Small Strictly Convex Quadrilateral Meshes of Point Sets

In this paper we give upper and lower bounds on the number of Steiner points required to construct a strictly convex quadrilateral mesh for a planar point set. In particular, we show that 3\lfloorn/2\rfloor internal Steiner points are always sufficient for a convex quadrilateral mesh of n points in the plane. Furthermore, for any given n\geq 4, there are point sets for which \lceil(n–3)/2\rceil–1 Steiner points are necessary for a convex quadrilateral mesh.     

Small Strictly Convex Quadrilateral Meshes of Point Sets

In this paper we give upper and lower bounds on the number of Steiner points required to construct a strictly convex quadrilateral mesh for a planar point set. In particular, we show that 3\lfloorn/2\rfloor internal Steiner points are always sufficient for a convex quadrilateral mesh of n points in the plane. Furthermore, for any given n\geq 4, there are point sets for which \lceil(n–3)/2\rceil–1 Steiner points are necessary for a convex quadrilateral mesh.     

Constant-Time Tree Algorithms on Reconfigurable Meshes on Size n × n

Recently, an elegant and powerful architecture called the reconfigurable mesh has been proposed in the literature. In essence, a reconfigurable mesh consists of a mesh-connected architecture enhanced by the addition of a dynamic bus system whose configuration changes in response to computational and communication needs. In this paper we show that the reconfigurable mesh architecture can be exploited to yield very simple constant-time algorithms to solve a number of important computational problems involving trees. Specifically, we address the problem of generating the computation tree form of an arithmetic expression, the problem of reconstructing a binary tree from its preorder and inorder traversals, and the problem of reconstructing an ordered forest from its preorder and postorder traversals. We show that with an input of size n, all these problems find constant-time solutions on a reconfigurable mesh of size n × n.     

An Efficient Algorithm to Generate all Maximal Cliques on Trapezoid Graphs

In this paper, to find all maximal cliques of a trapezoid graph a set of intervals have been constructed by projecting the geometrical representation of the graph on the bottom line. The proposed algorithm for this purpose takes O(n ² + yn) time, where n is the number of vertices of the graph and y is the output size.     

Skeleton-based computational method for the generation of a 3D finite element mesh sizing function

This paper focuses on the generation of a three-dimensional (3D) mesh sizing function for geometry-adaptive finite element (FE) meshing. The mesh size at a point in the domain of a solid depends on the geometric complexity of the solid. This paper proposes a set of tools that are sufficient to measure the geometric complexity of a solid. Discrete skeletons of the input solid and its surfaces are generated, which are used as tools to measure the proximity between geometric entities and feature size. The discrete skeleton and other tools, which are used to measure the geometric complexity, generate source points that determine the size and local sizing function at certain points in the domain of the solid. An octree lattice is used to store the sizing function as it reduces the meshing time. The size at every lattice-node is calculated by interpolating the size of the source points. The algorithm has been tested on many industrial models, and it can be extended to consider other non-geometric factors that influence the mesh size, such as physics, boundary conditions, etc.Sandia National Laboratory is a multiprogram laboratory operated by the Sandia Corporation, a Lockheed Martin Company, for the US Department of Energy under contract DE-AC04-94AL85000.     

A Fast Algorithm for k-Nearest Neighbor Problem on a Reconfigurable Mesh Computer

Given a set S of m points stored on a reconfigurable mesh computer of size n×n, one point per processing element (PE). In this paper we present a parallel method for solving the k-Nearest Neighbor problem (k-NN). This method permits each point of S to know its k-NN (0<k<m). The corresponding algorithm requires that each PE must have 2k registers where it stores the (x,y) coordinates of its k-NN in a given order. This algorithm has a complexity of O(logh+k ²) times, where h is a maximal number of points within a row of the mesh. This complexity is reduced to O(k ²) times, using an appropriate procedure which demonstrates the power of the reconfiguration operations carried out by the processors, and the polymorphic properties of the mesh.     

Finding the conditional location of a median path on a tree

In this paper, we study the problem of locating a median path of limited length on a tree under the condition that some existing facilities are already located. The existing facilities may be located at any subset of vertices. Upper and lower bounds are proposed for both the discrete and continuous models. In the discrete model, a median path is not allowed to contain partial edges. In the continuous model, a median path may contain partial edges. The proposed upper bounds for these two models are O(n log n) and O(n log nα(n)), respectively. They improve the previous known bounds from O(n log² n) and O(n²), respectively. The proposed lower bounds are both Ω(n log n).     

