arXiv subjects

Sariel Har-Peled

Publications and source records attributed to Sariel Har-Peled.

At least 91 records · Page 5Linked to original sources

arXiv2018

Grid peeling and the affine curve-shortening flow

In this paper we study an experimentally-observed connection between two seemingly unrelated processes, one from computational geometry and the other from differential geometry. The first one (which we call "grid peeling") is the convex-layer decomposition of subsets $G\subset \mathbb Z^2$ of the integer grid, previously studied for the particular case $G=\{1,\ldots,m\}^2$ by Har-Peled and Lidický (2013). The second one is the affine curve-shortening flow (ACSF), first studied by Alvarez et al. (1993) and Sapiro and Tannenbaum (1993). We present empirical evidence that, in a certain well-defined sense, grid peeling behaves at the limit like ACSF on convex curves. We offer some theoretical arguments in favor of this conjecture. We also pay closer attention to the simple case where $G=\mathbb N^2$ is a quarter-infinite grid. This case corresponds to ACSF starting with an infinite L-shaped curve, which when transformed using the ACSF becomes a hyperbola for all times $t>0$. We prove that, in the grid peeling of $\mathbb N^2$, (1) the number of grid points removed up to iteration $n$ is $Θ(n^{3/2}\log n)$; and (2) the boundary at iteration $n$ is sandwiched between two hyperbolas that are separated from each other by a constant factor.

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arXiv2018

A Simple Algorithm for Computing a Cycle Separator

We present a linear time algorithm for computing a cycle separator in a planar graph that is (arguably) simpler than previously known algorithms. Our algorithm builds on, and is somewhat similar to, previous algorithms for computing separators. The main new ingredient is a specific layered decomposition of the planar graph constructed differently from previous BFS-based layerings.

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arXiv2017

Decomposing arrangements of hyperplanes: VC-dimension, combinatorial dimension, and point location

$\renewcommand{\Re}{\mathbb{R}}$ We re-examine parameters for the two main space decomposition techniques---bottom-vertex triangulation, and vertical decomposition, including their explicit dependence on the dimension $d$, and discover several unexpected phenomena, which show that, in both techniques, there are large gaps between the VC-dimension (and primal shatter dimension), and the combinatorial dimension. For vertical decomposition, the combinatorial dimension is only $2d$, the primal shatter dimension is at most $d(d+1)$, and the VC-dimension is at least $1 + d(d+1)/2$ and at most $O(d^3)$. For bottom-vertex triangulation, both the primal shatter dimension and the combinatorial dimension are $Θ(d^2)$, but there seems to be a significant gap between them, as the combinatorial dimension is $\frac12d(d+3)$, whereas the primal shatter dimension is at most $d(d+1)$, and the VC-dimension is between $d(d+1)$ and $5d^2 \log{d}$ (for $d\ge 9$). Our main application is to point location in an arrangement of $n$ hyperplanes is $\Re^d$, in which we show that the query cost in Meiser's algorithm can be improved if one uses vertical decomposition instead of bottom-vertex triangulation, at the cost of some increase in the preprocessing cost and storage. The best query time that we can obtain is $O(d^3\log n)$, instead of $O(d^4\log d\log n)$ in Meiser's algorithm. For these bounds to hold, the preprocessing and storage are rather large (super-exponential in $d$). We discuss the tradeoff between query cost and storage (in both approaches, the one using bottom-vertex trinagulation and the one using vertical decomposition).

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arXiv2017

Sparse Approximation via Generating Point Sets

$ \newcommand{\kalg}{k_{\mathrm{alg}}} \newcommand{\kopt}{k_{\mathrm{opt}}} \newcommand{\algset}{T} \renewcommand{\Re}{\mathbb{R}} \newcommand{\eps}{\varepsilon} \newcommand{\pth}[2][\!]{#1\left({#2}\right)} \newcommand{\npoints}{n} \newcommand{\ballD}{\mathsf{b}} \newcommand{\dataset}{P} $ For a set $\dataset$ of $\npoints$ points in the unit ball $\ballD \subseteq \Re^d$, consider the problem of finding a small subset $\algset \subseteq \dataset$ such that its convex-hull $\eps$-approximates the convex-hull of the original set. We present an efficient algorithm to compute such a $\eps'$-approximation of size $\kalg$, where $\eps'$ is function of $\eps$, and $\kalg$ is a function of the minimum size $\kopt$ of such an $\eps$-approximation. Surprisingly, there is no dependency on the dimension $d$ in both bounds. Furthermore, every point of $\dataset$ can be $\eps$-approximated by a convex-combination of points of $\algset$ that is $O(1/\eps^2)$-sparse. Our result can be viewed as a method for sparse, convex autoencoding: approximately representing the data in a compact way using sparse combinations of a small subset $\algset$ of the original data. The new algorithm can be kernelized, and it preserves sparsity in the original input.

