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)
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)