arXiv subjects

Mirosław Kowaluk

Publications and source records attributed to Mirosław Kowaluk.

7 recordsLinked to original sources

arXiv2025

Fast approximate $\ell$-center clustering in high dimensional spaces

We study the design of efficient approximation algorithms for the $\ell$-center clustering and minimum-diameter $\ell$-clustering problems in high dimensional Euclidean and Hamming spaces. Our main tool is randomized dimension reduction. First, we present a general method of reducing the dependency of the running time of a hypothetical algorithm for the $\ell$-center problem in a high dimensional Euclidean space on the dimension size. Utilizing in part this method, we provide $(2+ε)$- approximation algorithms for the $\ell$-center clustering and minimum-diameter $\ell$-clustering problems in Euclidean and Hamming spaces that are substantially faster than the known $2$-approximation ones when both $\ell$ and the dimension are super-logarithmic. Next, we apply the general method to the recent fast approximation algorithms with higher approximation guarantees for the $\ell$-center clustering problem in a high dimensional Euclidean space. Finally, we provide a speed-up of the known $O(1)$-approximation method for the generalization of the $\ell$-center clustering problem to include $z$ outliers (i.e., $z$ input points can be ignored while computing the maximum distance of an input point to a center) in high dimensional Euclidean and Hamming spaces.

cs.DS↗

arXiv2021

Breaking the hegemony of the triangle method in clique detection

We consider the fundamental problem of detecting/counting copies of a fixed pattern graph in a host graph. The recent progress on this problem has not included complete pattern graphs, i.e., cliques (and their complements, i.e., edge-free pattern graphs, in the induced setting). The fastest algorithms for the aforementioned patterns are based on a straightforward reduction to triangle detection/counting. We provide an alternative method of detection/counting copies of fixed size cliques based on a multi-dimensional matrix product. It is at least as time efficient as the triangle method in cases of $K_4$ and $K_5.$ The complexity of the multi-dimensional matrix product is of interest in its own rights. We provide also another alternative method for detection/counting $K_r$ copies, again time efficient for $r\in \{ 4,\ 5 \}$.

cs.DS↗

arXiv2021

Quantum and approximation algorithms for maximum witnesses of Boolean matrix products

The problem of finding maximum (or minimum) witnesses of the Boolean product of two Boolean matrices (MW for short) has a number of important applications, in particular the all-pairs lowest common ancestor (LCA) problem in directed acyclic graphs (dags). The best known upper time-bound on the MW problem for n\times n Boolean matrices of the form O(n^{2.575}) has not been substantially improved since 2006. In order to obtain faster algorithms for this problem, we study quantum algorithms for MW and approximation algorithms for MW (in the standard computational model). Some of our quantum algorithms are input or output sensitive. Our fastest quantum algorithm for the MW problem, and consequently for the related problems, runs in time \tilde{O}(n^{2+λ/2})=\tilde{O}(n^{2.434}), where λsatisfies the equation ω(1, λ, 1) = 1 + 1.5 \, λand ω(1, λ, 1) is the exponent of the multiplication of an n \times n^λ$ matrix by an n^λ \times n matrix. Next, we consider a relaxed version of the MW problem (in the standard model) asking for reporting a witness of bounded rank (the maximum witness has rank 1) for each non-zero entry of the matrix product. First, by adapting the fastest known algorithm for maximum witnesses, we obtain an algorithm for the relaxed problem that reports for each non-zero entry of the product matrix a witness of rank at most \ell in time \tilde{O}((n/\ell)n^{ω(1,\log_n \ell,1)}). Then, by reducing the relaxed problem to the so called k-witness problem, we provide an algorithm that reports for each non-zero entry C[i,j] of the product matrix C a witness of rank O(\lceil W_C(i,j)/k\rceil ), where W_C(i,j) is the number of witnesses for C[i,j], with high probability. The algorithm runs in \tilde{O}(n^ωk^{0.4653} +n^2k) time, where ω=ω(1,1,1).

