The authors give an algorithm to compute the visibility polygon from a point among a set of h pairwise-disjoint polygonal obstacles with a total of n vertices. The algorithm uses $O(n)$ space and runs in optimal time $\Theta (n + h \log h)$, improving the previous upper bound of $O(n \log n)$. A direct consequence of the algorithm is an $O(n + h \log h)$ time algorithm for computing the convex hull of h disjoint simple polygons.

Let V be a set of n points in 3-dimensional Euclidean space. A subgraph of the complete Euclidean graph is a t-spanner if for any u and v in V, the length of the shortest path from u to v in the spanner is at most ttimes d(u, v). We show that for any t > 1, a greedy algorithm produces a t-spanner with O(n) edges, and total edge weight O(1).wt(MST), where MST is a minimum spanning tree of V.

In this paper we give optimal solutions for two versions of the two-guard problem. Given a simple polygon P with vertices s and t, the straight walk problem asks whether we can move two points monotonically on P from s to t, one clockwise and one counterclockwise, such that the points are always co-visible. In the counter walk problem, both points move clockwise, one from s to t and the other from t to s. We provide Θ(n) constructive algorithms for both problems. We obtain our results by examining the structure of the restrictions placed on the motion of the two points, and by employing properties of shortest paths and shortest path trees.

Approximate decision algorithms for point set congruence

A number of problems in computational geometry involving simple polygons can be solved in linear time once the polygon has been triangulated. Since the worst-case time bound for triangulating a general simple polygon is currently super-linear, these algorithms are not linear time in the worst case. In this paper we define the structured visibility profile of a polygonal path and show how to compute it in linear time. We apply this result to solve many problems in linear time that previously required triangulation. Our list of problems includes: translation separability of two simple polygons, computing the weak visibility region for a segment within a simple polygon, finding shortest monotone paths in a simple polygon, ray shooting from an edge, and the convex rope problem.