A New Lower Bound on Guard Placement for Wireless Localization

The problem of wireless localization asks to place and orient stations in the plane, each of which broadcasts a unique key within a fixed angular range, so that each point in the plane can determine whether it is inside or outside a given polygonal region. The primary goal is to minimize the number of stations. In this paper we establish a lower bound of ⌊2n/3⌋−1 stations for polygons in general position, for the case in which the placement of stations is restricted to polygon vertices, improving upon the existing ⌈n/2⌉ lower bound. The problem of wireless localization introduced in [EGS07] asks to place a set of fixed localizers (guards) in the plane so as to enable mobile communication devices to prove that they are inside or outside a secure region, defined by the interior of a polygon P . The guards are equipped with directional transmitters that can broadcast a key within a fixed angular range. The polygon P is virtual in the sense that it does not block broadcasts. A mobile device (henceforth, a point in the plane) determines whether it is inside or outside P from a monotone Boolean formula composed from the broadcasts using AND(·) and OR(+) operations only. The primary goal is to minimize the number of guards. Solutions for convex and orthogonal polygons were established [EGS07], but for general polygons, a considerable gap between a lower bound of ⌈n/2⌉ and an upper bound of n− 2 guards remains to be closed. See also [O’R07]. In this paper we establish a lower bound of ⌊2n/3⌋ − 1 guards for polygons in general position, for the case in which the placement of guards is restricted to polygon vertices (vertex guards). In [EGS07], the authors use vertex guards only, and leave open the question of whether general guards (i.e, guards placed at arbitrary points) are more efficient. In this paper we answer their question positively by establishing a solution with n/2 general guards for a polygon that requires no fewer than 2n/3− 1 vertex guards for localization. A vertex guard that broadcasts over the full internal or external angle at that vertex is called natural. Natural guards alone do not suffice to localize a region [EGS07], so non-natural guards must be employed as well. Theorem 1 There exist n-vertex simple polygons that require at least ⌊2n/3⌋ − 1 guards placed at polygon vertices for localization. Proof: The proof is by construction. Let n = 3m. Let P be a polygon consisting of m narrow spikes, as illustrated in Figure 1. P is parameterized in terms of w, h, and δ, where δ < h < w. The first m − 1 spikes each consists of three vertices li, ti, and ri, for 1 ≤ i < m. Edge tiri is vertical and of height h/2; edge rili+1 is horizontal. The vertical distance separating li and ri is h; the horizontal distance between li and ri is δ. The horizontal distance between ri and ri+1 is w. Dept. of Computer Science, Villanova Univ., Villanova, PA 19085, USA. [email protected]. Dept. of Computer Science, Siena College, Loudonville, NY 12211, USA. [email protected]. Dept. of Computer Science, Smith College, Northampton, MA 01063, USA. [email protected]. Dept. of Computer Science, Rutgers University, Camden, NJ 08102, USA. [email protected]. 1 To close the polygon, the mth spike deviates from this pattern slightly; its vertical edge tmrm has height 1.5h and the edge rml1 closes the polygon. 1 We now show that P cannot be localized with fewer than 2n/3− 1 guards placed at vertices.