP-hard (a further twist indicates that the problem does not admit any polynomial time approximation). The reduction is from possibly the simplest version of SAT that is NP-complete, $(\leq 2,1,\leq 3)$-SAT, where each variable appears at most twice positively and exact once negatively, and each clause contains at most three literals and some clauses must contain exactly two literals. (We hope that this technique will serve as a general tool to help us proving the NP-hardness for some more tricky sequence problems involving only one sequence -- much harder than with at least two input sequences, which we apply successfully at the end of the paper on some extra variations of the LLDS problem.) We then show that when each letter appears in $S$ at most 3 times, then the problem admits a factor $1.5-O(\frac{1}{n})$ approximation. Finally, we consider the weighted version, where the weight of a block $x_i^{d_i} (d_i\geq 2)$ could be any positive function which might not grow with $d_i$. We give a non-trivial $O(n^2)$ time dynamic programming algorithm for this version, i.e., computing an LD-subsequence of $S$ whose weight is maximized.

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