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A set <i>G</i> of points on a 1.5-dimensional terrain, also known as an <i>x</i>-monotone polygonal chain, is said to guard the terrain if every point on the terrain is seen by a point in <i>G</i>. Two points on the terrain see each other if and only if the line segment between them is never strictly below the terrain. The minimum terrain guarding problem(More)
We provide an O(log log OPT)-approximation algorithm for the problem of guarding a simple polygon with guards on the perimeter. We first design a polynomial-time algorithm for building ε-nets of size O 1 ε log log 1 ε for the instances of Hitting Set associated with our guarding problem. We then apply the technique of Brönnimann and Goodrich to build an(More)
A guarding problem can naturally be modeled as a set system (U, S) in which the universe U of elements is the set of points we need to guard and our collection S of sets contains, for each potential guard g, the set of points from U seen by g. We prove bounds on the maximum VC-dimension of set systems associated with guarding both 1.5D terrains (monotone(More)
Let P : R d → A be a query problem over R d for which there exists a data structure S that can compute P(q) in O(log n) time for any query point q ∈ R d. Let D be a probability measure over R d representing a distribution of queries. We describe a data structure T = T P,D , called the odds-on tree, of size O(n) that can be used as a filter that quickly(More)
Let G be a (possibly disconnected) planar subdivision and let D be a probability measure over R 2. The current paper shows how to preprocess (G, D) into an O(n) size data structure that can answer planar point location queries over G. The expected query time of this data structure, for a query point drawn according to D, is O(H + 1), where H is a lower(More)
A set G of points on a 1.5-dimensional terrain, also known as an x-monotone polygonal chain, is said to guard the terrain if any point on the terrain is seen by a point in G. Two points on the terrain see each other if and only if the line segment between them is never strictly below the terrain. The minimum terrain guarding problem asks for a minimum(More)