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arXiv2017

On Separating Points by Lines

Given a set $P$ of $n$ points in the plane, its separability is the minimum number of lines needed to separate all its pairs of points from each other. We show that the minimum number of lines needed to separate $n$ points, picked randomly (and uniformly) in the unit square, is $Θ( n^{2/3})$, where $Θ$ hides polylogarithmic factors. In addition, we provide a fast approximation algorithm for computing the separability of a given point set in the plane. Finally, we point out the connection between separability and partitions.

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arXiv2017

LSH on the Hypercube Revisited

LSH (locality sensitive hashing) had emerged as a powerful technique in nearest-neighbor search in high dimensions [IM98, HIM12]. Given a point set $P$ in a metric space, and given parameters $r$ and $\varepsilon > 0$, the task is to preprocess the point set, such that given a query point $q$, one can quickly decide if $q$ is in distance at most $\leq r$ or $\geq (1+\varepsilon)r$ from the point set $P$. Once such a near-neighbor data-structure is available, one can reduce the general nearest-neighbor search to logarithmic number of queries in such structures [IM98, Har01, HIM12]. In this note, we revisit the most basic settings, where $P$ is a set of points in the binary hypercube $\{0,1\}^d$, under the $L_1$/Hamming metric, and present a short description of the LSH scheme in this case. We emphasize that there is no new contribution in this note, except (maybe) the presentation itself, which is inspired by the authors recent work [HM17].

cs.CG↗

arXiv2017

Approximation Schemes for Independent Set and Sparse Subsets of Polygons

We present an $(1+\varepsilon)$-approximation algorithm with quasi-polynomial running time for computing the maximum weight independent set of polygons out of a given set of polygons in the plane (specifically, the running time is $n^{O( \mathrm{poly}( \log n, 1/\varepsilon))}$). Contrasting this, the best known polynomial time algorithm for the problem has an approximation ratio of~$n^{\varepsilon}$. Surprisingly, we can extend the algorithm to the problem of computing the maximum weight subset of the given set of polygons whose intersection graph fulfills some sparsity condition. For example, we show that one can approximate the maximum weight subset of polygons, such that the intersection graph of the subset is planar or does not contain a cycle of length $4$ (i.e., $K_{2,2}$). Our algorithm relies on a recursive partitioning scheme, whose backbone is the existence of balanced cuts with small complexity that intersect polygons from the optimal solution of a small total weight. For the case of large axis-parallel rectangles, we provide a polynomial time $(1+\varepsilon)$-approximation for the maximum weight independent set. Specifically, we consider the problem where each rectangle has one edge whose length is at least a constant fraction of the length of the corresponding edge of the bounding box of all the input elements. This is now the most general case for which a PTAS is known, and it requires a new and involved partitioning scheme, which should be of independent interest.

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arXiv2016

Depth contours in arrangements of halfplanes

Let $H$ be a set of $n$ halfplanes in $\mathbb{R}^2$ in general position, and let $k<n$ be a given parameter. We show that the number of vertices of the arrangement of $H$ that lie at depth exactly $k$ (i.e., that are contained in the interiors of exactly $k$ halfplanes of $H$) is $O(nk^{1/3} + n^{2/3}k^{4/3})$. The bound is tight when $k=Θ(n)$. This generalizes the study of Dey [Dey98], concerning the complexity of a single level in an arrangement of lines, and coincides with it for $k=O(n^{1/3})$.