cs.DS↗

arXiv2015

$β$-skeletons for a set of line segments in $R^2$

$β$-skeletons are well-known neighborhood graphs for a set of points. We extend this notion to sets of line segments in the Euclidean plane and present algorithms computing such skeletons for the entire range of $β$ values. The main reason of such extension is the possibility to study $β$-skeletons for points moving along given line segments. We show that relations between $β$-skeletons for $β> 1$, $1$-skeleton (Gabriel Graph), and the Delaunay triangulation for sets of points hold also for sets of segments. We present algorithms for computing circle-based and lune-based $β$-skeletons. We describe an algorithm that for $β\geq 1$ computes the $β$-skeleton for a set $S$ of $n$ segments in the Euclidean plane in $O(n^2 α(n) \log n)$ time in the circle-based case and in $O(n^2 λ_4(n))$ in the lune-based one, where the construction relies on the Delaunay triangulation for $S$, $α$ is a functional inverse of Ackermann function and $λ_4(n)$ denotes the maximum possible length of a $(n,4)$ Davenport-Schinzel sequence. When $0 < β< 1$, the $β$-skeleton can be constructed in a $O(n^3 λ_4(n))$ time. In the special case of $β= 1$, which is a generalization of Gabriel Graph, the construction can be carried out in a $O(n \log n)$ time.

cs.CG↗

arXiv2014

Generalized $β$-skeletons

$β$-skeletons, a prominent member of the neighborhood graph family, have interesting geometric properties and various applications ranging from geographic networks to archeology. This paper focuses on developing a new, more general than the present one, definition of $β$-skeletons based only on the distance criterion. It allows us to consider them in many different cases, e.g. for weighted graphs or objects other than points. Two types of $β$-skeletons are especially well-known: the Gabriel Graph (for $β= 1$) and the Relative Neighborhood Graph (for $β= 2$). The new definition retains relations between those graphs and the other well-known ones (minimum spanning tree and Delaunay triangulation). We also show several new algorithms finding $β$-skeletons.

cs.CG↗

arXiv2014

Planar $β$-skeletons via point location in monotone subdivisions of subset of lunes

We present a new algorithm for lune-based $β$-skeletons for sets of $n$ points in the plane, for $β\in (2,\infty]$, the only case when optimal algorithms are not known. The running time of the algorithm is $O(n^{3/2} \log^{1/2} n)$, which is the best known and is an improvement of Rao and Mukhopadhyay \cite{rm97} result. The method is based on point location in monotonic subdivisions of arrangements of curve segments.

cs.CG↗

arXiv2014

Multidimensional $β$-skeletons in $L_1$ and $L_{\infty}$ metric

The $β$-skeleton $\{G_β(V)\}$ for a point set V is a family of geometric graphs, defined by the notion of neighborhoods parameterized by real number $0 < β< \infty$. By using the distance-based version definition of $β$-skeletons we study those graphs for a set of points in $\mathbb{R}^d$ space with $l_1$ and $l_{\infty}$ metrics. We present algorithms for the entire spectrum of $β$ values and we discuss properties of lens-based and circle-based $β$-skeletons in those metrics. Let $V \in \mathbb{R}^d$ in $L_{\infty}$ metric be a set of $n$ points in general position. Then, for $β<2$ lens-based $β$-skeleton $G_β(V)$ can be computed in $O(n^2 \log^d n)$ time. For $β\geq 2$ there exists an $O(n \log^{d-1} n)$ time algorithm that constructs $β$-skeleton for the set $V$. We show that in $\mathbb{R}^d$ with $L_{\infty}$ metric, for $β<2$ $β$-skeleton $G_β(V)$ for $n$ points can be computed in $O(n^2 \log^d n)$ time. For $β\geq 2$ there exists an $O(n \log^{d-1} n)$ time algorithm. In $\mathbb{R}^d$ with $L_1$ metric for a set of $n$ points in arbitrary position $β$-skeleton $G_β(V)$ can be computed in $O(n^2 \log^{d+2} n)$ time.

cs.CG↗