WADS 2017: Algorithms and Data Structures pp 473-484

# Searching Edges in the Overlap of Two Plane Graphs

• John Iacono
• Elena Khramtcova
• Stefan Langerman
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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## Authors and Affiliations

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