I present a generalization of Chew's first algorithm for Delaunay mesh refinement. I split the line segments of an input planar straight line graph (PSLG) such that the lengths of split segments are asymptotically proportional to the local feature size at their endpoints. By employing prior algorithms, I then refine the truly or constrained Delaunay triangulation of the PSLG by inserting off-center Steiner vertices of "skinny" triangles while prioritizing triangles with shortest edges first. This technique inserts Steiner vertices in an advancing front manner such that we obtain a size-optimal, truly or constrained Delaunay mesh if the desired minimum angle is less than 30° (in the absence of small input angles). This is an improvement over prior algorithms that produce size-optimal meshes with minimum angles of about 26.4°and 28.6°for truly and constrained Delaunay meshes, respectively. Even in the presence of small input angles, the upper bound on the maximum angle is an angle strictly greater than 120° (an improvement from about 137°). The lower bound on the minimum angle in the presence of small angles is identical to prior bounds.

We investigate weighted straight skeletons from a geometric, graph-theoretical, and combinatorial point of view. We start with a thorough definition and shed light on some ambiguity issues in the procedural definition. We investigate the geometry, combinatorics, and topology of faces and the roof model, and we discuss in which cases a weighted straight skeleton is connected. Finally, we show that the weighted straight skeleton of even a simple polygon may be non-planar and may contain cycles, and we discuss under which restrictions on the weights and/or the input polygon the weighted straight skeleton still behaves similar to its unweighted counterpart. In particular, we obtain a non-procedural description and a linear-time construction algorithm for the straight skeleton of strictly convex polygons with arbitrary weights.

Many properties of finite point sets only depend on the relative position of the points, e.g., on the order type of the set. However, many fundamental algorithms in computational geometry rely on coordinate representations. This includes the straightforward algorithms for finding a halving line for a given planar point set, as well as finding a point on the convex hull, both in linear time. In his monograph , Knuth asks whether these problems can be solved in linear time in a more abstract setting, given only the orientation of each point triple, i.e., the set's chirotope, as a source of information. We answer this question in the affirmative. More precisely, we can find a halving line through any given point, as well as the vertices of the convex hull edges that are intersected by the supporting line of any two given points of the set in linear time. We first give a proof for sets realizable in the Euclidean plane and then extend the result to non-realizable abstract order types.

In this paper, we study an interesting geometric partition problem, called , which arises in (IMRT). In current clinical practice, a multi-leaf collimator (MLC) with a constraint is used to deliver the prescribed intensity maps (IMs). However, the maximum leaf spread of an MLC may require to split a large intensity map into several overlapping sub-IMs with each being delivered separately. We develop a close-to-linear time algorithm for solving the field splitting problem while minimizing the total complexity of the resulting sub-IMs, thus improving the treatment delivery efficiency. Meanwhile, our algorithm strives to minimize the maximum beam-on time of those sub-IMs. Our basic idea is to formulate the field splitting problem as computing a shortest path in a directed acyclic graph, which expresses a special "layered" structure. The edge weights of the graph satisfy the Monge property, which enables us to solve this shortest path problem by examining only a small portion of the graph, yielding a close-to-linear time algorithm. To minimize the maximum beam-on time of the resulting sub-IMs, we consider an interesting min-max slope path problem in a monotone polygon which is solvable in linear time. The min-max slope path problem may be of interest in its own right. Experimental results based on real medical data and computer generated IMs showed that our new algorithm runs fast and produces high quality field splitting results.

Given a set of black points in general position, we say that a set of white points blocks if in the Delaunay triangulation of [Formula: see text] there is no edge connecting two black points. We give the following bounds for the size of the smallest set blocking : (i) [Formula: see text] white points are always sufficient to block a set of black points, (ii) if is in convex position, [Formula: see text] white points are always sufficient to block it, and (iii) at least [Formula: see text] white points are always necessary to block a set of black points.

We study the problem how to draw a planar graph crossing-free such that every vertex is incident to an angle greater than . In general a plane straight-line drawing cannot guarantee this property. We present algorithms which construct such drawings with either tangent-continuous biarcs or quadratic Bézier curves (parabolic arcs), even if the positions of the vertices are predefined by a given plane straight-line drawing of the graph. Moreover, the graph can be drawn with circular arcs if the vertices can be placed arbitrarily. The topic is related to non-crossing drawings of multigraphs and vertex labeling.

In this paper we present a package for implementing exact kinetic data structures built on objects which move along polynomial trajectories. We discuss how the package design was influenced by various considerations, including extensibility, support for multiple kinetic data structures, access to existing data structures and algorithms in CGAL, as well as debugging. Due to the similarity between the operations involved, the software can also be used to compute arrangements of polynomial objects using a sweepline approach. The package consists of three main parts, the kinetic data structure framework support code, an algebraic kernel which implements the set of algebraic operations required for kinetic data structure processing, and kinetic data structures for Delaunay triangulations in one and two dimensions, and Delaunay and regular triangulations in three dimensions. The models provided for the algebraic kernel support both exact operations and inexact approximations with heuristics to improve numerical stability.

For a set S of points in ℝ(d), an s-spanner is a subgraph of the complete graph with node set S such that any pair of points is connected via some path in the spanner whose total length is at most s times the Euclidean distance between the points. In this paper we propose a new sparse (1 + ε)-spanner with O(n/ε(d)) edges, where ε is a specified parameter. The key property of this spanner is that it can be efficiently maintained under dynamic insertion or deletion of points, as well as under continuous motion of the points in both the kinetic data structures setting and in the more realistic blackbox displacement model we introduce. Our deformable spanner succinctly encodes all proximity information in a deforming point cloud, giving us efficient kinetic algorithms for problems such as the closest pair, the near neighbors of all points, approximate nearest neighbor search (aka approximate Voronoi diagram), well-separated pair decompositions, and approximate k-centers.