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Sariel Har-Peled

Publications and source records attributed to Sariel Har-Peled.

At least 37 records · Page 2Linked to original sources
arXiv2025

Bifurcation: How to Explore a Tree

Avraham et al. [AFK+15] presented an alternative approach to parametric search, called \emph{bifurcation}, that performs faster under certain circumstances. Intuitively, when the underlying decider execution can be rolled back cheaply and the decider has a near-linear running time. For some problems, this leads to fast algorithms that beat the seemingly natural lower bound arising from distance selection. Bifurcation boils down to a tree exploration problem. You are given a binary (unfortunately implicit) tree of height $n$ and $k$ internal nodes with two children (all other internal nodes have a single child), and assume each node has an associated parameter value. These values are sorted in the inorder traversal of the tree. Assume there is (say) a node (not necessarily a leaf) that is the target node that the exploration needs to discover. The player starts from the root. At each step, the player can move to adjacent nodes to the current location (i.e., one of the children or the parent). Alternatively, the player can call an oracle on the current node, which returns either that it is the target (thus, mission accomplished!) or whether the target value is strictly smaller or larger than the current one. A naive algorithm explores the whole tree, in $O(n k)$ time, then performs $O(\log k n)$ calls to the oracle to find the desired leaf. Avraham \etal showed that this can be improved to $O(n \sqrt{k} )$ time, and $O( \sqrt{k} \log n)$ oracle calls. Here, we improve this to $O(n \sqrt{k} )$ time, with only $ O( \sqrt{k} + \log n)$ oracle calls. We also show matching lower bounds, under certain assumptions. We believe our interpretation of bifurcation as a tree exploration problem, and the associated algorithm, are of independent interest.

cs.CG↗
arXiv2025

Well-Separated Pairs Decomposition Revisited

We revisit the notion of WSPD (i.e., well-separated pairs-decomposition), presenting a new construction of WSPD for any finite metric space, and show that it is asymptotically instance-optimal in size. Next, we describe a new WSPD construction for the weighted unit-distance metric in the plane, and show a bound $O( \varepsilon^{-2} n \log n)$ on its size, improving by a factor of $1/\varepsilon^2$ over previous work. The new construction is arguably simpler and more elegant. We point out that using WSPD, one can approximate, in near-linear time, the distortion of a bijection between two point sets in low dimensions. As a new application of WSPD, we show how to shortcut a polygonal curve such that its dilation is below a prespecified quantity. In particular, we show a near-linear time algorithm for computing a simple subcurve for a given polygonal curve in the plane so that the new subcurve has no self-intersection.

cs.CG↗
arXiv2025

Near-Optimal Euclidean Locality-Sensitive Orderings

$ \newcommand{\Re}{\mathbb{R}} \newcommand{\reals}{\mathbb{R}} \newcommand{\SetX}{\mathsf{X}} \newcommand{\rad}{r} \newcommand{\Eps}{\Mh{\mathcal{E}}} \newcommand{\p}{\Mh{p}} \newcommand{\q}{\Mh{q}} \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{\pts}{s}$ For a parameter $\eps \in (0,1)$, we present a new construction of $\eps$-locality-sensitive orderings (<LSOs) in $\Re^d$ of size $M = O(\Eps^{d-1} \log \Eps)$, where $\Eps = 1/\eps$. This improves over previous work by a factor of $\Eps$, and is optimal up to a factor of $\log \Eps$. Such a set of LSOs has the property that for any two points, $\p, \q \in [0,1]^d$, there exist an order in the set such that all the points between $\p$ and $\q$ in the order are $\eps$-close to either $\p$ or $\q$. The existence of such LSOs is a fundamental property of low dimensional Euclidean space, conceptually similar to the existence of well-separated pairs decomposition, so the question of how to compute (near) optimal construction of LSOs is quite natural. As a consequence we get a flotilla of improved dynamic geometric algorithms, such as maintaining bichromatic closest pair, and spanners, among others. In particular, for geometric dynamic spanners the new result matches (up to the aforementioned $\log \Eps$ factor) the lower bound, Thus offering a near-optimal simple dynamic data-structure for maintaining spanners under insertions and deletions.

cs.CG↗
arXiv2025

Edge Nearest Neighbor in Sampling-Based Motion Planning

Neighborhood finders and nearest neighbor queries are fundamental parts of sampling based motion planning algorithms. Using different distance metrics or otherwise changing the definition of a neighborhood produces different algorithms with unique empiric and theoretical properties. In \cite{l-pa-06} LaValle suggests a neighborhood finder for the Rapidly-exploring Random Tree RRT algorithm \cite{l-rrtnt-98} which finds the nearest neighbor of the sampled point on the swath of the tree, that is on the set of all of the points on the tree edges, using a hierarchical data structure. In this paper we implement such a neighborhood finder and show, theoretically and experimentally, that this results in more efficient algorithms, and suggest a variant of the Rapidly-exploring Random Graph RRG algorithm \cite{f-isaom-10} that better exploits the exploration properties of the newly described subroutine for finding narrow passages.

