arXiv subjects

Matthew J. Katz

Publications and source records attributed to Matthew J. Katz.

28 records · Page 2Linked to original sources

arXiv2019

Efficient Nearest-Neighbor Query and Clustering of Planar Curves

We study two fundamental problems dealing with curves in the plane, namely, the nearest-neighbor problem and the center problem. Let $\mathcal{C}$ be a set of $n$ polygonal curves, each of size $m$. In the nearest-neighbor problem, the goal is to construct a compact data structure over $\mathcal{C}$, such that, given a query curve $Q$, one can efficiently find the curve in $\mathcal{C}$ closest to $Q$. In the center problem, the goal is to find a curve $Q$, such that the maximum distance between $Q$ and the curves in $\mathcal{C}$ is minimized. We use the well-known discrete Frechet distance function, both under~$L_\infty$ and under $L_2$, to measure the distance between two curves. For the nearest-neighbor problem, despite discouraging previous results, we identify two important cases for which it is possible to obtain practical bounds, even when $m$ and $n$ are large. In these cases, either $Q$ is a line segment or $\mathcal{C}$ consists of line segments, and the bounds on the size of the data structure and query time are nearly linear in the size of the input and query curve, respectively. The returned answer is either exact under $L_\infty$, or approximated to within a factor of $1+\varepsilon$ under~$L_2$. We also consider the variants in which the location of the input curves is only fixed up to translation, and obtain similar bounds, under $L_\infty$. As for the center problem, we study the case where the center is a line segment, i.e., we seek the line segment that represents the given set as well as possible. We present near-linear time exact algorithms under $L_\infty$, even when the location of the input curves is only fixed up to translation. Under $L_2$, we present a roughly $O(n^2m^3)$-time exact algorithm.

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arXiv2018

A PTAS for vertex guarding weakly-visible polygons - An extended abstract

In this extended abstract, we present a PTAS for guarding the vertices of a weakly-visible polygon $P$ from a subset of its vertices, or in other words, a PTAS for computing a minimum dominating set of the visibility graph of the vertices of $P$. We then show how to obtain a PTAS for vertex guarding $P$'s boundary.

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arXiv2017

Batched Point Location in SINR Diagrams via Algebraic Tools

The SINR model for wireless networks has been extensively studied recently. It tries to model whether a particular transmitter is heard at a specific location, with $n$ transmitting simultaneously. The SINR diagram consists of $n$ regions where each transmitter can be heard and the remaining space where no one can be heard. Efficient point location in the SINR diagram, i.e., building a data structure to determine, for a query point, whether any transmitter is heard there, and if so, which one, has been recently investigated. Previous such planar data structures are constructed in time at least quadratic in $n$ and support logarithmic-time approximate queries. Moreover, the performance of these depends not only on the number $n$ of transmitters and on the approximation parameter $\varepsilon$, but also on some geometric parameters that cannot be bounded a priori as a function of $n$ or $\varepsilon$. We address the question of batched point location queries, i.e., answering many queries simultaneously. Specifically, in one dimension, we can answer $n$ queries exactly in amortized polylogarithmic time per query, while in the plane we can do it approximately. All these results can handle arbitrary power assignments to the transmitters. Moreover, the amortized query time in these results depends only on $n$ and $\varepsilon$. We also show how to speed up the preprocessing in a previously proposed point-location structure in SINR diagram for uniform-power sites, by almost a full order of magnitude. For this we obtain results on the sensitivity of the reception regions to slight changes in the reception threshold, which are of independent interest. Finally, these results demonstrate the (so far underutilized) power of combining algebraic tools with those of computational geometry and other fields.

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arXiv2016

The Discrete and Semi-continuous Fréchet Distance with Shortcuts via Approximate Distance Counting and Selection Techniques

