Searching Edges in the Overlap of Two Plane Graphs

Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 10389)

Abstract

Consider a pair of plane straight-line graphs whose edges are colored red and blue, respectively, and let n be the total complexity of both graphs. We present a \(O(n\log {n})\)-time O(n)-space technique to preprocess such a pair of graphs, that enables efficient searches among the red-blue intersections along edges of one of the graphs. Our technique has a number of applications to geometric problems. This includes: (1) a solution to the batched red-blue search problem [Dehne et al. 2006] in \(O(n\log {n})\) queries to the oracle; (2) an algorithm to compute the maximum vertical distance between a pair of 3D polyhedral terrains, one of which is convex, in \(O(n\log {n})\) time, where n is the total complexity of both terrains; (3) an algorithm to construct the Hausdorff Voronoi diagram of a family of point clusters in the plane in \(O((n+m)\log ^3{n})\) time and \(O(n+m)\) space, where n is the total number of points in all clusters and m is the number of crossings between all clusters; (4) an algorithm to construct the farthest-color Voronoi diagram of the corners of n disjoint axis-aligned rectangles in \(O(n\log ^2{n})\) time; (5) an algorithm to solve the stabbing circle problem for n parallel line segments in the plane in optimal \(O(n\log {n})\) time. All these results are new or improve on the best known algorithms.

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Copyright information

© Springer International Publishing AG 2017

Authors and Affiliations

  • John Iacono
    • 1
  • Elena Khramtcova
    • 2
  • Stefan Langerman
    • 2
  1. 1.Department of Computer Science and EngineeringNew York UniversityNew YorkUSA
  2. 2.Computer Science DepartmentUniversité Libre de Bruxelles (ULB)BrusselsBelgium

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