arXiv subjects
Publications and source records attributed to Sariel Har-Peled.
In this note, we revisit the algorithm of Har-Peled et. al. [HRZ07] for computing a linear maximum margin classifier. Our presentation is self contained, and the algorithm itself is slightly simpler than the original algorithm. The algorithm itself is a simple Perceptron like iterative algorithm. For more details and background, the reader is referred to the original paper.
$\newcommand{\eps}{\varepsilon}$ In this paper, we consider two important problems defined on finite metric spaces, and provide efficient new algorithms and approximation schemes for these problems on inputs given as graph shortest path metrics or high-dimensional Euclidean metrics. The first of these problems is the greedy permutation (or farthest-first traversal) of a finite metric space: a permutation of the points of the space in which each point is as far as possible from all previous points. We describe randomized algorithms to find $(1+\eps)$-approximate greedy permutations of any graph with $n$ vertices and $m$ edges in expected time $O(\eps^{-1}(m+n)\log n\log(n/\eps))$, and to find $(1+\eps)$-approximate greedy permutations of points in high-dimensional Euclidean spaces in expected time $O(\eps^{-2} n^{1+1/(1+\eps)^2 + o(1)})$. Additionally we describe a deterministic algorithm to find exact greedy permutations of any graph with $n$ vertices and treewidth $O(1)$ in worst-case time $O(n^{3/2}\log^{O(1)} n)$. The second of the two problems we consider is distance selection: given $k \in [ \binom{n}{2} ]$, we are interested in computing the $k$th smallest distance in the given metric space. We show that for planar graph metrics one can approximate this distance, up to a constant factor, in near linear time.
$\renewcommand{\Re}{{\rm I\!\hspace{-0.025em} R}} \newcommand{\eps}{\varepsilon} \newcommand{\SetX}{\mathsf{X}} \newcommand{\VorX}[1]{\mathcal{V} \pth{#1}} \newcommand{\Polygon}{\mathsf{P}} \newcommand{\Space}{\overline{\mathsf{m}}} \newcommand{\pth}[2][\!]{#1\left({#2}\right)}$ We revisit the problem of computing Fréchet distance between polygonal curves under $L_1$, $L_2$, and $L_\infty$ norms, focusing on discrete Fréchet distance, where only distance between vertices is considered. We develop efficient algorithms for two natural classes of curves. In particular, given two polygonal curves of $n$ vertices each, a $\eps$-approximation of their discrete Fréchet distance can be computed in roughly $O(nκ^3\log n/\eps^3)$ time in three dimensions, if one of the curves is \emph{$κ$-bounded}. Previously, only a $κ$-approximation algorithm was known. If both curves are the so-called \emph{\backbone~curves}, which are widely used to model protein backbones in molecular biology, we can $\eps$-approximate their Fréchet distance in near linear time in two dimensions, and in roughly $O(n^{4/3}\log nm)$ time in three dimensions. In the second part, we propose a pseudo--output-sensitive algorithm for computing Fréchet distance exactly. The complexity of the algorithm is a function of a quantity we call the \emph{\bwnumber{}}, which is quadratic in the worst case, but tends to be much smaller in practice.
In this paper, we provide an $O(n \mathrm{polylog} n)$ bound on the expected complexity of the randomly weighted Voronoi diagram of a set of $n$ sites in the plane, where the sites can be either points, interior-disjoint convex sets, or other more general objects. Here the randomness is on the weight of the sites, not their location. This compares favorably with the worst case complexity of these diagrams, which is quadratic. As a consequence we get an alternative proof to that of Agarwal etal [AHKS13] of the near linear complexity of the union of randomly expanded disjoint segments or convex sets (with an improved bound on the latter). The technique we develop is elegant and should be applicable to other problems.
We investigate what computational tasks can be performed on a point set in $\Re^d$, if we are only given black-box access to it via nearest-neighbor search. This is a reasonable assumption if the underlying point set is either provided implicitly, or it is stored in a data structure that can answer such queries. In particular, we show the following: (A) One can compute an approximate bi-criteria $k$-center clustering of the point set, and more generally compute a greedy permutation of the point set. (B) One can decide if a query point is (approximately) inside the convex-hull of the point set. We also investigate the problem of clustering the given point set, such that meaningful proximity queries can be carried out on the centers of the clusters, instead of the whole point set.
