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THE HAUSDORFF VORONOI DIAGRAM OF POLYGONAL OBJECTS: A DIVIDE AND CONQUER APPROACH
 INTERNATIONAL JOURNAL OF COMPUTATIONAL GEOMETRY & APPLICATIONS
, 2004
"... We study the Hausdorff Voronoi diagram of a set S of polygonal objects in the plane, a generalization of Voronoi diagrams based on the maximum distance of a point from a polygon, and show that it is equivalent to the Voronoi diagram of S under the Hausdorff distance function. We investigate the st ..."
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We study the Hausdorff Voronoi diagram of a set S of polygonal objects in the plane, a generalization of Voronoi diagrams based on the maximum distance of a point from a polygon, and show that it is equivalent to the Voronoi diagram of S under the Hausdorff distance function. We investigate the structural and combinatorial properties of the Hausdorff Voronoi diagram and give a divide and conquer algorithm for the construction of this diagram that improves upon previous results. As a byproduct we introduce the Hausdorff hull, a structure that relates to the Hausdorff Voronoi diagram in the same way as a convex hull relates to the ordinary Voronoi diagram. The Hausdorff Voronoi diagram finds direct application in the problem of computing the critical area of a VLSI Layout, a measure reflecting the sensitivity of a VLSI design to random manufacturing defects, described in a companion paper. 13 Keywords: Voronoi diagram; Hausdorff distance; Hausdorffhull; divide and conquer; VLSI yield; VLSI critical area; viablock defects.
FarthestPolygon Voronoi Diagrams
, 2007
"... Given a family of k disjoint connected polygonal sites of total complexity n, we consider the farthestsite Voronoi diagram of these sites, where the distance to a site is the distance to a closest point on it. We show that the complexity of this diagram is O(n), and give an O(n log 3 n) time algori ..."
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Given a family of k disjoint connected polygonal sites of total complexity n, we consider the farthestsite Voronoi diagram of these sites, where the distance to a site is the distance to a closest point on it. We show that the complexity of this diagram is O(n), and give an O(n log 3 n) time algorithm to compute it. We also prove a number of structural properties of this diagram. In particular, a Voronoi region may consist of k − 1 connected components, but if one component is bounded, then it is equal to the entire region.
Stabbing Segments with Rectilinear Objects
 MEXICAN CONFERENCE ON DISCRETE MATHEMATICS AND COMPUTATIONAL GEOMETRY
, 2013
"... Given a set of n line segments in the plane, we say that a region R ⊆ R2 is a stabber if R contains exactly one endpoint of each segment of the set. In this paper we provide efficient algorithms for determining whether or not a stabber exists for several shapes of stabbers. Specifically, we consider ..."
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Given a set of n line segments in the plane, we say that a region R ⊆ R2 is a stabber if R contains exactly one endpoint of each segment of the set. In this paper we provide efficient algorithms for determining whether or not a stabber exists for several shapes of stabbers. Specifically, we consider the case in which the stabber can be described as the intersection of isothetic halfplanes (thus the stabbers are halfplanes, strips, quadrants, 3sided rectangles, or rectangles). We provide efficient algorithms reporting all combinatorially different stabbers of that shape. The algorithms run in O(n) time (for the halfplane case), O(n logn) time (for strips and quadrants), O(n2) (for 3sided rectangles), or O(n3) time (for rectangles).
Spanning Colored Points with Intervals
"... We study a variant of the problem of spanning colored objects where the goal is to span colored objects with two similar regions. We dedicate our attention in this paper to the case where objects are points lying on the real line and regions are intervals. Precisely, the goal is to compute two inter ..."
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We study a variant of the problem of spanning colored objects where the goal is to span colored objects with two similar regions. We dedicate our attention in this paper to the case where objects are points lying on the real line and regions are intervals. Precisely, the goal is to compute two intervals together spanning all colors. As the main ingredient of our algorithm, we first introduce a kinetic data structure to keep track of minimal intervals spanning all colors. Then we present a novel algorithm using the proposed KDS to compute a pair of intervals which together span all the colors with the property that the largest one is as small as possible. The algorithm runs in O(n2 log n) using O(n) space where n is the number of points. 1
A Randomized Incremental Approach for the Hausdorff Voronoi Diagram of Noncrossing Clusters ∗
"... Abstract. In the Hausdorff Voronoi diagram of a set of pointclusters in the plane, the distance between a point t and a cluster P is measured as the maximum distance between t and any point in P while the diagram is defined in a nearest sense. This diagram finds direct applications in VLSI comput ..."
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Abstract. In the Hausdorff Voronoi diagram of a set of pointclusters in the plane, the distance between a point t and a cluster P is measured as the maximum distance between t and any point in P while the diagram is defined in a nearest sense. This diagram finds direct applications in VLSI computeraided design. In this paper, we consider “noncrossing ” clusters, for which the combinatorial complexity of the diagram is linear in the total number n of points on the convex hulls of all clusters. We present a randomized incremental construction, based on pointlocation, to compute the diagram in expected O(n log2 n) time and expected O(n) space, which considerably improves previous results. Our technique efficiently handles nonstandard characteristics of generalized Voronoi diagrams, such as sites of nonconstant complexity, sites that are not enclosed in their Voronoi regions, and empty Voronoi regions. 1