Covering With Inexactly Placed Sensors Amotz Bar-Noy Theodore Brown Matthew P. Johnson Ou Liu City University of New York Abstract—We consider a class of geometric coverage problems in which the objects used to cover the region of interest are, because of practical difficulties, positioned with only approximate accuracy. This changes the character of some coverage problems that solve for optimal disk positions or disk sizes, ordinarily assuming the disks can be placed precisely in their chosen positions, and motivates new problems. These include guaranteed and probabilistic coverage of a region using few inexactly placed unit disks, maximizing the sum of the allowable placement errors given a fixed set of disk positions, and choosing error bounds that satisfy max-min fairness, which we do in O(n log n) time. The allowable placement areas need not be disks, however; we conclude with a general method of constructing them. I. I NTRODUCTION In sensor networks, a disk is typically used to model the coverage range of a sensor [3]. There are a variety of coverage problems which solve for optimal locations for sensor place- ment, or conversely, given a chosen set of sensor locations, solve for optimal (disk-shaped) coverage ranges, according to some objective function [1]. In all these problems, it is typically assumed that sensors can be (or have been) placed exactly in their intended positions. Since sensors are physical objects, however, this assumption of exact placement is only an ideal, which may in realistic situations be violated to various degrees. There may be physical obstructions preventing place- ment in particular locations, in both urban environments and in nature. More fundamentally, the placement of any physical object is necessarily performed with some amount of error. This simple observation yields two classes of problems: 1) positioning sensors given bounds on the placement error and 2) determining the amount of room for error given a chosen set of intended sensor positions. Referring to the actual or empirical position of a disk’s center, the disk’s wiggle region is the point set of locations for the disk center that are consistent with the global coverage guarantee (see the shaded areas of Fig. 1.b). The wiggle radius is the radius of the largest disk centered on the a priori center contained in this region [4]. Summary of results: We solve the placement problem for unit disks with fixed uniform wiggle radii, for guaranteed and probabilistic coverage, by reduction to coverage solutions for boolean disks and probabilistic disks, respectively. Given a set of disk sensors, maximizing the sum of wiggle radii reduces to minimizing the sum of disk radii. Finding a max-min fair wiggle radii assignment (equivalently, min-max fair disk radii) is NP-hard for the discrete problem (of covering a finite set of points), but is solvable in O(n log n) time for the continuous setting (of covering a region). Finally, we characterize the set of maximal wiggle regions in terms of regions of 3-coverage. (a) Exact placement (b) Wiggle areas shaded Fig. 1. Triangular lattice for exact and inexact placement II. POSITIONING SENSORS Deterministic setting: Covering a large convex region (ignore edges, or assume the region of interest is the convex hull of the set of server locations) using a minimal number of disks of radius R, is solved optimally by arranging the disks in a triangular lattice (see Fig. 1.a [6], [8]). Suppose each disk has wiggle radius w, and we again seek positions for covering the area with few sensors. Let us indicate a disk with radius X and wiggle radius y as an (X, y) disk. Then it is easy to see that an (R, w) is functionally equivalent to an (R - w, 0) in the sense that in a given sensor configuration, one (R, w) disk can be replaced with an (R - w, 0) disk with affecting the coverage guarantee. The problem of covering with (R, w) disks is equivalent to covering with (R - w, 0) disks. Probabilistic setting: The probabilistic setting is more compli- cated. Wiggle radius w again determines a wiggle disk, but we assume that the center’s empirical location within the wiggle disk is chosen according to some probability distribution, such as uniform of Gaussian. A binary disk (R, w) with wiggle disk pdf f will act as a probabilistic (R + w, 0) disk with a certain probability function f * of distance. We will have f * (d)=1 for all d ≤ R-w, f * (R + w)=0, and a decreasing function over the range [R - w, R + w]. We find the function f * exactly or by calculation, depending on the underlying wiggle disc pdf. A full coverage can be found using just the guaranteed portion of these probabilistic disks, but a more appropriate coverage problem is to guarantee for each particular point p that it is covered with probability 1 - ǫ. This problem was considered recently for coverage probabilities that stay fixed at 1 up to a certain distance and then drop exponentially fast [5]. A similar, but more complicated analysis can be given for our setting, since the probability may decline more slowly than exponential. (A uniform wiggle disk pdf e.g. yields an f * nearly linear over [R - w, R + w].) The optimal granularity of the triangular lattice can then be found by calculation.