Watchman, Safari, and Zookeeper Routes in a Simple Polygon with Rubberband Algorithms

So far, the best result in running time for solving the “floating” watchman route problem (WRP) is O(n log n); the best result in running time for solving the “fixed” safari route problem (SRP) is O(n log n) published in 2003 by M. Dror, A. Efrat, A. Lubiw, and J. Mitchell. The best result in running time for solving the “floating” zookeeper route problem (ZRP) is O(n) published in 2001 by X. Tan. This paper provides an algorithm for the “floating” WRP with κ(ε) · O(kn) runtime, where n is the number of vertices of the given simple polygonΠ, and k the number of essential cuts; κ(ε) defines the numerical accuracy in dependency of a selected constant ε > 0. This paper also provides an algorithm for the “floating” SRP with κ(ε) · O(kn + mk) runtime, where n is the number of vertices of the given simple polygon Π, k the number of convex polygon Pis, andmk is the number of vertices of Pis. This paper also provides an algorithm for the “floating” ZRP with κ(ε) · O(kn) runtime, where n is the number of vertices of all polygons involved, k the number of the “cages”. Moreover, our algorithms are significantly simpler, easier to understand and implement than previous ones for solving the WRP, SRP and ZRP. Finally, our algorithms can solve more general SRP and ZRP where each convex polygon is replaced by a convex region such as convex polybezier (beziergon) or ellipse.