Covering moving points with anchored disks

Proceedings of the …, 2006

Algorithmica, 2007

The minimum disc cover can be used to construct a dominating set on the fly for energy-efficient communications in mobile ad hoc networks, but the approach used to compute the minimum disc cover proposed in previous studies is computationally relatively expensive. In this paper, we show that the disc cover problem is in fact a special case of the general α-hull problem. In spite of being a special case, the disc cover problem is not easier than the general α-hull problem. In addition to applying the existing α-hull algorithm to solve the disc cover problem, we present a simple, yet optimal divideand-conquer algorithm that constructs the minimum disc cover for arbitrary cases, including those degenerate cases where the α-hull approach would fail.

2013

We consider the following multi-covering problem with disks. We are given two point sets Y (servers) and X (clients) in the plane, a coverage function κ : X → N, and a constant α ≥ 1. Centered at each server is a single disk whose radius we are free to set. The requirement is that each client x ∈ X be covered by at least κ(x) of the server disks. The objective function we wish to minimize is the sum of the α-th powers of the disk radii. We present a polynomial-time algorithm for this problem achieving an O(1) approximation.

2008

We propose a new randomized algorithm for maintaining a set of clusters among moving nodes in the plane. Given a specified cluster radius, our algorithm selects and maintains a variable subset of the nodes as cluster centers. This subset has the property that (1) balls of the given radius centered at the chosen nodes cover all the others and (2) the number of centers selected is a constant-factor approximation of the minimum possible. As the nodes move, an event-based kinetic data structure updates the clustering as necessary. This kinetic data structure is shown to be responsive, efficient, local, and compact. The produced cover is also smooth, in the sense that wholesale cluster re-arrangements are avoided. The algorithm can be implemented without exact knowledge of the node positions, if each node is able to sense its distance to other nodes up to the cluster radius. Such a kinetic clustering can be used in numerous applications where mobile devices must be interconnected into an...

Discrete & Computational Geometry, 2003

We propose a new randomized algorithm for maintaining a set of clusters among moving nodes in the plane. Given a specified cluster radius, our algorithm selects and maintains a variable subset of the nodes as cluster centers. This subset has the property that (1) balls of the given radius centered at the chosen nodes cover all the others and (2) the number of centers selected is a constantfactor approximation of the minimum possible. As the nodes move, an event-based kinetic data structure updates the clustering as necessary. This kinetic data structure is shown to be responsive, efficient, local, and compact. The produced cover is also smooth, in the sense that wholesale cluster re-arrangements are avoided. This clustering algorithm is distributed in nature and can enable numerous applications in ad hoc wireless networks, where mobile devices must be interconnected to collaboratively perform various tasks.

2016

In the metric multi-cover problem (MMC), we are given two point sets $Y$ (servers) and $X$ (clients) in an arbitrary metric space $(X \cup Y, d)$, a positive integer $k$ that represents the coverage demand of each client, and a constant $\alpha \geq 1$. Each server can have a single ball of arbitrary radius centered on it. Each client $x \in X$ needs to be covered by at least $k$ such balls centered on servers. The objective function that we wish to minimize is the sum of the $\alpha$-th powers of the radii of the balls. In this article, we consider the MMC problem as well as some non-trivial generalizations, such as (a) the non-uniform MMC, where we allow client-specific demands, and (b) the $t$-MMC, where we require the number of open servers to be at most some given integer $t$. For each of these problems, we present an efficient algorithm that reduces the problem to several instances of the corresponding $1$-covering problem, where the coverage demand of each client is $1$. Our ...

Let P be a set of points in the plane. The goal is to place two unit disks in the plane such that the number of points from P covered by the disks is maximized. In addition, the distance between the centers of the two disks should not exceed a specified constant R c ≥ 0. We propose two algorithms to solve this problem. The first algorithm is a simple exhaustive algorithm which runs in O(n 4) time. We then improve this algorithm by a constructing connectivity region and building a segment tree to compute two optimal disks. The resulting algorithm has O(n 3 log n) time complexity.

Lecture Notes in Computer Science, 2005

The minimum disc cover set can be used to construct the dominating set on the fly for energy-efficient communications in mobile ad hoc networks. The approach used to compute the minimum disc cover set proposed in previous study has been considered too expensive. In this paper, we show that the disc cover set problem is in fact a special case of the general α-hull problem. In addition, we prove that the disc cover set problem is not any easier than the α-hull problem by linearly reducing the element uniqueness problem to the disc cover set problem. In addition to the known α-hull approach, we provide an optimal divide-and-conquer algorithm that constructs the minimum disc cover set for arbitrary cases, including the degenerate ones where the traditional α-hull algorithm incapable of handling.

Geographical Analysis, 2010

The location set-covering problem is extended to a p p l y to three new situations. (1) The demands are assumed to occur continuously along arcs of a network. (2) A mobile unit departs from one of the locations to be chosen and picks up the demand, providing service at a still more distant point. (3) New demands and sites occur over time. N , = (j E Jlt, I s) for all p E P. Charles ReVelle is professor of geography and enuironmental engineering, The. Johns Hopkins Uniuersity. Constantine Toregas is affiliated with Public Technology, Inc. Louis Fa1 kson is economist, Cornell Uniuersity.

2003

Continuum percolation models where each point of a two-dimensional Poisson point process is the center of a disc of given (or random) radius r, have been extensively studied. In this paper, we consider the generalization in which a deterministic algorithm (given the points of the point process) places the discs on the plane, in such a way that AMS 1991 subject classi cations. Primary 60D05, 60K35, 82B26, 82B43, 94C99. 1 each disc covers at least one point of the point process and that each point is covered by at least one disc. This gives a model for wireless communication networks, which was the original motivation to study this class of problems.