Matt Gibson

Decomposing Coverings and the Planar Sensor Cover Problem

We show that a k-fold covering using translates of an arbitrary convex polygon can be decomposed into Omega(k) covers (using an efficient algorithm). We generalize this result to obtain a constant factor approximation to the sensor cover problem where the ranges of the sensors are translates of a given convex polygon. The crucial ingredient in this generalization is a constant factor approximation algorithm for a one-dimensional version of the sensor cover problem, called the Restricted Strip Cover (RSC) problem, where sensors are intervals of possibly different lengths. Our algorithm for RSC improves on the previous O(log log log n) approximation.

Algorithms for dominating set in disk graphs: breaking the log n Barrier

We consider the problem of finding a lowest cost dominating set in a given disk graph containing n disks. The problem has been extensively studied on subclasses of disk graphs, yet the best known approximation for disk graphs has remained O(log n) - a bound that is asymptotically no better than the general case. We improve the status quo in two ways: for the unweighted case, we show how to obtain a PTAS using the framework recently proposed (independently) by Mustafa and Ray [16] and by Chan and Har-Peled [4]; for the weighted case where each input disk has an associated rational weight with the objective of finding a minimum cost dominating set, we give a randomized algorithm that obtains a dominating set whose weight is within a factor 2O(log* n) of a minimum cost solution, with high probability - the technique follows the framework proposed recently by Varadarajan [19].

An Approximation Scheme for Terrain Guarding

We obtain a polynomial time approximation scheme for the terrain guarding problem improving upon several recent constant factor approximations. Our algorithm is a local search algorithm inspired by the recent results of Chan and Har-Peled [2] and Mustafa and Ray [15]. Our key contribution is to show the existence of a planar graph that appropriately relates the local and global optimum.

GUARDING TERRAINS VIA LOCAL SEARCH

We obtain a polynomial time approximation scheme for the terrain guarding problem improving upon several recent constant factor approximations. Our algorithm is a local search algorithm inspired by the recent results of Chan and Har-Peled (SoCG 2009) and Mustafa and Ray (DCG 2010). Our key contribution is to show the existence of a planar graph that appropriately relates the local and global optimum.

On Metric Clustering to Minimize the Sum of Radii

Given an n-point metric (P,d) and an integer k> 0, we consider the problem of covering Pby kballs so as to minimize the sum of the radii of the balls. We present a randomized algorithm that runs in nO(logn·logΔ)time and returns with high probability the optimal solution. Here, Δis the ratio between the maximum and minimum interpoint distances in the metric space. We also show that the problem is NP-hard, even in metrics induced by weighted planar graphs and in metrics of constant doubling dimension.

Optimally Decomposing Coverings with Translates of a Convex Polygon

We show that any k-fold covering using translates of an arbitrary convex polygon can be decomposed into Ω(k) covers. Such a decomposition can be computed using an efficient (polynomial-time) algorithm.

On isolating points using disks

In this paper, we consider the problem of choosing disks (that we can think of as corresponding to wireless sensors) so that given a set of input points in the plane, there exists no path between any pair of these points that is not intercepted by some disk. We try to achieve this separation using a minimum number of a given set of unit disks. We show that a constant factor approximation to this problem can be found in polynomial time using a greedy algorithm. To the best of our knowledge we are the first to study this optimization problem.

On Clustering to Minimize the Sum of Radii

Given a metric <i>d</i> defined on a set <i>V</i> of points (a metric space), we define the ball B(<i>v, r</i>) centered at <i>u</i> ∈ <i>V</i> and having radius <i>r</i> ≥ 0 to be the set {<i>q</i> ∈ <i>V/d(v, q)</i> ≤<i>r</i>}. In this work, we consider the problem of computing a minimum cost <i>k</i>-cover for a given set <i>P</i> ⊆ <i>V</i> of <i>n</i> points, where <i>k</i> > 0 is some given integer which is also part of the input. For <i>k</i> ≥ 0, a <i>k</i>-cover for subset <i>Q</i> ⊆ <i>P</i> is a set of at most <i>k</i> balls, each centered at a point in <i>P</i>, whose union covers (contains) <i>Q.</i> The cost of a set <i>D</i> of balls, denoted cost(<i>D</i>), is the sum of the radii of those balls.

A Characterization of Visibility Graphs for Pseudo-polygons

In this paper, we give a characterization of the visibility graphs of pseudo-polygons. We first identify some key combinatorial properties of pseudo-polygons, and we then give a set of five necessary conditions based off our identified properties. We then prove that these necessary conditions are also sufficient via a reduction to a characterization of vertex-edge visibility graphs given by O’Rourke and Streinu.

Choosing thresholds for density-based map construction algorithms

Due to the ubiquitous use of various positioning technologies in smart phones and other devices, geospatial tracking data has become a routine data source. One of its uses that has gained recent popularity is the construction of street maps from vehicular tracking data. Due to the inherent noise in the data, many map construction algorithms are based on thresholding a density function. While kernel density estimation provides a firm theoretical foundation for computing the density from the measurements, the thresholds are generally picked in a heuristic, and often brute-force way, which results in slow algorithms with no guarantees on the map construction quality. In this paper, we formalize the selection of thresholds in a density-based street map construction algorithm. We propose a new thresholding technique that uses persistent homology combined with statistical analysis to determine a small set of thresholds that captures all relevant topological features. We formally prove that when the samples are drawn uniformly from the street map, a constant number of thresholds suffices to recover the street map. We also provide algorithms to compute the thresholds for different sampling assumptions. Finally, we show the effectiveness of our algorithms in several experiments on artificially generated data and on real GPS trajectory data.