Time-Optimal Nearest-Neighbor Computations on Enhanced Meshes

The All Nearest Neighbor problem (ANN, for short) is stated as follows: given a setSof points in the plane, determine for every point inS, a point that lies closest to it. The ANN problem is central to VLSI design, computer graphics, pattern recognition, and image processing, among others. In this paper we propose time-optimal algorithms to solve the ANN problem for an arbitrary set of points in the plane and also for the special case where the points are vertices of a convex polygon. Both our algorithms run on meshes with multiple broadcasting. Our first main contribution is to establish an Ω(logn) time lower bound for the task of solving an arbitraryn-point instance of the ANN problem, even if the points are the vertices of a convex polygon. We obtain our time lower bound results for the CREW-PRAM by using a novel technique involving geometric constructions. These constructions allow us to reduce the well-known OR problem to each of the geometric problems of interest. We then port these time lower bounds to the mesh with multiple broadcasting using simulation results. Our second main contribution is to show that the time lower bound obtained is tight, by exhibiting algorithms solving the problem inO(logn) time on a mesh with multiple broadcasting of sizen×n.     

Mandatory Critical Points of 2D Uncertain Scalar Fields

This paper introduces a novel, non‐local characterization of critical points and their global relation in 2D uncertain scalar fields. The characterization is based on the analysis of the support of the probability density functions (PDF) of the input data. Given two scalar fields representing reliable estimations of the bounds of this support, our strategy identifies mandatory critical points: spatial regions and function ranges where critical points have to occur in any realization of the input. The algorithm provides a global pairing scheme for mandatory critical points which is used to construct mandatory join and split trees. These trees enable a visual exploration of the common topological structure of all possible realizations of the uncertain data. To allow multi‐scale visualization, we introduce a simplification scheme for mandatory critical point pairs revealing the most dominant features. Our technique is purely combinatorial and handles parametric distribution models and ensemble data. It does not depend on any computational parameter and does not suffer from numerical inaccuracy or global inconsistency. The algorithm exploits ideas of the established join/split tree computation. It is therefore simple to implement, and its complexity is output‐sensitive. We illustrate, evaluate, and verify our method on synthetic and real‐world data.     

On the All-Farthest-Segments problem for a planar set of points

In this note, we outline a very simple algorithm for the following problem: Given a set S of n points p₁,p₂,p₃,…,pn in the plane, we have O(n²) segments implicitly defined on pairs of these n points. For each point pi, find a segment from this set of implicitly defined segments that is farthest from pi. The time complexity of our algorithm is in O(nh+nlogn), where n is the number of input points, and h is the number of vertices on the convex hull of S.     

Online uniformity of integer points on a line

This Letter presents algorithms for computing a uniform sequence of n integer points in a given interval [0,m] where m and n are integers such that m>n>0. The uniformity of a point set is measured by the ratio of the minimum gap over the maximum gap. We prove that we can insert n integral points one by one into the interval [0,m] while keeping the uniformity of the point set at least 1/2. If we require uniformity strictly greater than 1/2, such a sequence does not always exist, but we can prove a tight upper bound on the length of the sequence for given values of n and m.     

On finding a widest empty 1-corner corridor

A 1-corner corridor through a set S of points is an open subset of CH(S) containing no points from S and bounded by a pair of parallel polygonal lines each of which contains two segments. Given a set of n points in the plane, we consider the problem of computing a widest empty 1-corner corridor. We describe an algorithm that solves the problem in O(n⁴logn) time and O(n) space. We also present an approximation algorithm that computes in time a solution with width at least a fraction (1−ε) of the optimal, for any small enough ε>0.     

On the symmetric angle-restricted nearest neighbor problem

We present an O(nlogn) time divide-and-conquer algorithm for solving the symmetric angle-restricted nearest neighbor (SARNN) problem for a set of n points in the plane under any Lp metric, 1?p?∞. This algorithm is asymptotically optimal (within a multiplicative constant) for any constant p?1.     

