Related papers: Simple Quantum Algorithm for Approximate $k$-Misma…
$k$-Clustering in $\mathbb{R}^d$ (e.g., $k$-median and $k$-means) is a fundamental machine learning problem. While near-linear time approximation algorithms were known in the classical setting for a dataset with cardinality $n$, it remains…
We consider the recovery of a nonnegative vector x from measurements y = Ax, where A is an m-by-n matrix whos entries are in {0, 1}. We establish that when A corresponds to the adjacency matrix of a bipartite graph with sufficient…
This study investigates whether reoptimization can help in solving the closest substring problem. We are dealing with the following reoptimization scenario. Suppose, we have an optimal l-length closest substring of a given set of sequences…
We study the Closest Pair Problem in Hamming metric, which asks to find the pair with the smallest Hamming distance in a collection of binary vectors. We give a new randomized algorithm for the problem on uniformly random input…
In the paper, we investigate two problems on strings. The first one is the String matching problem, and the second one is the String comparing problem. We provide a quantum algorithm for the String matching problem that uses exponentially…
Given a set $P$ of $n$ points in $\mathbf{R}^d$, and a positive integer $k \leq n$, the $k$-dispersion problem is that of selecting $k$ of the given points so that the minimum inter-point distance among them is maximized (under Euclidean…
We study the problem of estimating the size of a maximum matching in sublinear time. The problem has been studied extensively in the literature and various algorithms and lower bounds are known for it. Our result is a $0.5109$-approximation…
We study the internal dictionary matching (IDM) problem where a dictionary $\mathcal{D}$ containing $d$ substrings of a text $T$ is given, and each query concerns the occurrences of patterns in $\mathcal{D}$ in another substring of $T$. We…
We provide a new quantum algorithm that efficiently determines the quality of a least-squares fit over an exponentially large data set by building upon an algorithm for solving systems of linear equations efficiently (Harrow et al., Phys.…
Sequence classification algorithms, such as SVM, require a definition of distance (similarity) measure between two sequences. A commonly used notion of similarity is the number of matches between $k$-mers ($k$-length subsequences) in the…
The subspace approximation problem Subspace($k$,$p$) asks for a $k$-dimensional linear subspace that fits a given set of points optimally, where the error for fitting is a generalization of the least squares fit and uses the $\ell_{p}$ norm…
Error Tree is a novel tree structure that is mainly oriented to solve the approximate pattern matching problems, Hamming and edit distances, as well as the wildcards matching problem. The input is a text of length $n$ over a fixed alphabet…
Given $d$ strings over the alphabet $\{0,1,\ldots,\sigma{-}1\}$, the classical Aho--Corasick data structure allows us to find all $occ$ occurrences of the strings in any text $T$ in $O(|T| + occ)$ time using $O(m\log m)$ bits of space,…
We consider the problem of counting the copies of a length-$k$ pattern $\sigma$ in a sequence $f \colon [n] \to \mathbb{R}$, where a copy is a subset of indices $i_1 < \ldots < i_k \in [n]$ such that $f(i_j) < f(i_\ell)$ if and only if…
In the problem of the longest common substring with $k$ mismatches we are given two strings $X, Y$ and must find the maximal length $\ell$ such that there is a length-$\ell$ substring of $X$ and a length-$\ell$ substring of $Y$ that differ…
We present a quantum algorithm solving the $k$-distinctness problem in $O(n^{1-2^{k-2}/(2^k-1)})$ queries with a bounded error. This improves the previous $O(n^{k/(k+1)})$-query algorithm by Ambainis. The construction uses a modified…
We consider the longest common subsequence (LCS) problem with the restriction that the common subsequence is required to consist of at least $k$ length substrings. First, we show an $O(mn)$ time algorithm for the problem which gives a…
In the Proper Interval Completion problem we are given a graph G and an integer k, and the task is to turn G using at most k edge additions into a proper interval graph, i.e., a graph admitting an intersection model of equal-length…
For any given $\epsilon>0$ we provide an algorithm for the Densest $k$-Subhypergraph Problem with an approximation ratio of at most $O(n^{\theta_m+2\epsilon})$ for $\theta_m=\frac{1}{2}m-\frac{1}{2}-\frac{1}{2m}$ and run time at most…
Here we study the complexity of string problems as a function of the size of a program that generates input. We consider straight-line programs (SLP), since all algorithms on SLP-generated strings could be applied to processing…