arXiv subjects

Sariel Har-Peled

Publications and source records attributed to Sariel Har-Peled.

At least 55 records · Page 3Linked to original sources

arXiv2023

Revisiting Random Points: Combinatorial Complexity and Algorithms

Consider a set $P$ of $n$ points picked uniformly and independently from $[0,1]^d$ for a constant dimension $d$ -- such a point set is extremely well behaved in many aspects. For example, for a fixed $r \in [0,1]$, we prove a new concentration result on the number of pairs of points of $P$ at a distance at most $r$ -- we show that this number lies in an interval that contains only $O(n \log n)$ numbers. We also present simple linear time algorithms to construct the Delaunay triangulation, Euclidean MST, and the convex hull of the points of $P$. The MST algorithm is an interesting divide-and-conquer algorithm which might be of independent interest. We also provide a new proof that the expected complexity of the Delaunay triangulation of $P$ is linear -- the new proof is simpler and more direct, and might be of independent interest. Finally, we present a simple $\tilde{O}(n^{4/3})$ time algorithm for the distance selection problem for $d=2$.

cs.CG↗

arXiv2023

On the Budgeted Hausdorff Distance Problem

$ \newcommand{\Re}{\mathbb{R}} \newcommand{\reals}{\mathbb{R}} \newcommand{\SetX}{\mathsf{X}} \newcommand{\optX}[1]{#1^\star} \newcommand{\Qopt}{\Mh{\optX{Q}}} \newcommand{\rad}{r} \newcommand{\Mh}[1]{#1} \newcommand{\query}{q} \newcommand{\eps}{\varepsilon} \newcommand{\VorX}[1]{\mathcal{V} \pth{#1}} \newcommand{Polygon}{\mathsf{P}} \newcommand{\IntRange}[1]{[ #1 ]} \newcommand{\Space}{\overline{\mathsf{m}}} \newcommand{\pth}[2][\!]{#1\left({#2}\right)} \newcommand{\polylog}{\mathrm{polylog}} \newcommand{\N}{\mathbb N} \newcommand{\Z}{\mathbb Z} \newcommand{\pt}{p} \newcommand{\distY}[2]{\left\| {#1} - {#2} \right\|} \newcommand{\ptq}{q} \newcommand{\RunningTime}{O\bigl(n^{3/2} \sqrt{k} \log^{3/2} n + kn \log^2 n\bigr)} \newcommand{\pts}{s}$ Given a set $P$ of $n$ points in the plane, and a parameter $k$, we present an algorithm, whose running time is $\RunningTime$, with high probability, that computes a subset $\Qopt \subseteq P$ of $k$ points, that minimizes the Hausdorff distance between the convex-hulls of $\Qopt$ and $P$. This is the first subquadratic algorithm for this problem if $k$ is small.

cs.CG↗

arXiv2023

Improved Approximation Algorithms for Tverberg Partitions

$\newcommand{\floor}[1]{\left\lfloor {#1} \right\rfloor} \renewcommand{\Re}{\mathbb{R}}$ Tverberg's theorem states that a set of $n$ points in $\Re^d$ can be partitioned into $\floor{n/(d+1)}$ sets with a common intersection. A point in this intersection (aka Tverberg point) is a centerpoint of the input point set, and the Tverberg partition provides a compact proof of this, which is algorithmically useful. Unfortunately, computing a Tverberg point exactly requires $n^{O(d^2)}$ time. We provide several new approximation algorithms for this problem, which improve either the running time or quality of approximation, or both. In particular, we provide the first strongly polynomial (in both $n$ and $d$) approximation algorithm for finding a Tverberg point.

