Simeon C. Ntafos

Shortest watchman routes in simple polygons

In this paper we present an O(n4, log logn) algorithm to find a shortest watchman route in a simple polygon through a point,s, in its boundary. A watchman route is a route such that each point in the interior of the polygon is visible from at least one point along the route.

Watchman routes under limited visibility

We consider the motion planning problem for a point constrained to move along a smooth closed convex path of bounded curvature. The workspace of the moving point is bounded by a convex polygon with m vertices, containing an obstacle in a form of a simple polygon with n vertices. We present an O(m+n) time algorithm finding the path, going around the obstacle, whose curvature is the smallest possible.

Optimum watchman routes

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.

The zookeeper route problem

Abstract The { S , T }-route problem inside a polygon P with a set S of of sights (edge segments) and a set T of point threats is to find a route such that each point in S is visible from at least one point in the route and the route is not visible to any of the points in T [7]. The requirement that exposure to the threats is not allowed at all is usually too severe and solutions do not exist in many cases, especially those arising in typical applications where the sights and threats overlap (e.g., defense applications). An alternative way to model the risk posed by the threats is to enclose each threat location in a polygonal envelope (within which the risk is unacceptably high) resulting in the zookeeper route problem. We are given a polygon P and a collection P ′ of convex polygons inside P and we want to find a shortest route that visits (without entering) the polygons in P ′ (e.g., design a route for a zookeeper that wants to feed animals in enclosures). We show that the general zookeeper route problem is NP-hard, we present necessary and sufficient conditions for the existence of a zookeeper route and give an O( n 2 ) algorithm for the case where P is a simple polygon and the polygons in P ′ are attached to the boundary of P .

External Watchman Routes

We consider the problem of finding a shortest watchman route from which the exterior of a polygon is visible (external watchman route). We present an O (n4 log logn) algorithm to find shortest external watchman routes for simple polygons by transforming the external watchman route problem to a set of internal watchman route problems. Also, we present faster external watchman route algorithms for special cases. These include optimal O (n) algorithms for convex, monotone, star and spiral polygons and an O (n log logn) algorithm for rectilinear polygons.

The robber route problem

Abstract We are given a polygon P together with a point x on the boundary of P , a set S of edges of P (the sights), and a set T of points in P (the threats). The robber route (or {{ S, T }-route) problem is to find a shortest route (if one exists) from x to x and such that every point in S is visible from some point along the route while the route is not visible to any of the points in T . We present necessary and sufficient conditions for the existence of a robber route and results on the complexity of the robber route problem and its relationship to the watchman route problem.

Watchman routes in the presence of a pair of convex polygons

Given a set of polygonal obstacles in the plane, the shortest watchman route problem asks for a closed route from which each point in the exterior of the polygons is visible to some point along the route. This problem is known to be NP-hard and the development of an efficient approximation algorithm is still open. We present an O(n2) time algorithm for computing the shortest watchman route for a pair of convex polygons, where n is the total number of edges in the polygons. We also show that the algorithm can be easily extended, without increasing its complexity, to compute the shortest watchman route when the polygons are enclosed by a third convex polygon.

Software Testing: Theory and Practice

Imtrnduction. One of the major problems in computing is that of developing reliable software. The main method for reaching some level of confidence on the reliability of a software product is testing, Program testing strategies execute the program on a (usually very small) subset of its inputs. Herein lies the problem and challenge of soflware testing. How do we select the test cases so that we can be confident about the reliability of the program from a very small sample? Suggested approaches (testing strategies) are plentiful. Most strategies use some kind of knowledge about the program. This may be the structure of the program, the functional characteristics of the program, information on commonly occurring error types or various combinations of them. Automation of the testing process is an important factor in the applicability of a method. This involves tools for generating test cases to satisfy a testing criterion, tools that run and monitor the test cases and tools that check the outputs for correctness. Another important issue is the type of conclusions that can be reached about the reliability of the program from the test outcomes. In every turn, one faces very difficult problems.