Improved Bounds for Wireless Localization

We consider a novel class of art gallery problems inspired by wireless localization that has recently been introduced by Eppstein, Goodrich, and Sitchinava. Given a simple polygonP, place and orient guards each of which broadcasts a unique key within a fixed angular range. In contrast to the classical art gallery setting, broadcasts are not blocked by the edges ofP. At any point in the plane one must be able to tell whether or not one is located inside P only by looking at the set of keys received. In other words, the interior of the polygon must be described by a monotone Boolean formula composed from the keys. We improve both upper and lower bounds for the general problem where guards may be placed anywhere by showing that the maximum number of guards to describe any simple polygon on n vertices is between roughly $\frac{3}{5}n$ and $\frac{4}{5}n$ . Aguarding that uses at most $\frac{4}{5}n$ guards can be obtained in O(nlog n) time. For the natural setting where guards may be placed aligned to one edge or two consecutive edges ofP only, we prove that n−2 guards are always sufficient and sometimes necessar

Large bichromatic point sets admit empty monochromatic 4-gons

We consider a variation of a problem stated by Erd˝os and Szekeres in 1935 about the existence of a number fES(k) such that any set S of at least fES(k) points in general position in the plane has a subset of k points that are the vertices of a convex k-gon. In our setting the points of S are colored, and we say that a (not necessarily convex) spanned polygon is monochromatic if all its vertices have the same color. Moreover, a polygon is called empty if it does not contain any points of S in its interior. We show that any bichromatic set of n ≥ 5044 points in R2 in general position determines at least one empty, monochromatic quadrilateral (and thus linearly many).Postprint (published version

Improved bounds for wireless localization

Abstract. We consider a novel class of art gallery problems inspired by wireless localization. Given a simple polygon P, place and orient guards each of which broadcasts a unique key within a fixed angular range. Broadcasts are not blocked by the edges of P. The interior of the polygon must be described by a monotone Boolean formula composed from the keys. We improve both upper and lower bounds for the general setting by showing that the maximum number of guards to describe any simple polygon on n vertices is between roughly 3 5 4 n and n. For the natura