cs.CG↗

arXiv2023

Computing Optimal Kernels in Two Dimensions

Let $P$ be a set of $n$ points in $\Re^2$. For a parameter $\varepsilon\in (0,1)$, a subset $C\subseteq P$ is an \emph{$\varepsilon$-kernel} of $P$ if the projection of the convex hull of $C$ approximates that of $P$ within $(1-\varepsilon)$-factor in every direction. The set $C$ is a \emph{weak $\varepsilon$-kernel} of $P$ if its directional width approximates that of $P$ in every direction. Let $\mathsf{k}_{\varepsilon}(P)$ (resp.\ $\mathsf{k}^{\mathsf{w}}_{\varepsilon}(P)$) denote the minimum-size of an $\varepsilon$-kernel (resp. weak $\varepsilon$-kernel) of $P$. We present an $O(n\mathsf{k}_{\varepsilon}(P)\log n)$-time algorithm for computing an $\varepsilon$-kernel of $P$ of size $\mathsf{k}_{\varepsilon}(P)$, and an $O(n^2\log n)$-time algorithm for computing a weak $\varepsilon$-kernel of $P$ of size ${\mathsf{k}}^{\mathsf{w}}_{\varepsilon}(P)$. We also present a fast algorithm for the Hausdorff variant of this problem. In addition, we introduce the notion of \emph{$\varepsilon$-core}, a convex polygon lying inside $\mathsf{ch}(P)$, prove that it is a good approximation of the optimal $\varepsilon$-kernel, present an efficient algorithm for computing it, and use it to compute an $\varepsilon$-kernel of small size.

cs.CG↗

arXiv2023

On the Number of Incidences When Avoiding an Induced Biclique in Geometric Settings

Given a set of points $P$ and a set of regions $\mathcal{O}$, an incidence is a pair $(p,o ) \in P \times \mathcal{O}$ such that $p \in o$. We obtain a number of new results on a classical question in combinatorial geometry: What is the number of incidences (under certain restrictive conditions)? We prove a bound of $O\bigl( k n(\log n/\log\log n)^{d-1} \bigr)$ on the number of incidences between $n$ points and $n$ axis-parallel boxes in $\mathbb{R}^d$, if no $k$ boxes contain $k$ common points, that is, if the incidence graph between the points and the boxes does not contain $K_{k,k}$ as a subgraph. This new bound improves over previous work, by Basit, Chernikov, Starchenko, Tao, and Tran (2021), by more than a factor of $\log^d n$ for $d >2$. Furthermore, it matches a lower bound implied by the work of Chazelle (1990), for $k=2$, thus settling the question for points and boxes. We also study several other variants of the problem. For halfspaces, using shallow cuttings, we get a linear bound in two and three dimensions. We also present linear (or near linear) bounds for shapes with low union complexity, such as pseudodisks and fat triangles.

cs.CG↗

arXiv2023

On the Width of the Regular $n$-Simplex

Consider the regular $n$-simplex $Δ_n$ - it is formed by the convex-hull of $n+1$ points in Euclidean space, with each pair of points being in distance exactly one from each other. We prove an exact bound on the width of $Δ_n$ which is $\approx \sqrt{2/n}$. Specifically, $ \mathrm{width}(Δ_n) = \sqrt{\frac{2}{n + 1}}$ if $n$ is odd, and $ \mathrm{width}(Δ_n) = \sqrt{\frac{2(n+1)}{n(n+2)}} $ if $n$ is even. While this bound is well known [GK92, Ale77], we provide a self-contained elementary proof that might (or might not) be of interest.

cs.CG↗

arXiv2022

Reliable Spanners for Metric Spaces

A spanner is reliable if it can withstand large, catastrophic failures in the network. More precisely, any failure of some nodes can only cause a small damage in the remaining graph in terms of the dilation, that is, the spanner property is maintained for almost all nodes in the residual graph. Constructions of reliable spanners of near linear size are known in the low-dimensional Euclidean settings. Here, we present new constructions of reliable spanners for planar graphs, trees and (general) metric spaces.

cs.CG↗

arXiv2022

Halving by a Thousand Cuts or Punctures

$\newcommand{\Arr}{\mathcal{A}} \newcommand{\numS}{k} \newcommand{\ArrX}[1]{\Arr(#1)} \newcommand{\eps}{\varepsilon} \newcommand{\opt}{\mathsf{o}}$ For point sets $P_1, \ldots, P_\numS$, a set of lines $L$ is halving if any face of the arrangement $\ArrX{L}$ contains at most $|P_i|/2$ points of $P_i$, for all $i$. We study the problem of computing a halving set of lines of minimal size. Surprisingly, we show a polynomial time algorithm that outputs a halving set of size $O(\opt^{3/2})$, where $\opt$ is the size of the optimal solution. Our solution relies on solving a new variant of the weak $\eps$-net problem for corridors, which we believe to be of independent interest. We also study other variants of this problem, including an alternative setting, where one needs to introduce a set of guards (i.e., points), such that no convex set avoiding the guards contains more than half the points of each point set.