cs.RO↗
arXiv2025

Improving the average dilation of a metric graph by adding edges

For a graph $G$ spanning a metric space, the dilation of a pair of points is the ratio of their distance in the shortest path graph metric to their distance in the metric space. Given a graph $G$ and a budget $k$, a classic problem is to augment $G$ with $k$ additional edges to reduce the maximum dilation. In this note, we consider a variant of this problem where the goal is to reduce the average dilation for pairs of points in $G$. We provide an $O(k)$ approximation algorithm for this problem, matching the approximation ratio given by prior work for the maximum dilation variant.

cs.CG↗
arXiv2025

A Practical Approach for Computing the Diameter of a Point Set

We present an approximation algorithm for computing the diameter of a point-set in $\Re^d$. The new algorithm is sensitive to the ``hardness'' of computing the diameter of the given input, and for most inputs it is able to compute the exact diameter extremely fast. The new algorithm is simple, robust, has good empirical performance, and can be implemented quickly. As such, it seems to be the algorithm of choice in practice for computing/approximating the diameter.

cs.CG↗
arXiv2025

Orthogonal Emptiness Queries for Random Points

We present a data-structure for orthogonal range searching for random points in the plane. The new data-structure uses (in expectation) $O\bigl(n \log n ( \log \log n)^2 \bigr)$ space, and answers emptiness queries in constant time. As a building block, we construct a data-structure of expected linear size, that can answer predecessor/rank queries, in constant time, for random numbers sampled uniformly from $[0,1]$. While the basic idea we use is known [Dev89], we believe our results are still interesting.

cs.CG↗
arXiv2025

Fast Approximation Algorithms for Piercing Boxes by Points

$\newcommand{\popt}{\mathcal{p}} \newcommand{\Re}{\mathbb{R}}\newcommand{\N}{\mathcal{N}} \newcommand{\BX}{\mathcal{B}} \newcommand{\bb}{\mathsf{b}} \newcommand{\eps}{\varepsilon} \newcommand{\polylog}{\mathrm{polylog}} $ Let $\mathcal{B}=\{\mathsf{b}_1, \ldots ,\mathsf{b}_n\}$ be a set of $n$ axis-aligned boxes in $\Re^d$ where $d\geq2$ is a constant. The \emph{piercing problem} is to compute a smallest set of points $\N \subset \Re^d$ that hits every box in $\mathcal{B}$, i.e., $\N\cap \mathsf{b}_i\neq \emptyset$, for $i=1,\ldots, n$. Let $\popt=\popt(\mathcal{B})$, the \emph{piercing number} be the minimum size of a piercing set of $\mathcal{B}$. We present a randomized $O(d^2\log\log \popt)$-approximation algorithm with expected running time $O(n^{d/2}\polylog n)$. Next, we present a faster $O(n^{\log d+1})$-time algorithm but with a slightly inferior approximation factor of $O(2^{4d}\log\log\popt)$. The running time of both algorithms can be improved to near-linear using a sampling-based technique, if $\popt = O(n^{1/d})$. For the dynamic version of the problem in the plane, we obtain a randomized $O(\log\log\popt)$-approximation algorithm with $O(n^{1/2}\polylog n )$ amortized expected update time for insertion or deletion of boxes. For squares in $\Re^2$, the update time can be improved to $O(n^{1/3}\polylog n )$.

cs.CG↗
arXiv2025

Quickly Avoiding a Random Catastrophe

We study the problem of constructing simulations of a given randomized search algorithm \texttt{alg} with expected running time $O( \mathcal{O} \log \mathcal{O})$, where $\mathcal{O}$ is the optimal expected running time of any such simulation. Counterintuitively, these simulators can be dramatically faster than the original algorithm in getting alg to perform a single successful run, and this is done without any knowledge about alg, its running time distribution, etc. For example, consider an algorithm that randomly picks some integer $t$ according to some distribution over the integers, and runs for $t$ seconds. then with probability $1/2$ it stops, or else runs forever (i.e., a catastrophe). The simulators described here, for this case, all terminate in constant expected time, with exponentially decaying distribution on the running time of the simulation. Luby et al. studied this problem before -- and our main contribution is in offering several additional simulation strategies to the one they describe. In particular, one of our (optimal) simulation strategies is strikingly simple: Randomly pick an integer $t>0$ with probability $c/t^2$ (with $c= 6/π^2$). Run the algorithm for $t$ seconds. If the run of alg terminates before this threshold is met, the simulation succeeded and it exits. Otherwise, the simulator repeat the process till success.

cs.DS↗
arXiv2025

How Packed Is It, Really?

