Cao An Wang

Finding the medial axis of a simple polygon in linear time

Abstract. We give a linear-time algorithm for computing the medial axis of a simple polygon P . This answers a long-standing open question—previously, the best deterministic algorithm ran in O(n log n) time. We decompose P into pseudonormal histograms, then influence histograms, then xy monotone histograms. We can compute the medial axes for xy monotone histograms and merge to obtain the medial axis for P .

Finding the Medial Axis of a Simple Polygon in Linear Time

We give a linear-time algorithm for computing the medial axis of a simple polygon P, This answers a long-standing open question—previously, the best deterministic algorithm ran in O(n log n) time. We decompose P into pseudo-normal histograms, then influence histograms and xy monotone histograms. We can compute the medial axes for xy monotone histograms and merge to obtain the medial axis for P.

Finding the Constrained Delaunay Triangulation and Constrained Voronoi Diagram of a Simple Polygon in Linear Time

In this paper, we present an

On constrained minimum pseudotriangulations

\Theta (n)

Finding the constrained Delaunay triangulation and constrained Voronoi diagram of a simple polygon in linear-time

time worst-case deterministic algorithm for finding the constrained Delaunay triangulation and constrained Voronoi diagram of a simple n-sided polygon in the plane. Up to now, only an O(n log n) worst-case deterministic and an O(n) expected time bound have been shown, leaving an O(n) deterministic solution open to conjecture.

Maximum weight triangulation and graph drawing

In this paper, we show some properties of a pseudotriangle and present three combinatorial bounds: the ratio of the size of minimum pseudotriangulation of a point set S and the size of minimal pseudotriangulation contained in a triangulation T, the ratio of the size of the best minimal pseudotriangulation and the worst minimal pseudotriangulation both contained in a given triangulation T, and the maximum number of edges in any settings of S and T. We also present a linear-time algorithm for finding a minimal pseudotriangulation contained in a given triangulation. We finally study the minimum pseudotriangulation containing a given set of non-crossing line segments.

Finding constrained and weighted Voronoi diagrams in the plane

In this paper, we present a θ(n) time worst-case deterministic algorithm for finding the constrained Delaunay triangulation and constrained Voronoi diagram of a simple n-sided polygon in the plane. Up to now, only an O(n log n) worst-case deterministic and an O(n) expected time bound have been shown, leaving an O(n) deterministic solution open to conjecture.

Approximation for Minimum Triangulations of Simplicial Convex 3-Polytopes

Abstract In this paper, we investigate the maximum weight triangulation of a convex polygon and its application to graph drawing. We can find the maximum weight triangulation of a special n -gon which inscribed on a circle in O (n 2 ) time. The complexity of this algorithm can be reduced to O (n) if the polygon is regular. The algorithm also produces a triangulation approximating the maximum weight triangulation of a convex n -gon with weight ratio 0.5. We further show that a tree always admits a maximum weight drawing if the internal nodes of the tree connect to at most 2 non-leaf nodes, and the drawing can be done in O (n) time. Finally, we prove a property of maximum planar graphs which do not admit a maximum weight drawing on any convex point set.

Three-dimensional weak visibility: complexity and applications

The Voronoi diagram of a set of weighted points (sites) whose visibilities are constrained by a set of line segments (obstacles) on the plane is studied. The diagram is called constrained and weighted Voronoi diagram. When all the sites are of the same weight, it becomes the constrained Voronoi diagram in which the endpoints of the obstacles need not be sites. An Ω(m2n2) lower bound on the combinatorial complexity of both constrained Voronoi diagram and constrained and weighted Voronoi diagram is established, where n is the number of sites and m is the number of obstacles. For constrained Voronoi diagram, an O(m2n2+n4) time and space algorithm is presented. The algorithm is optimal when m ≥ cn, for any positive constant c. For constrained and weighted Voronoi diagram, an O(m2n2 + n42α(n)) time and O(m2n2 + n4) space algorithm (where α(n) is the functional inverse of the Ackermanns function) is presented. The algorithm is near-optimal when m ≥ cn, for any positive constant c.

Construction of the Nearest Neighbor Embracing Graph of a Point Set

A minimum triangulation of a convex 3-polytope is a triangulation that contains the minimum number of tetrahedra over all its possible triangulations. Since finding minimum triangulations of convex 3-polytopes was recently shown to be NP-hard, it becomes significant to find algorithms that give good approximation. In this paper we give a new triangulation algorithm with an improved approximation ratio 2 - Ω(1/\sqrt n ) , where n is the number of vertices of the polytope. We further show that this is the best possible for algorithms that only consider the combinatorial structure of the polytopes.