cs.CG↗

arXiv2022

Few Cuts Meet Many Point Sets

We study the problem of how to breakup many point sets in $\mathbb{R}^d$ into smaller parts using a few splitting (shared) hyperplanes. This problem is related to the classical Ham-Sandwich Theorem. We provide a logarithmic approximation to the optimal solution using the greedy algorithm for submodular optimization.

cs.CG↗

arXiv2022

A Note on Stabbing Convex Bodies with Points, Lines, and Flats

$\newcommand{\eps}{\varepsilon}\newcommand{\tldO}{\widetilde{O}}$Consider the problem of constructing weak $\eps$-nets where the stabbing elements are lines or $k$-flats instead of points. We study this problem in the simplest setting where it is still interesting -- namely, the uniform measure of volume over the hypercube $[0,1]^d\bigr.$. Specifically, a $(k,\eps)$-net is a set of $k$-flats, such that any convex body in $[0,1]^d$ of volume larger than $\eps$ is stabbed by one of these $k$-flats. We show that for $k \geq 1$, one can construct $(k,\eps)$-nets of size $O(1/\eps^{1-k/d})$. We also prove that any such net must have size at least $Ω(1/\eps^{1-k/d})$. As a concrete example, in three dimensions all $\eps$-heavy bodies in $[0,1]^3$ can be stabbed by $Θ(1/\eps^{2/3})$ lines. Note, that these bounds are \emph{sublinear} in $1/\eps$, and are thus somewhat surprising. The new construction also works for points providing a weak $\eps$-net of size $O(\tfrac{1}{\eps}\log^{d-1} \tfrac{1}{\eps} )$.

cs.CG↗

arXiv2022

Sometimes Reliable Spanners of Almost Linear Size

Reliable spanners can withstand huge failures, even when a linear number of vertices are deleted from the network. In case of failures, a reliable spanner may have some additional vertices for which the spanner property no longer holds, but this collateral damage is bounded by a fraction of the size of the attack. It is known that $Ω(n\log n)$ edges are needed to achieve this strong property, where $n$ is the number of vertices in the network, even in one dimension. Constructions of reliable geometric $(1+\varepsilon)$-spanners, for $n$ points in $\Re^d$, are known, where the resulting graph has $O( n \log n \log \log^{6}n )$ edges. Here, we show randomized constructions of smaller size spanners that have the desired reliability property in expectation or with good probability. The new construction is simple, and potentially practical -- replacing a hierarchical usage of expanders (which renders the previous constructions impractical) by a simple skip-list like construction. This results in a $1$-spanner, on the line, that has linear number of edges. Using this, we present a construction of a reliable spanner in $\Re^d$ with $O( n \log \log^{2} n \log \log \log n )$ edges.

cs.CG↗

arXiv2022

Approximation Algorithms for Maximum Matchings in Geometric Intersection Graphs

We present a $(1- \varepsilon)$-approximation algorithms for maximum cardinality matchings in disk intersection graphs -- all with near linear running time. We also present estimation algorithm that returns $(1\pm \varepsilon)$-approximation to the size of such matchings -- this algorithms run in linear time for unit disks, and $O(n \log n)$ for general disks (as long as the density is relatively small).

cs.CG↗

arXiv2022

Local Spanners Revisited

$\newcommand{\Emph}[1]{\it{#1}} \newcommand{\FF}{\mathcal{F}}\newcommand{\region}{\mathsf{r}}\newcommand{\restrictY}[2]{#1 \cap {#2}}$For a set of points $P \subseteq \mathbb{R}^2$, and a family of regions $\FF$, a $\Emph{local~t-spanner}$ of $P$, is a sparse graph $G$ over $P$, such that, for any region $\region \in \FF$, the subgraph restricted to $\region$, denoted by $\restrictY{G}{\region} = G_{P \cap \region}$, is a $t$-spanner for all the points of $\region \cap P$. We present algorithms for the construction of local spanners with respect to several families of regions, such as homothets of a convex region. Unfortunately, the number of edges in the resulting graph depends logarithmically on the spread of the input point set. We prove that this dependency can not be removed, thus settling an open problem raised by Abam and Borouny. We also show improved constructions (with no dependency on the spread) of local spanners for fat triangles, and regular $k$-gons. In particular, this improves over the known construction for axis parallel squares. We also study a somewhat weaker notion of local spanner where one allows to shrink the region a "bit". Any spanner is a weak local spanner if the shrinking is proportional to the diameter. Surprisingly, we show a near linear size construction of a weak spanner for axis-parallel rectangles, where the shrinkage is $\Emph{multiplicative}$.