$\newcommand{\MatA}{\mathcal{M}}$ $\newcommand{\eps}{\varepsilon}$ $\newcommand{\NSize}{\mathsf{N}{}}$ $\newcommand{\MatB}{\mathcal{B}}$ $\newcommand{\Fnorm}[1]{\left\| {#1} \right\|_F}$ $\newcommand{\PrcOpt}[2]{μ_{\mathrm{opt}}\pth{#1, #2}}$ $\newcommand{\pth}[1]{\left(#1\right)}$ Given a matrix $\MatA$ with $n$ rows and $d$ columns, and fixed $k$ and $\eps$, we present an algorithm that in linear time (i.e., $O(\NSize )$) computes a $k$-rank matrix $\MatB$ with approximation error $\Fnorm{\MatA - \MatB}^2 \leq (1+\eps) \PrcOpt{\MatA}{k}$, where $\NSize = n d$ is the input size, and $\PrcOpt{\MatA}{k}$ is the minimum error of a $k$-rank approximation to $\MatA$. This algorithm succeeds with constant probability, and to our knowledge it is the first linear-time algorithm to achieve multiplicative approximation.
Given a set of n disjoint balls b1, . . ., bn in IRd, we provide a data structure, of near linear size, that can answer (1 \pm ε)-approximate kth-nearest neighbor queries in O(log n + 1/ε^d) time, where k and εare provided at query time. If k and εare provided in advance, we provide a data structure to answer such queries, that requires (roughly) O(n/k) space; that is, the data structure has sublinear space requirement if k is sufficiently large.
Given a set $P$ of $n$ points in $\mathbb{R}^3$, we show that, for any $\varepsilon >0$, there exists an $\varepsilon$-net of $P$ for halfspace ranges, of size $O(1/\varepsilon)$. We give five proofs of this result, which are arguably simpler than previous proofs \cite{msw-hnlls-90, cv-iaags-07, pr-nepen-08}. We also consider several related variants of this result, including the case of points and pseudo-disks in the plane.
We study the convex-hull problem in a probabilistic setting, motivated by the need to handle data uncertainty inherent in many applications, including sensor databases, location-based services and computer vision. In our framework, the uncertainty of each input site is described by a probability distribution over a finite number of possible locations including a \emph{null} location to account for non-existence of the point. Our results include both exact and approximation algorithms for computing the probability of a query point lying inside the convex hull of the input, time-space tradeoffs for the membership queries, a connection between Tukey depth and membership queries, as well as a new notion of $\some$-hull that may be a useful representation of uncertain hulls.
Let $P$ and $Q$ be two simple polygons in the plane of total complexity $n$, each of which can be decomposed into at most $k$ convex parts. We present an $(1-\varepsilon)$-approximation algorithm, for finding the translation of $Q$, which maximizes its area of overlap with $P$. Our algorithm runs in $O(c n)$ time, where $c$ is a constant that depends only on $k$ and $\varepsilon$. This suggest that for polygons that are "close" to being convex, the problem can be solved (approximately), in near linear time.
Given a set $\mathsf{P}$ of $n$ points in $\mathbb{R}^d$, we show how to insert a set $\mathsf{X}$ of $O( n^{1-1/d} )$ additional points, such that $\mathsf{P}$ can be broken into two sets $\mathsf{P}_1$ and $\mathsf{P}_2$, of roughly equal size, such that in the Voronoi diagram $\mathcal{V}( \mathsf{P} \cup \mathsf{X} )$, the cells of $\mathsf{P}_1$ do not touch the cells of $\mathsf{P}_2$; that is, $\mathsf{X}$ separates $\mathsf{P}_1$ from $\mathsf{P}_2$ in the Voronoi diagram. Given such a partition $(\mathsf{P}_1,\mathsf{P}_2)$ of $\mathsf{P}$, we present approximation algorithms to compute the minimum size separator realizing this partition. Finally, we present a simple local search algorithm that is a PTAS for geometric hitting set of fat objects (which can also be used to approximate the optimal Voronoi partition).
Let $S$ be a set of $n$ points in $\mathbb{R}^d$. A Steiner convex partition is a tiling of ${\rm conv}(S)$ with empty convex bodies. For every integer $d$, we show that $S$ admits a Steiner convex partition with at most $\lceil (n-1)/d\rceil$ tiles. This bound is the best possible for points in general position in the plane, and it is best possible apart from constant factors in every fixed dimension $d\geq 3$. We also give the first constant-factor approximation algorithm for computing a minimum Steiner convex partition of a planar point set in general position. Establishing a tight lower bound for the maximum volume of a tile in a Steiner convex partition of any $n$ points in the unit cube is equivalent to a famous problem of Danzer and Rogers. It is conjectured that the volume of the largest tile is $ω(1/n)$. Here we give a $(1-\varepsilon)$-approximation algorithm for computing the maximum volume of an empty convex body amidst $n$ given points in the $d$-dimensional unit box $[0,1]^d$.
We describe how to approximate, in quasi-polynomial time, the largest independent set of polygons, in a given set of polygons. Our algorithm works by extending the result of Adamaszek and Wiese \cite{aw-asmwi-13, aw-qmwis-14} to polygons of arbitrary complexity. Surprisingly, the algorithm also works or computing the largest subset of the given set of polygons that has some sparsity condition. For example, we show that one can approximate the largest subset of polygons, such that the intersection graph of the subset does not contain a cycle of length $4$ (i.e., $K_{2,2}$).