The \emph{Fréchet distance} is a well studied similarity measures between curves. The \emph{discrete Fréchet distance} is an analogous similarity measure, defined for a sequence $A$ of $m$ points and a sequence $B$ of $n$ points, where the points are usually sampled from input curves. In this paper we consider a variant, called the \emph{discrete Fréchet distance with shortcuts}, which captures the similarity between (sampled) curves in the presence of outliers. For the \emph{two-sided} case, where shortcuts are allowed in both curves, we give an $O((m^{2/3}n^{2/3}+m+n)\log^3 (m+n))$-time algorithm for computing this distance. When shortcuts are allowed only in one noise-containing curve, we give an even faster randomized algorithm that runs in $O((m+n)^{6/5+\varepsilon})$ expected time, for any $\varepsilon>0$. Our techniques are novel and may find further applications. One of the main new technical results is: Given two sets of points $A$ and $B$ and an interval $I$, we develop an algorithm that decides whether the number of pairs $(x,y)\in A\times B$ whose distance ${\rm dist}(x,y)$ is in $I$, is less than some given threshold $L$. The running time of this algorithm decreases as $L$ increases. In case there are more than $L$ pairs of points whose distance is in $I$, we can get a small sample of pairs that contains a pair at approximate median distance (i.e., we can approximately "bisect" $I$). We combine this procedure with additional ideas to search, with a small overhead, for the optimal one-sided Fréchet distance with shortcuts, using a very fast decision procedure. We also show how to apply this technique for approximating distance selection (with respect to rank), and for computing the semi-continuous Fréchet distance with one-sided shortcuts.

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arXiv2015

The Discrete Fréchet Gap

We introduce the discrete Fréchet gap and its variants as an alternative measure of similarity between polygonal curves. We believe that for some applications the new measure (and its variants) may better reflect our intuitive notion of similarity than the discrete Fréchet distance (and its variants), since the latter measure is indifferent to (matched) pairs of points that are relatively close to each other. Referring to the frogs analogy by which the discrete Fréchet distance is often described, the discrete Fréchet gap is the minimum difference between the longest and shortest positions of the leash needed for the frogs to traverse their point sequences. We present an optimization scheme, which is suitable for any monotone function defined for pairs of distances such as the gap and ratio functions. We apply this scheme to two variants of the discrete Fréchet gap, namely, the one-sided discrete Fréchet gap with shortcuts and the weak discrete Fréchet gap, to obtain $O(n^2 \log^2 n)$-time algorithms for computing them.

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arXiv2014

On the Chain Pair Simplification Problem

The problem of efficiently computing and visualizing the structural resemblance between a pair of protein backbones in 3D has led Bereg et al. to pose the Chain Pair Simplification problem (CPS). In this problem, given two polygonal chains $A$ and $B$ of lengths $m$ and $n$, respectively, one needs to simplify them simultaneously, such that each of the resulting simplified chains, $A'$ and $B'$, is of length at most $k$ and the discrete \frechet\ distance between $A'$ and $B'$ is at most $δ$, where $k$ and $δ$ are given parameters. In this paper we study the complexity of CPS under the discrete \frechet\ distance (CPS-3F), i.e., where the quality of the simplifications is also measured by the discrete \frechet\ distance. Since CPS-3F was posed in 2008, its complexity has remained open. However, it was believed to be \npc, since CPS under the Hausdorff distance (CPS-2H) was shown to be \npc. We first prove that the weighted version of CPS-3F is indeed weakly \npc\, even on the line, based on a reduction from the set partition problem. Then, we prove that CPS-3F is actually polynomially solvable, by presenting an $O(m^2n^2\min\{m,n\})$ time algorithm for the corresponding minimization problem. In fact, we prove a stronger statement, implying, for example, that if weights are assigned to the vertices of only one of the chains, then the problem remains polynomially solvable. We also study a few less rigid variants of CPS and present efficient solutions for them. Finally, we present some experimental results that suggest that (the minimization version of) CPS-3F is significantly better than previous algorithms for the motivating biological application.

