Geometric Searching and Link Distance (Extended Abstract)

Given n orthogonal line segments on the plane, their intersection graph is defined such that each vertex corresponds to a segment, and each edge corresponds to a pair of intersecting segments. Although this graph can have f~(n 2) edges, we show that breadth first search can be accomplished in O(n log n) time and O(n) space. As an application, we show that the minimum link rectilinear path between two points s and t amidst rectilinear polygonal obstacles can be computed in O(nlog n) time and O(n) space, which is optimal. We mention other related results in the paper. 1 I n t r o d u c t i o n Given a set S of n orthogonal line segments on the plane, the intersection graph of S is defined as follows: each segment corresponds to a vertex in the graph, and an edge connects a pair of vertices if the two corresponding segments intersect. Clearly this graph is bipartite with the horizontal and vertical segments forming two independent sets. This graph can potentially have ~(n 2) edges. Given an initial segment h, the main result of this paper is an efficient algorithm to label every segment by its shortest distance from h in the graph. This is equivalent to breadth first search. Our algorithm does not explicitly generate all the edges, and in fact runs in O(n log n) time and O(n) space. It uses elementary data structures such as binary trees and priority queues. Using this algorithm we solve several other problems efficiently. Asano and Imai have shown how to perform breadth first and even depth first search on such graphs in O(nlogn) time and O(nlog n) space ([IA86], [IA87]). They use fairly complex data structures. However their techniques have wider applications. The primary application of our result is motivated by a motion planning problem. Suppose we are given a collection of disjoint rectilinear polygonal obstacles inside a