Searching Edges in the Overlap of Two Plane Graphs

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.