Computing simple paths among obstacles

AbstractGiven a set X of points in the plane, two distinguished points s,t∈X, and a set Φ of obstacles represented by line segments, we wish to compute a simple polygonal path from s to t that uses only points in X as vertices and avoids the obstacles in Φ. We present two results: (1) we show that finding such simple paths among arbitrary obstacles is NP-complete, and (2) we give a polynomial-time algorithm that computes simple paths when the obstacles form a simple polygon P and X is inside P. Our algorithm runs in time O(m2n2), where m is the number of vertices of P and n is the number of points in X

Computing Simple Paths Among Obstacles

Given a set X of points in the plane, two distinguished points s; t 2 X , and a set \Phi of obstacles represented by line segments, we wish to compute a simple polygonal path from s to t that uses only points in X as vertices and avoids the obstacles in \Phi. We present two results: (1) We show that finding such simple paths among arbitrary obstacles is NP-complete. (2) We give a polynomial-time algorithm that computes simple paths when the obstacles form a simple polygon P and X is inside P . Our algorithm runs in time O(m 2 n 2 ), where m is the number of vertices of P and n is the number of points in X . 1 Introduction The research we describe in this paper was motivated by polygon generation problems. Suppose that given a set X of points in the plane, we wish to generate all simple polygons whose vertices are in X. A simple, iterative approach to this problem is to start at an arbitrary point x 2 X and successively extend the path. Given the current path L ending at a point y..