We consider the situation where one is given a set S of points in the plane and a collection D of unit disks embedded in the plane. We show that finding a minimum cardinality subset of D such that any path between any two points in S is intersected by at least one disk is NP-complete. This settles an open problem raised in . Using a similar reduction, we… (More)
Let S be a set of n points in the plane. We present data structures that solve range-aggregate query problems on three geometric extent measure problems. Using these data structures, we can report , for any axis-parallel query rectangle Q, the area/perimeter of the convex hull, the width, and the radius of the smallest enclosing disk of the points in S ∩ Q.
Given a polygon P , for two points s and t contained in the polygon, their geodesic distance is the length of the shortest st-path within P. A geodesic disk of radius r centered at a point v ∈ P is the set of points in P whose geo-desic distance to v is at most r. We present a polynomial time 2-approximation algorithm for finding a densest geodesic unit… (More)
In this paper, we present an algorithm for exploring an unknown graph with opaque edges by multiple robots. We show that this algorithm is near optimal on graphs with n vertices and superlinear number of edges (i.e., ω(n) edges), and give an adversarial construction to show that the algorithm does not perform well on cyclic graphs with O(n) edges.
I, Ivo Vigan, declare that this thesis titled, 'Geometric Separation and Packing Problems' and the work presented in it are my own. I confirm that: This work was done wholly or mainly while in candidature for a research degree at this University. Where any part of this thesis has previously been submitted for a degree or any other qualification at this… (More)
We propose to study combinatorial and algorithmic aspects of geometric separation problems in the plane. In such a situation one is given a set of points, line segments or polygons in the plane and a set of separators such as lines, line segments, disks or polygons and the goal is to select a small subset of those separators such that every path between any… (More)