# Searching edges in the overlap of two plane graphs

## TL;DR

This paper introduces an efficient O(n log n) preprocessing technique for pairs of plane graphs with red and blue edges, enabling fast intersection searches and solving several geometric problems with improved algorithms.

## Contribution

The paper presents a novel O(n log n)-time preprocessing method for red-blue plane graph pairs, enabling efficient geometric computations and improving existing algorithms.

## Key findings

- Efficient intersection search among red-blue edges in O(n log n) time.
- New algorithms for maximum vertical distance between 3D terrains and Hausdorff Voronoi diagrams.
- Improved solutions for the farthest-color Voronoi diagram and stabbing circle problems.

## 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 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 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.