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arXiv2016

From Proximity to Utility: A Voronoi Partition of Pareto Optima

We present an extension of Voronoi diagrams where when considering which site a client is going to use, in addition to the site distances, other site attributes are also considered (for example, prices or weights). A cell in this diagram is then the locus of all clients that consider the same set of sites to be relevant. In particular, the precise site a client might use from this candidate set depends on parameters that might change between usages, and the candidate set lists all of the relevant sites. The resulting diagram is significantly more expressive than Voronoi diagrams, but naturally has the drawback that its complexity, even in the plane, might be quite high. Nevertheless, we show that if the attributes of the sites are drawn from the same distribution (note that the locations are fixed), then the expected complexity of the candidate diagram is near linear. To this end, we derive several new technical results, which are of independent interest. In particular, we provide a high-probability, asymptotically optimal bound on the number of Pareto optima points in a point set uniformly sampled from the $d$-dimensional hypercube. To do so we revisit the classical backward analysis technique, both simplifying and improving relevant results in order to achieve the high-probability bounds.

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arXiv2016

Computing the k Nearest-Neighbors for all Vertices via Dijkstra

We are given a directed graph $G = (V,E)$ with $n$ vertices and $m$ edges, with positive weights on the edges, and a parameter $k >0$. We show how to compute, for every vertex $v \in V$, its $k$ nearest-neighbors. The algorithm runs in $O( k ( n \log n + m ) )$ time, and follows by a somewhat careful modification of Dijkstra's shortest path algorithm. This result is probably folklore, but we were unable to find a reference to it -- thus, this note.

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arXiv2016

Nearest-Neighbor Searching Under Uncertainty II

Nearest-neighbor search, which returns the nearest neighbor of a query point in a set of points, is an important and widely studied problem in many fields, and it has wide range of applications. In many of them, such as sensor databases, location-based services, face recognition, and mobile data, the location of data is imprecise. We therefore study nearest-neighbor queries in a probabilistic framework in which the location of each input point is specified as a probability distribution function. We present efficient algorithms for - computing all points that are nearest neighbors of a query point with nonzero probability; and - estimating the probability of a point being the nearest neighbor of a query point, either exactly or within a specified additive error.

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arXiv2016

Approximation Algorithms for Polynomial-Expansion and Low-Density Graphs

We study the family of intersection graphs of low density objects in low dimensional Euclidean space. This family is quite general, and includes planar graphs. We prove that such graphs have small separators. Next, we present efficient $(1+\varepsilon)$-approximation algorithms for these graphs, for Independent Set, Set Cover, and Dominating Set problems, among others. We also prove corresponding hardness of approximation for some of these optimization problems, providing a characterization of their intractability in terms of density.

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arXiv2016

Approximating the $k$-Level in Three-Dimensional Plane Arrangements

$\renewcommand{\Re}{{\rm I\!\hspace{-0.025em} R}} \newcommand{\SetX}{\mathsf{X}} \newcommand{\eps}{\varepsilon} \newcommand{\VorX}[1]{\mathcal{V} \pth{#1}} \newcommand{\Polygon}{\mathsf{P}} \newcommand{\IntRange}[1]{[ #1 ]} \newcommand{\Space}{\ovebarline{\mathsf{m}}} \newcommand{\pth}[2][\!]{#1\left({#2}\right)} \newcommand{\Arr}{\cal A}$ Let $H$ be a set of $n$ planes in three dimensions, and let $r \leq n$ be a parameter. We give a simple alternative proof of the existence of a $(1/r)$-cutting of the first $n/r$ levels of $\Arr(H)$, which consists of $O(r)$ semi-unbounded vertical triangular prisms. The same construction yields an approximation of the $(n/r)$-level by a terrain consisting of $O(r/\eps^3)$ triangular faces, which lies entirely between the levels $(1\pm\eps)n/r$. The proof does not use sampling, and exploits techniques based on planar separators and various structural properties of levels in three-dimensional arrangements and of planar maps. The proof is constructive, and leads to a simple randomized algorithm, with expected near-linear running time. An application of this technique allows us to mimic Matousek's construction of cuttings in the plane, to obtain a similar construction of "layered" $(1/r)$-cutting of the entire arrangement $\Arr(H)$, of optimal size $O(r^3)$. Another application is a simplified optimal approximate range counting algorithm in three dimensions, competing with that of Afshani and Chan.

