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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)

- James King
- LATIN
- 2006

A mistake in the paper was found after publication. In Lemma 10, the statement " d is the leftmost exposed vertex seen by R (d), so no exposed vertex to the left of d can be seen by R (d) " is not always true. This mistake can be fixed by changing Lemma 10 to " Any guard in [L(R(p)), p) that sees d is dominated by {L(d), c, R (d)} " and proceeding according… (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)

- James King
- CCCG
- 2008

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)

A hyperplane search tree is a binary tree used to store a set S of n d-dimensional data points. In a random hyperplane search tree for S, the root represents a hyperplane defined by d data points drawn uniformly at random from S. The remaining data points are split by the hyperplane, and the definition is used recursively on each subset. We assume that the… (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)

Given a set S of n ≥ d points in general position in R d , a random hyperplane split is obtained by sampling d points uniformly at random without replacement from S and splitting based on their affine hull. A random hyperplane search tree is a binary space partition tree obtained by recursive application of random hyperplane splits. We investigate the… (More)