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arXiv2014

Bounded-Angle Spanning Tree: Modeling Networks with Angular Constraints

We introduce a new structure for a set of points in the plane and an angle $α$, which is similar in flavor to a bounded-degree MST. We name this structure $α$-MST. Let $P$ be a set of points in the plane and let $0 < α\le 2π$ be an angle. An $α$-ST of $P$ is a spanning tree of the complete Euclidean graph induced by $P$, with the additional property that for each point $p \in P$, the smallest angle around $p$ containing all the edges adjacent to $p$ is at most $α$. An $α$-MST of $P$ is then an $α$-ST of $P$ of minimum weight. For $α< π/3$, an $α$-ST does not always exist, and, for $α\ge π/3$, it always exists. In this paper, we study the problem of computing an $α$-MST for several common values of $α$. Motivated by wireless networks, we formulate the problem in terms of directional antennas. With each point $p \in P$, we associate a wedge $W_p$ of angle $α$ and apex $p$. The goal is to assign an orientation and a radius $r_p$ to each wedge $W_p$, such that the resulting graph is connected and its MST is an $α$-MST. (We draw an edge between $p$ and $q$ if $p \in W_q$, $q \in W_p$, and $|pq| \le r_p, r_q$.) Unsurprisingly, the problem of computing an $α$-MST is NP-hard, at least for $α=π$ and $α=2π/3$. We present constant-factor approximation algorithms for $α= π/2, 2π/3, π$. One of our major results is a surprising theorem for $α= 2π/3$, which, besides being interesting from a geometric point of view, has important applications. For example, the theorem guarantees that given any set $P$ of $3n$ points in the plane and any partitioning of the points into $n$ triplets, one can orient the wedges of each triplet {\em independently}, such that the graph induced by $P$ is connected. We apply the theorem to the {\em antenna conversion} problem.

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arXiv2012

Approximation Schemes for Covering and Packing

The local search framework for obtaining PTASs for NP-hard geometric optimization problems was introduced, independently, by Chan and Har-Peled (2009) and Mustafa and Ray (2010). In this paper, we generalize the framework by extending its analysis to additional families of graphs, beyond the family of planar graphs. We then present several applications of the generalized framework, some of which are very different from those presented to date (using the original framework). These applications include PTASs for finding a maximum l-shallow set of a set of fat objects, for finding a maximum triangle matching in an l-shallow unit disk graph, and for vertex-guarding a (not-necessarily-simple) polygon under an appropriate shallowness assumption. We also present a PTAS (using the original framework) for the important problem where one has to find a minimum-cardinality subset of a given set of disks (of varying radii) that covers a given set of points, and apply it to a class cover problem (studied in Bereg et al., 2012) to obtain an improved solution.

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arXiv2012

Bottleneck Non-Crossing Matching in the Plane

Let $P$ be a set of $2n$ points in the plane, and let $M_{\rm C}$ (resp., $M_{\rm NC}$) denote a bottleneck matching (resp., a bottleneck non-crossing matching) of $P$. We study the problem of computing $M_{\rm NC}$. We first prove that the problem is NP-hard and does not admit a PTAS. Then, we present an $O(n^{1.5}\log^{0.5} n)$-time algorithm that computes a non-crossing matching $M$ of $P$, such that $bn(M) \le 2\sqrt{10} \cdot bn(M_{\rm NC})$, where $bn(M)$ is the length of a longest edge in $M$. An interesting implication of our construction is that $bn(M_{\rm NC})/bn(M_{\rm C}) \le 2\sqrt{10}$.

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arXiv2011

Symmetric Connectivity with Directional Antennas

Let P be a set of points in the plane, representing directional antennas of angle a and range r. The coverage area of the antenna at point p is a circular sector of angle a and radius r, whose orientation is adjustable. For a given orientation assignment, the induced symmetric communication graph (SCG) of P is the undirected graph that contains the edge (u,v) iff v lies in u's sector and vice versa. In this paper we ask what is the smallest angle a for which there exists an integer n=n(a), such that for any set P of n antennas of angle a and unbounded range, one can orient the antennas so that the induced SCG is connected, and the union of the corresponding wedges is the entire plane. We show that the answer to this problem is a=π/2, for which n=4. Moreover, we prove that if Q_1 and Q_2 are quadruplets of antennas of angle π/2 and unbounded range, separated by a line, to which one applies the above construction, independently, then the induced SCG of Q_1 \cup Q_2 is connected. This latter result enables us to apply the construction locally, and to solve the following two further problems. In the first problem, we are given a connected unit disk graph (UDG), corresponding to a set P of omni-directional antennas of range 1, and the goal is to replace these antennas by directional antennas of angle π/2 and range r=O(1) and to orient them, such that the induced SCG is connected, and, moreover, is an O(1)-spanner of the UDG, w.r.t. hop distance. In our solution r = 14\sqrt{2} and the spanning ratio is 8. In the second problem, we are given a set P of directional antennas of angle π/2 and adjustable range. The goal is to assign to each antenna p, an orientation and a range r_p, such that the resulting SCG is connected, and \sum_{p \in P} r_p^βis minimized, where β\ge 1 is a constant. We present an O(1)-approximation algorithm.

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