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arXiv2016

Towards Tight Bounds for the Streaming Set Cover Problem

We consider the classic Set Cover problem in the data stream model. For $n$ elements and $m$ sets ($m\geq n$) we give a $O(1/δ)$-pass algorithm with a strongly sub-linear $\tilde{O}(mn^δ)$ space and logarithmic approximation factor. This yields a significant improvement over the earlier algorithm of Demaine et al. [DIMV14] that uses exponentially larger number of passes. We complement this result by showing that the tradeoff between the number of passes and space exhibited by our algorithm is tight, at least when the approximation factor is equal to $1$. Specifically, we show that any algorithm that computes set cover exactly using $({1 \over 2δ}-1)$ passes must use $\tildeΩ(mn^δ)$ space in the regime of $m=O(n)$. Furthermore, we consider the problem in the geometric setting where the elements are points in $\mathbb{R}^2$ and sets are either discs, axis-parallel rectangles, or fat triangles in the plane, and show that our algorithm (with a slight modification) uses the optimal $\tilde{O}(n)$ space to find a logarithmic approximation in $O(1/δ)$ passes. Finally, we show that any randomized one-pass algorithm that distinguishes between covers of size 2 and 3 must use a linear (i.e., $Ω(mn)$) amount of space. This is the first result showing that a randomized, approximate algorithm cannot achieve a space bound that is sublinear in the input size. This indicates that using multiple passes might be necessary in order to achieve sub-linear space bounds for this problem while guaranteeing small approximation factors.

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arXiv2016

Notes on Approximation Algorithms for Polynomial-Expansion and Low-Density Graphs

This write-up contains some minor results and notes related to our work [HQ15] (some of them already known in the literature). In particular, it shows the following: - We show that a graph with polynomial expansion have sublinear separators. - We show that hereditary sublinear separators imply that a graph have small divisions. - We show a natural condition on a set of segments, such that they have low density. This might be of independent interest in trying to define a realistic input model for a set of segments. Unlike the previous two results, this is new. For context and more details, see the main paper.

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arXiv2015

Shortest Path in a Polygon using Sublinear Space

$\renewcommand{\Re}{{\rm I\!\hspace{-0.025em} R}} \newcommand{\SetX}{\mathsf{X}} \newcommand{\VorX}[1]{\mathcal{V} \pth{#1}} \newcommand{\Polygon}{\mathsf{P}} \newcommand{\Space}{\overline{\mathsf{m}}} \newcommand{\pth}[2][\!]{#1\left({#2}\right)}$ We resolve an open problem due to Tetsuo Asano, showing how to compute the shortest path in a polygon, given in a read only memory, using sublinear space and subquadratic time. Specifically, given a simple polygon $\Polygon$ with $n$ vertices in a read only memory, and additional working memory of size $\Space$, the new algorithm computes the shortest path (in $\Polygon$) in $O( n^2 /\, \Space )$ expected time. This requires several new tools, which we believe to be of independent interest.

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arXiv2015

Proximity in the Age of Distraction: Robust Approximate Nearest Neighbor Search

We introduce a new variant of the nearest neighbor search problem, which allows for some coordinates of the dataset to be arbitrarily corrupted or unknown. Formally, given a dataset of $n$ points $P=\{ x_1,\ldots, x_n\}$ in high-dimensions, and a parameter $k$, the goal is to preprocess the dataset, such that given a query point $q$, one can compute quickly a point $x \in P$, such that the distance of the query to the point $x$ is minimized, when ignoring the "optimal" $k$ coordinates. Note, that the coordinates being ignored are a function of both the query point and the point returned. We present a general reduction from this problem to answering ANN queries, which is similar in spirit to LSH (locality sensitive hashing) [IM98]. Specifically, we give a sampling technique which achieves a bi-criterion approximation for this problem. If the distance to the nearest neighbor after ignoring $k$ coordinates is $r$, the data-structure returns a point that is within a distance of $O(r)$ after ignoring $O(k)$ coordinates. We also present other applications and further extensions and refinements of the above result. The new data-structures are simple and (arguably) elegant, and should be practical -- specifically, all bounds are polynomial in all relevant parameters (including the dimension of the space, and the robustness parameter $k$).

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arXiv2015

How to Walk Your Dog in the Mountains with No Magic Leash

We describe a $O(\log n )$-approximation algorithm for computing the homotopic \Frechet distance between two polygonal curves that lie on the boundary of a triangulated topological disk. Prior to this work, algorithms were known only for curves on the Euclidean plane with polygonal obstacles. A key technical ingredient in our analysis is a $O(\log n)$-approximation algorithm for computing the minimum height of a homotopy between two curves. No algorithms were previously known for approximating this parameter. Surprisingly, it is not even known if computing either the homotopic \Frechet distance, or the minimum height of a homotopy, is in NP.

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