We introduce a new approach for drawing diagrams. Our approach is to use a technique we call confluent drawing for visualizing non-planar graphs in a planar way. This approach allows us to draw, in a crossing-free manner, graphs—such as software interaction diagrams—that would normally have many crossings. The main idea of this approach is quite simple: we… (More)
Results The greedy triangulation (GT) of a set S of n points in the plane is the triangulation obtained by starting with the empty set and at each step adding the shortest compatible edge between two of the points, where a compatible edge is defined to be an edge that crosses none of the previously added edges. In this paper we present a simple, practical… (More)
We present an O(n 4)-time and O(n 2)-space algorithm that computes a sub-graph of the minimum weight triangulation (MWT) of a general point set. The algorithm works by finding a collection of edges guaranteed to be in any locally minimal triangulation. We call this subgraph the LMT-skeleton. We also give a variant called the modified LMT-skeleton that is… (More)
We show that a popular variant of the well known k-d tree data structure satisfies an important packing lemma. This variant is a binary spatial partitioning tree T defined on a set of n points in IR d , for fixed d ≥ 1, using the simple rule of splitting each node's hyper-rectangular region with a hyperplane that cuts the longest side. An interesting… (More)
A three-dimensional polygon is triangulable if it has a non-self-intersecting triangulation which deenes a simply-connected 2-manifold. We show that the problem of deciding whether a 3D polygon is triangulable is an NP-complete problem. We then establish some necessary conditions and some suucient conditions for a polygon to be tri-angulable, providing… (More)
We present algorithms for five interdistance enumeration problems that take as input a set S of n points in I R d (for a fixed but arbitrary dimension d) and as output enumerate pairs of points in S satisfying various conditions. We present: an O(n log n + k) time and O(n) space algorithm that takes as additional input a distance δ and outputs all k pairs… (More)
1 Overview of Result We present a new polynomial time algorithm that computes a subgraph of the exact minimum weight triangu-lation of a set of points. The subgraph contains more edges than previously known subgraphs 6, 4, 1]. In fact, it is been tested on many hundreds of uniformly distributed random point sets containing up to 220 points and in every… (More)
In this paper we investigate a new type of Voronoi diagrams in which every region is deened by a pair of point sites and some distance function from a point to two points. We analyze the complexity of the respective nearest-and furthest-neighbor diagrams of several such distance functions, and show how to compute the diagrams eeciently.