A graph drawn in the plane with n vertices is k-fan-crossing free for k > 1 if there are no k+1 edges $g,e_1,...e_k$, such that $e_1,e_2,...e_k$ have a common endpoint and $g$ crosses all $e_i$. We prove a tight bound of 4n-8 on the maximum number of edges of a 2-fan-crossing free graph, and a tight 4n-9 bound for a straight-edge drawing. For k > 2, we prove an upper bound of 3(k-1)(n-2) edges. We also discuss generalizations to monotone graph properties.
Let $\mathcal{C}=\{C_1,\ldots,C_n\}$ be a set of $n$ pairwise-disjoint convex sets of constant description complexity, and let $π$ be a probability density function (pdf for short) over the non-negative reals. For each $i$, let $K_i$ be the Minkowski sum of $C_i$ with a disk of radius $r_i$, where each $r_i$ is a random non-negative number drawn independently from the distribution determined by $π$. We show that the expected complexity of the union of $K_1, \ldots, K_n$ is $O(n^{1+\varepsilon})$ for any $\varepsilon > 0$; here the constant of proportionality depends on $\varepsilon$ and on the description complexity of the sets in $\mathcal{C}$, but not on $π$. If each $C_i$ is a convex polygon with at most $s$ vertices, then we show that the expected complexity of the union is $O(s^2n\log n)$. Our bounds hold in the stronger model in which we are given an arbitrary multi-set $R=\{r_1,\ldots,r_n\}$ of expansion radii, each a non-negative real number. We assign them to the members of $\mathcal{C}$ by a random permutation, where all permutations are equally likely to be chosen; the expectations are now with respect to these permutations. We also present an application of our results to a problem that arises in analyzing the vulnerability of a network to a physical attack. %
The similarity of two polygonal curves can be measured using the Fréchet distance. We introduce the notion of a more robust Fréchet distance, where one is allowed to shortcut between vertices of one of the curves. This is a natural approach for handling noise, in particular batched outliers. We compute a (3+\eps)-approximation to the minimum Fréchet distance over all possible such shortcuts, in near linear time, if the curve is c-packed and the number of shortcuts is either small or unbounded. To facilitate the new algorithm we develop several new tools: (A) A data structure for preprocessing a curve (not necessarily c-packed) that supports (1+\eps)-approximate Fréchet distance queries between a subcurve (of the original curve) and a line segment. (B) A near linear time algorithm that computes a permutation of the vertices of a curve, such that any prefix of 2k-1 vertices of this permutation, form an optimal approximation (up to a constant factor) to the original curve compared to any polygonal curve with k vertices, for any k > 0. (C) A data structure for preprocessing a curve that supports approximate Fréchet distance queries between a subcurve and query polygonal curve. The query time depends quadratically on the complexity of the query curve, and only (roughly) logarithmically on the complexity of the original curve. To our knowledge, these are the first data structures to support these kind of queries efficiently.
We study the \LowerBoundedCenter (\lbc) problem, which is a clustering problem that can be viewed as a variant of the \kCenter problem. In the \lbc problem, we are given a set of points P in a metric space and a lower bound λ, and the goal is to select a set C \subseteq P of centers and an assignment that maps each point in P to a center of C such that each center of C is assigned at least λpoints. The price of an assignment is the maximum distance between a point and the center it is assigned to, and the goal is to find a set of centers and an assignment of minimum price. We give a constant factor approximation algorithm for the \lbc problem that runs in O(n \log n) time when the input points lie in the d-dimensional Euclidean space R^d, where d is a constant. We also prove that this problem cannot be approximated within a factor of 1.8-εunless P = \NP even if the input points are points in the Euclidean plane R^2.
For a set of $n$ points in $\Re^d$, and parameters $k$ and $\eps$, we present a data structure that answers $(1+\eps,k)$-\ANN queries in logarithmic time. Surprisingly, the space used by the data-structure is $\Otilde (n /k)$; that is, the space used is sublinear in the input size if $k$ is sufficiently large. Our approach provides a novel way to summarize geometric data, such that meaningful proximity queries on the data can be carried out using this sketch. Using this, we provide a sublinear space data-structure that can estimate the density of a point set under various measures, including: \begin{inparaenum}[(i)] \item sum of distances of $k$ closest points to the query point, and \item sum of squared distances of $k$ closest points to the query point. \end{inparaenum} Our approach generalizes to other distance based estimation of densities of similar flavor. We also study the problem of approximating some of these quantities when using sampling. In particular, we show that a sample of size $\Otilde (n /k)$ is sufficient, in some restricted cases, to estimate the above quantities. Remarkably, the sample size has only linear dependency on the dimension.