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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(More)
The native extracellular matrix (ECM) consists of an integrated fibrous protein network and proteoglycan-based ground (hydrogel) substance. We designed a novel electrospinning technique to engineer a three dimensional fiber-hydrogel composite that mimics the native ECM structure, is injectable, and has practical macroscale dimensions for clinically relevant(More)
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> &#8712; <i>V</i> and having radius <i>r</i> &#8805; 0 to be the set {<i>q</i> &#8712; <i>V/d(v, q)</i> &#8804;<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>(More)
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(More)
We obtain a polynomial time approximation scheme for the 1.5D 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 [3] and Mustafa and Ray [18]. Our key contribution is to show the existence of a planar graph that appropriately(More)
Given an n-point metric (P,d) and an integer k>0, we consider the problem of covering P by k balls so as to minimize the sum of the radii of the balls. We present a randomized algorithm that runs in n O(log n⋅log Δ) time and returns with high probability the optimal solution. Here, Δ is the ratio between the maximum and minimum interpoint distances in the(More)
We consider an optimization version of the image segmentation problem, in which we are given a grid graph with weights on the grid cells. We are interested in finding the maximum weight subgraph such that the subgraph can be decomposed into two ”star-shaped” images. We show that this problem can be reduced to the problem of finding a maximum-weight closed(More)
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(More)