cs.CG↗

arXiv2022

Sparsifying Disk Intersection Graphs for Reliable Connectivity

The intersection graph induced by a set $\Disks$ of $n$ disks can be dense. It is thus natural to try and sparsify it, while preserving connectivity. Unfortunately, sparse graphs can always be made disconnected by removing a small number of vertices. In this work, we present a sparsification algorithm that maintains connectivity between two disks in the computed graph, if the original graph remains ``well-connected'' even after removing an arbitrary ``attack'' set $\BSet \subseteq \Disks$ from both graphs. Thus, the new sparse graph has similar reliability to the original disk graph, and can withstand catastrophic failure of nodes while still providing a connectivity guarantee for the remaining graph. The new graphs has near linear complexity, and can be constructed in near linear time. The algorithm extends to any collection of shapes in the plane, such that their union complexity is near linear.

cs.CG↗

arXiv2021

Optimal Algorithms for Geometric Centers and Depth

$\renewcommand{\Re}{\mathbb{R}}$ We develop a general randomized technique for solving "implic it" linear programming problems, where the collection of constraints are defined implicitly by an underlying ground set of elements. In many cases, the structure of the implicitly defined constraints can be exploited in order to obtain efficient linear program solvers. We apply this technique to obtain near-optimal algorithms for a variety of fundamental problems in geometry. For a given point set $P$ of size $n$ in $\Re^d$, we develop algorithms for computing geometric centers of a point set, including the centerpoint and the Tukey median, and several other more involved measures of centrality. For $d=2$, the new algorithms run in $O(n\log n)$ expected time, which is optimal, and for higher constant $d>2$, the expected time bound is within one logarithmic factor of $O(n^{d-1})$, which is also likely near optimal for some of the problems.

cs.CG↗

arXiv2021

On Competitive Permutations for Set Cover by Intervals

We revisit the problem of computing an optimal partial cover of points by intervals. We show that the greedy algorithm computes a permutation $Π= π_1, π_2,\ldots$ of the intervals that is $3/4$-competitive for any prefix of $k$ intervals. That is, for any $k$, the intervals $π_1 \cup \cdots \cup π_k$ covers at least $3/4$-fraction of the points covered by the optimal solution using $k$ intervals. We also provide an approximation algorithm that, in $O(n + m/\varepsilon)$ time, computes a cover by $(1+\varepsilon)k$ intervals that is as good as the optimal solution using $k$ intervals, where $n$ is the number of input points, and $m$ is the number of intervals (we assume here the input is presorted). Finally, we show a counter example illustrating that the optimal solutions for set cover do not have the diminishing return property -- that is, the marginal benefit from using more sets is not monotonically decreasing. Fortunately, the diminishing returns does hold for intervals.

cs.DS↗

arXiv2021

On Clusters that are Separated but Large

$\renewcommand{\Re}{\mathbb{R}}$Given a set $P$ of $n$ points in $\Re^d$, consider the problem of computing $k$ subsets of $P$ that form clusters that are well-separated from each other, and each of them is large (cardinality wise). We provide tight upper and lower bounds, and corresponding algorithms, on the quality of separation, and the size of the clusters that can be computed, as a function of $n,d,k,s$, and $Φ$, where $s$ is the desired separation, and $Φ$ is the spread of the point set $P$.

cs.CG↗

arXiv2021

On Undecided LP, Clustering and Active Learning

We study colored coverage and clustering problems. Here, we are given a colored point set where the points are covered by (unknown) $k$ clusters, which are monochromatic (i.e., all the points covered by the same cluster, have the same color). The access to the colors of the points (or even the points themselves) is provided indirectly via various queries (such as nearest neighbor, or separation queries). We show that if the number of clusters is a constant, then one can correctly deduce the color of all the points (i.e., compute a monochromatic clustering of the points) using a polylogarithmic number of queries. We investigate several variants of this problem, including Undecided Linear Programming, covering of points by $k$ monochromatic balls, covering by $k$ triangles/simplices, and terrain simplification. For the later problem, we present the first near linear time approximation algorithm. While our approximation is slightly worse than previous work, this is the first algorithm to have subquadratic complexity if the terrain has "small" complexity.

cs.CG↗