Feature Preserving Mesh Generation from 3D Point Clouds

We address the problem of generating quality surface triangle meshes from 3D point clouds sampled on piecewise smooth surfaces. Using a feature detection process based on the covariance matrices of Voronoi cells, we first extract from the point cloud a set of sharp features. Our algorithm also runs on the input point cloud a reconstruction process, such as Poisson reconstruction, providing an implicit surface. A feature preserving variant of a Delaunay refinement process is then used to generate a mesh approximating the implicit surface and containing a faithful representation of the extracted sharp edges. Such a mesh provides an enhanced trade‐off between accuracy and mesh complexity. The whole process is robust to noise and made versatile through a small set of parameters which govern the mesh sizing, approximation error and shape of the elements. We demonstrate the effectiveness of our method on a variety of models including laser scanned datasets ranging from indoor to outdoor scenes.     

Competitive On-Line Switching Policies

Abstract. Consider the following problem. A switch connecting n input channels to a single output channel must deliver all incoming messages through this channel. Messages are composed of packets , and in each time slot the switch can deliver a single packet from one of the input queues to the output channel. In order to prevent packet loss, a buffer is maintained for each input channel. The goal of a switching policy is to minimize the maximum buffer size. The setting is on-line; decisions must be made based on the current state without knowledge of future events. This general scenario models multiplexing tasks in various systems such as communication networks, cable modem systems, and traffic control. Traditionally, researchers analyzed the performance of a given policy assuming some distribution on the arrival rates of messages at the input queues, or assuming that the service rate is at least the aggregate of all the input rates. We use competitive analysis, avoiding any prior assumptions on the input. We show O(log n )-competitive switching policies for the problem and demonstrate matching lower bounds.     

Finding the convex hull of a sorted point set in parallel

We present a parallel algorithm for finding the convex hull of a sorted planar point set. Our algorithm runs in O(log n) time using O(n/log n) processors in the CREW PRAM computational model, which is optimal. One of the techniques we use to achieve these optimal bounds is the use of a parallel data structure which we call the hull tree.     

Aspect-ratio Voronoi diagram and its complexity bounds

This Letter first defines an aspect ratio of a triangle by the ratio of the longest side over the minimal height. Given a set of line segments, any point p in the plane is associated with the worst aspect ratio for all the triangles defined by the point and the line segments. When a line segment si gives the worst ratio, we say that p is dominated by si. Now, an aspect-ratio Voronoi diagram for a set of line segments is a partition of the plane by this dominance relation. We first give a formal definition of the Voronoi diagram and give O(n2+ε) upper bound and Ω(n²) lower bound on the complexity, where ε is any small positive number. The Voronoi diagram is interesting in itself, and it also has an application to a problem of finding an optimal point to insert into a simple polygon to have a triangulation that is optimal in the sense of the aspect ratio.     

On Exact Solutions for the Rectilinear Steiner Tree Problem Part I: Theoretical Results

The rectilinear Steiner tree problem asks for a shortest tree connecting given points in the plane with rectilinear distance. The best theoretically analyzed algorithms for this problem are based on dynamic programming and have a running time of O(n ² . . . 2.62 ⁿ ) (Ganley and Cohoon), resp. (Smith). The first algorithm can solve problems of size 27, the second one is highly impractical because of the large constant in the exponent. The best implementations perform poorly even on small problem instances; the best practical results can be reached using a Branch \& Bound approach (Salowe and Warme); this implementation can solve random problems of size 35 within a day, while the dynamic programming approach of Ganley and Cohoon can handle only 27 point examples. In this paper we improve the theoretical worst-case time bound to O(n ² . . . 2.38 ⁿ ) , for random problem instances we prove a running time of α ⁿ with a constant α < 2 . We have implemented our algorithms and can now solve problems of 40 points in a day using a provably good dynamic programming approach, and can solve problems of 55 points with a Branch \& Bound strategy. For exponential-time algorithms, this is an enormous improvement. Received April 2, 1997; revised January 5, 1998.     

Fast linear expected-time algorithms for computing maxima and convex hulls

This paper examines the expected complexity of boundary problems on a set ofN points inK-space. We assume that the points are chosen from a probability distribution in which each component of a point is chosen independently of all other components. We present an algorithm to find the maximal points usingKN + O (N^1–1/K log^1/K N) expected scalar comparisons, for fixedK 2. A lower bound shows that the algorithm is optimal in the leading term. We describe a simple maxima algorithm that is easy to code, and present experimental evidence that it has similar running time. For fixedK 2, an algorithm computes the convex hull of the set in 2KN + O(N^1–1/K log^1/KN) expected scalar comparisons. The history of the algorithms exhibits interesting interactions among consulting, algorithm design, data analysis, and mathematical analysis of algorithms.This work was performed while this author was visiting AT&T Bell Laboratories.