Paz Carmi

Covering points by unit disks of fixed location

Given a set P of points in the plane, and a set D of unit disks of fixed location, the discrete unit disk cover problem is to find a minimum-cardinality subset D′ ⊆ D that covers all points of P. This problem is a geometric version of the general set cover problem, where the sets are defined by a collection of unit disks. It is still NP-hard, but while the general set cover problem is not approximable within c log |P|, for some constant c, the discrete unit disk cover problem was shown to admit a constant-factor approximation. Due to its many important applications, e.g., in wireless network design, much effort has been invested in trying to reduce the constant of approximation of the discrete unit disk cover problem. In this paper we significantly improve the best known constant from 72 to 38, using a novel approach. Our solution is based on a 4-approximation that we devise for the subproblem where the points of P are located below a line l and contained in the subset of disks of D centered above l. This problem is of independent interest.

Connectivity guarantees for wireless networks with directional antennas

We study a combinatorial geometric problem related to the design of wireless networks with directional antennas. Specifically, we are interested in necessary and sufficient conditions on such antennas that enable one to build a connected communication network, and in efficient algorithms for building such networks when possible. We formulate the problem by a set P of n points in the plane, indicating the positions of n transceivers. Each point is equipped with an @a-degree directional antenna, and one needs to adjust the antennas (represented as wedges), by specifying their directions, so that the resulting (undirected) communication graph G is connected. (Two points p,q@?P are connected by an edge in G, if and only if q lies in p@?s wedge and p lies in q@?s wedge.) We prove that if @a=60^o, then it is always possible to adjust the wedges so that G is connected, and that @a>=60^o is sometimes necessary to achieve this. Our proof is constructive and yields an O(nlogk) time algorithm for adjusting the wedges, where k is the size of the convex hull of P. Sometimes it is desirable that the communication graph G contain a Hamiltonian path. By a result of Fekete and Woeginger (1997) [8], if @a=90^o, then it is always possible to adjust the wedges so that G contains a Hamiltonian path. We give an alternative proof to this, which is interesting, since it produces paths of a different nature than those produced by the construction of Fekete and Woeginger. We also show that for any n and @e>0, there exist sets of points such that G cannot contain a Hamiltonian path if @a=90^o-@e.

On the stretch factor of convex Delaunay graphs

Let C be a compact and convex set in the plane that contains the origin in its interior, and let S be a finite set of points in the plane. The Delaunay graph DG C ( S ) of S is defined to be the dual of the Voronoi diagram of S with respect to the convex distance function defined by C . We prove that DG C ( S ) is a t -spanner for S , for some constant t that depends only on the shape of the set C . Thus, for any two points p and q in S , the graph DG C ( S ) contains a path between p and q whose Euclidean length is at most t times the Euclidean distance between p and q .

Private approximation of search problems

Many approximation algorithms have been presented in the last decades for hard search problems. The focus of this paper is on cryptographic applications, where it is desired to design algorithms which do not leak unnecessary information. Specifically, we are interested in private approximation algorithms -- efficient algorithms whose output does not leak information not implied by the optimal solutions to the search problems. Privacy requirements add constraints on the approximation algorithms; in particular, known approximation algorithms usually leak a lot of information.For functions, [Feigenbaum et al., ICALP 2001] presented a natural requirement that a private algorithm should not leak information not implied by the original function. Generalizing this requirement to search problems is not straightforward as an input may have many different outputs. We present a new definition that captures a minimal privacy requirement from such algorithms -- applied to an input instance, it should not leak any information that is not implied by its collection of exact solutions. Although our privacy requirement seems minimal, we show that for well studied problems, as vertex cover and 3SAT, private approximation algorithms are unlikely to exist even for poor approximation ratios. Similar to [Halevi et al., STOC 2001], we define a relaxed notion of approximation algorithms that leak (little) information, and demonstrate the applicability of this notion by showing near optimal approximation algorithms for 3SAT that leak little information.

On the Fermat--Weber center of a convex object

We show that for any convex object Q in the plane, the average distance from the Fermat-Weber center of Q to the points in Q is at least @D(Q)/7, where @D(Q) is the diameter of Q, and that there exists a convex object P for which this distance is @D(P)/6. We use this result to obtain a linear-time approximation scheme for finding an approximate Fermat-Weber center of a convex polygon Q.

Computing the Greedy Spanner in Near-Quadratic Time

AbstractThe greedy algorithm produces high-quality spanners and, therefore, is used in several applications. However, even for points in d-dimensional Euclidean space, the greedy algorithm has near-cubic running time. In this paper, we present an algorithm that computes the greedy spanner for a set of n points in a metric space with bounded doubling dimension in

Approximating the Visible Region of a Point on a Terrain

\ensuremath {\mathcal {O}}(n^{2}\log n)

Minimum-cost load-balancing partitions

time. Since computing the greedy spanner has an Ω(n2) lower bound, the time complexity of our algorithm is optimal within a logarithmic factor.

Switching to directional antennas with constant increase in radius and hop distance

Given a terrain and a point p on or above it, we wish to compute the region Rp that is visible from p. We present a generic radar-like algorithm for computing an approximation of Rp. The algorithm interpolates the visible region between two consecutive rays (emanating from p) whenever the rays are close enough; that is, whenever the difference between the sets of visible segments along the cross sections in the directions specified by the rays is below some threshold. Thus the density of the sampling by rays is sensitive to the shape of the visible region. We suggest a specific way to measure the resemblance (difference) and to interpolate the visible region between two consecutive rays. We also present an alternative algorithm, which uses circles of increasing radii centered at p instead of rays emanating from p. Both algorithms compute a representation of the (approximated) visible region that is especially suitable for is-visible-from-p queries, i.e., given a query point q on the terrain determine whether q is visible from p. Finally, we report on the experiments that we performed with these algorithms and with their corresponding fixed versions, using a natural error measure. Our main conclusion is that the radar-like algorithm is significantly better than the others.

Fault-tolerant power assignment and backbone in wireless networks

We consider the problem of balancing the load among several service-providing facilities, while keeping the total cost low. Let <i>D</i> be the underlying demand region, and let <i>p</i>₁, …, <i>p_m</i> be <i>m</i> points representing <i>m</i> facilities. We consider the following problem: Subdivide <i>D</i> into <i>m</i> equal-area regions <i>R</i>₁, …, <i>R_m</i>, so that region <i>R_i</i> is served by facility <i>p_i</i>, and the average distance between a point <i>q</i> in <i>D</i> and the facility that serves <i>q</i> is minimal.We present constant-factor approximation algorithms for this problem, with the additional requirement that the resulting regions must be convex. As an intermediate result we show how to partition a convex polygon into <i>m</i>=2^<i>k</i> equal-area convex subregions so that the fatness of the resulting regions is within a constant factor of the fatness of the original polygon. We also prove that our partition is, up to a constant factor, the best one can get if ones goal is to maximize the fatness of the least fat subregion.We also discuss the structure of the optimal partition for the aforementioned load balancing problem: indeed, we argue that it is always induced by an additive-weighted Voronoi diagram for an appropriate choice of weights.