The congestion of a curve is a measure of how much it zigzags around locally. More precisely, a curve $π$ is $c$-packed if the length of the curve lying inside any ball is at most $c$ times the radius of the ball, and its congestion is the minimum $c$ for which $π$ is $c$-packed. This paper presents a randomized $42$-approximation algorithm for computing the congestion of a curve (or any set of segments in the plane). It runs in $O( n \log^2 n)$ time and succeeds with high probability. Although the approximation factor is large, the running time improves over the previous fastest constant approximation algorithm, which took $\widetilde{O}(n^{4/3})$ time. We carefully combine new ideas with known techniques to obtain our new near-linear time algorithm.

cs.CG↗
arXiv2025

Dependable Spanners via Unreliable Edges

Let $P$ be a set of $n$ points in $\mathbb{R}^d$, and let $\varepsilon,ψ\in (0,1)$ be parameters. Here, we consider the task of constructing a $(1+\varepsilon)$-spanner for $P$, where every edge might fail (independently) with probability $1-ψ$. For example, for $ψ=0.1$, about $90\%$ of the edges of the graph fail. Nevertheless, we show how to construct a spanner that survives such a catastrophe with near linear number of edges. The measure of reliability of the graph constructed is how many pairs of vertices lose $(1+\varepsilon)$-connectivity. Surprisingly, despite the spanner constructed being of near linear size, the number of failed pairs is close to the number of failed pairs if the underlying graph was a clique. Specifically, we show how to construct such an exact dependable spanner in one dimension of size $O(\tfrac{n}ψ \log n)$, which is optimal. Next, we build an $(1+\varepsilon)$-spanners for a set $P \subseteq \mathbb{R}^d$ of $n$ points, of size $O( C n \log n )$, where $C \approx 1/\bigl(\varepsilon^{d} ψ^{4/3}\bigr)$. Surprisingly, these new spanners also have the property that almost all pairs of vertices have a $\leq 4$-hop paths between them realizing this short path.

cs.CG↗
arXiv2025

No-dimensional Tverberg Partitions Revisited

$ \newcommand{\epsA}{\Mhδ} \newcommand{\Re}{\mathbb{R}} \newcommand{\reals}{\mathbb{R}} \newcommand{\SetX}{\mathsf{X}} \renewcommand¶{P} \newcommand{\diam}Δ \newcommand{\Mh}[1]{#1} \newcommand{\query}{q} \newcommand{\eps}{\varepsilon} \newcommand{\VorX}[1]{\mathcal{V} \pth{#1}} \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{\PP}{P} \newcommand{\ptq}{q} \newcommand{\pts}{s}$Given a set $P \subset \Re^d$ of $n$ points, with diameter $\diam$, and a parameter $\epsA \in (0,1)$, it is known that there is a partition of $P$ into sets $P_1, \ldots, P_t$, each of size $O(1/\epsA^2)$, such that their convex-hulls all intersect a common ball of radius $\epsA \diam$. We prove that a random partition, with a simple alteration step, yields the desired partition, resulting in a (randomized) linear time algorithm. We also provide a deterministic algorithm with running time $O( dn \log n)$. Previous proofs were either existential (i.e., at least exponential time), or required much bigger sets. In addition, the algorithm and its proof of correctness are significantly simpler than previous work, and the constants are slightly better. We also include a number of applications and extensions using the same central ideas. For example, we provide a linear time algorithm for computing a ``fuzzy'' centerpoint, and prove a no-dimensional weak $\eps$-net theorem with an improved constant.

cs.CG↗
arXiv2024

SPITE: Simple Polyhedral Intersection Techniques for modified Environments

Motion planning in modified environments is a challenging task, as it compounds the innate difficulty of the motion planning problem with a changing environment. This renders some algorithmic methods such as probabilistic roadmaps less viable, as nodes and edges may become invalid as a result of these changes. In this paper, we present a method of transforming any configuration space graph, such as a roadmap, to a dynamic data structure capable of updating the validity of its nodes and edges in response to discrete changes in obstacle positions. We use methods from computational geometry to compute 3D swept volume approximations of configuration space points and curves to achieve 10-40 percent faster updates and up to 60 percent faster motion planning queries than previous algorithms while requiring a significantly shorter pre-processing phase, requiring minutes instead of hours needed by the competing method to achieve somewhat similar update times.

