We study unique coverage problems with rectangle and half-strip regions, motivated by wireless networks in the context of coverage using directional antennae without interference. Given a set [Formula: see text] of points (clients) and a set [Formula: see text] of directional antennae in the plane, the goal is to assign a direction to each directional antenna in [Formula: see text], such that the number of clients in [Formula: see text] that are uniquely covered by the directional antennae is maximized. A client is covered uniquely if it is covered by exactly one antenna. We consider two types of rectangular regions representing half-strip directional antennae: unbounded half-strips and half-strips bounded by a range [Formula: see text] (i.e., [Formula: see text]-sided rectangular regions and rectangular regions). The directional antennae can be directed up or down. We present two polynomial time algorithms: an optimal solution for the problem with the [Formula: see text]-sided rectangular regions, and a constant factor approximation for the rectangular regions.

We introduce the SINRk model, which is a practical version of the SINR model. In the SINRk model, in order to determine whether s's signal is received at c, where s is a sender and c is a receiver, one only considers the k most significant senders w.r.t. to c (other than s). Assuming uniform power, these are the k closest senders to c (other than s). Under this model, we consider the well-studied scheduling problem: Given a set L of sender-receiver requests, find a partition of L into a minimum number of subsets (rounds), such that in each subset all requests can be satisfied simultaneously. We present an O(1)-approximation algorithm for the scheduling problem (under the SINRk model). For comparison, the best known approximation ratio under the SINR model is O(log n). We also present an O(1)- approximation algorithm for the maximum capacity problem (i.e., for the single round problem), obtaining a constant of approximation which is considerably better than those obtained under the SINR model. Finally, for the special case where k = 1, we present a PTAS for the maximum capacity problem. Our algorithms are based on geometric analysis of the SINRk model.

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 ≤ 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 ∈ 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 ≥ π/3, it always exists [1,2,9]. 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 ∈ P, we associate a wedge ∧ p of angle α and apex p. The goal is to assign an orientation and a radius r p to each wedge ∧ 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 ∈ ∧ q, q ∈ ∧ p, and |pq| ≤ 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 independently, such that the graph induced by P is connected. We apply the theorem to the antenna conversion problem.

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=\pi/2, for which n=4. Moreover, we prove that if Q_1 and Q_2 are quadruplets of antennas of angle \pi/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 \pi/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 \pi/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^\beta is minimized, where \beta \ge 1 is a constant. We present an O(1)-approximation algorithm.

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.

The coverage area of a directional antenna located at point p is a circular sector of angle α, whose orientation and radius can be adjusted. The interference at p, denoted I(p), is the number of antennas that cover p, and the interference of a communication graph G = (P,E) is I(G) = max I(p) : p ∈ P. In this paper we address the question in its title. That is, we study several variants of the following problem: What is the minimum interference I, such that for any set P of n points in the plane, representing transceivers equipped with a directional antenna of angle α, one can assign orientations and ranges to the points in P, so that the induced communication graph G is either connected or strongly connected and I(G) ≤ I. In the asymmetric model (i.e., G is required to be strongly connected), we prove that I = O(1) for α < 2π/3, in contrast with I = Θ(logn) for α = 2π, proved by Korman [12]. In the symmetric model (i.e., G is required to be connected), the situation is less clear. We show that I = Θ(n) for α < π/2, and prove that I=O(√(n)) for π/2 ≤ α ≤ 3π/2, by applying the Erdös-Szekeres theorem. The corresponding result for α = 2π is I=Θ(√(n)) , proved by Halldórsson and Tokuyama [10]. As in [12] and [10] who deal with the case α = 2π, in both models, we assign ranges that are bounded by some constant c, assuming that UDG(P) (i.e., the unit disk graph over P) is connected. Moreover, the O(√(n)) bound in the symmetric model reduces to O(√(Δ)) , where Δ is the maximum degree in UDG(P).

A power assignment is an assignment of transmission power to each of the nodes of a wireless network, so that the induced communication graph has some desired properties. The cost of a power assignment is the sum of the powers. The energy of a transmission path from node u to node v is the sum of the squares of the distances between adjacent nodes along the path. For a constant t > 1, an energy t-spanner is a graph G', such that for any two nodes u and v, there exists a path from u to v in G', whose energy is at most t times the energy of a minimum-energy path from a ton in the complete Euclidean graph. In this paper, we study the problem of finding a power assignment, such that (i) its induced communication graph is a 'good' energy spanner, and (ii) its cost is 'low'. We show that for any constant t > 1, one can find a power assignment, such that its induced communication graph is an energy t-spanner, and its cost is bounded by some constant times the cost of an optimal power assignment (where the sole requirement is strong connectivity of the induced communication graph). This is a very significant improvement over the best current result due to Shpungin and Segal, presented in last year's conference.

Symmetric disk graphs are often used to model wireless communication networks. Given a set S of n points in ℝ d (representing n transceivers) and a transmission range assignment r: S →ℝ, the symmetric disk graph of S (denoted SDG(S)) is the undirected graph over S whose set of edges is E = {(u,v) | r(u) ≥ |uv| and r(v) ≥ |uv|}, where |uv| denotes the Euclidean distance between points u and v. We prove that the weight of the MST of any connected symmetric disk graph over a set S of n points in the plane, is only O(logn) times the weight of the MST of the complete Euclidean graph over S. We then show that this bound is tight, even for points on a line. Next, we prove that if the number of different ranges assigned to the points of S is only k, k < < n, then the weight of the MST of SDG(S) is at most 2k times the weight of the MST of the complete Euclidean graph. Moreover, in this case, the MST of SDG(S) can be computed efficiently in time O(knlogn). We also present two applications of our main theorem, including an alternative and simpler proof of the Gap Theorem, and a result concerning range assignment in wireless networks. Finally, we show that in the non-symmetric model (where E = {(u,v) | r(u) ≥ |uv|}), the weight of a minimum spanning subgraph might be as big as Ω(n) times the weight of the MST of the complete Euclidean graph.