Optimum watchman routes

  • Authors:

  • W Chin;S Ntafos

  • Affiliations:

  • Computer Science Program, The University of Texas at Dallas Richardson, TX;Computer Science Program, The University of Texas at Dallas Richardson, TX

  • Venue:
  • SCG '86 Proceedings of the second annual symposium on Computational geometry
  • Year:
  • 1986

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Abstract

In this paper we consider the problem of finding shortest routes from which every point in a given space is visible (watchman routes). We show that the problem is NP-hard when the space is a polygon with holes even if the polygon and the holes are convex or rectilinear. The problem remains NP-hard for simple polyhedra. We present O(n) and O(nlogn) algorithms to find a shortest route in a simple rectilinear monotone polygon and a simple rectilinear polygon respectively, where n is the number of vertices in the polygon. Finding optimum watchman routes in simple polygons is closely related to the problem of finding shortest routes that visit a set of convex polygons in the plane in the presence of obstacles. We show that finding a shortest route that visits a set of convex polygons is NP-hard even when there are no obstacles. We present an O(logn) algorithm to find the shortest route that visits a point and two convex polygons, where n is the total number of vertices.