cs.RO↗
arXiv2024

Proof of Dudley's Convex Approximation

We provide a self contained proof of a result of Dudley [Dud64]} which shows that a bounded convex-body in $\Re^d$ can be $\varepsilon$-approximated, by the intersection of $O_d\bigl(\varepsilon^{-(d-1)/2} \bigr)$ halfspaces, where $O_d$ hides constants that depends on $d$.

cs.CG↗
arXiv2024

Approximating Densest Subgraph in Geometric Intersection Graphs

$ \newcommand{\cardin}[1]{\left| {#1} \right|}% \newcommand{\Graph}{\Mh{\mathsf{G}}}% \providecommand{\G}{\Graph}% \renewcommand{\G}{\Graph}% \providecommand{\GA}{\Mh{H}}% \renewcommand{\GA}{\Mh{H}}% \newcommand{\VV}{\Mh{\mathsf{V}}}% \newcommand{\VX}[1]{\VV\pth{#1}}% \providecommand{\EE}{\Mh{\mathsf{E}}}% \renewcommand{\EE}{\Mh{\mathsf{E}}}% \newcommand{\Re}{\mathbb{R}} \newcommand{\reals}{\mathbb{R}} \newcommand{\SetX}{\mathsf{X}} \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{\pts}{s}$ For an undirected graph $\mathsf{G}=(\mathsf{V}, \mathsf{E})$, with $n$ vertices and $m$ edges, the \emph{densest subgraph} problem, is to compute a subset $S \subseteq \mathsf{V}$ which maximizes the ratio $|\mathsf{E}_S| / |S|$, where $\mathsf{E}_S \subseteq \mathsf{E}$ is the set of all edges of $\mathsf{G}$ with endpoints in $S$. The densest subgraph problem is a well studied problem in computer science. Existing exact and approximation algorithms for computing the densest subgraph require $Ω(m)$ time. We present near-linear time (in $n$) approximation algorithms for the densest subgraph problem on \emph{implicit} geometric intersection graphs, where the vertices are explicitly given but not the edges. As a concrete example, we consider $n$ disks in the plane with arbitrary radii and present two different approximation algorithms.

cs.CG↗
arXiv2024

Oracle-Augmented Prophet Inequalities

In the classical prophet inequality settings, a gambler is given a sequence of $n$ random variables $X_1, \dots, X_n$, taken from known distributions, observes their values in this (potentially adversarial) order, and select one of them, immediately after it is being observed, so that its value is as high as possible. The classical \emph{prophet inequality} shows a strategy that guarantees a value at least half of that an omniscience prophet that picks the maximum, and this ratio is optimal. Here, we generalize the prophet inequality, allowing the gambler some additional information about the future that is otherwise privy only to the prophet. Specifically, at any point in the process, the gambler is allowed to query an oracle $\mathcal{O}$. The oracle responds with a single bit answer: YES if the current realization is greater than the remaining realizations, and NO otherwise. We show that the oracle model with $m$ oracle calls is equivalent to the \textsc{Top-$1$-of-$(m+1)$} model when the objective is maximizing the probability of selecting the maximum. This equivalence fails to hold when the objective is maximizing the competitive ratio, but we still show that any algorithm for the oracle model implies an equivalent competitive ratio for the \textsc{Top-$1$-of-$(m+1)$} model. We resolve the oracle model for any $m$, giving tight lower and upper bound on the best possible competitive ratio compared to an almighty adversary. As a consequence, we provide new results as well as improvements on known results for the \textsc{Top-$1$-of-$m$} model.

cs.GT↗
arXiv2023

On the Rectangles Induced by Points

$ \newcommand{\Re}{\mathbb{R}} \newcommand{\reals}{\mathbb{R}} \newcommand{\SetX}{\mathsf{X}} \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{\pts}{s}$ A set $P$ of $n$ points in the plane, induces a set of Delaunay-type axis-parallel rectangles $\mathcal{R}$, potentially of quadratic size, where an axis-parallel rectangle is in $\mathcal{R}$, if it has two points of $P$ as corners, and no other point of $P$ in it. We study various algorithmic problems related to this set of rectangles, including how to compute it, in near linear time, and handle various algorithmic tasks on it, such as computing its union and depth. The set of rectangles $\mathcal{R}$ induces the rectangle influence graph $G = (P,\mathcal{R})$, which we also study. Potentially our most interesting result is showing that this graph can be described as the union of $O(n)$ bicliques, where the total weight of the bicliques is $O(n \log^2 n)$. Here, the weight of a bicliques is the cardinality of its vertices.

cs.CG↗