Improved Algorithms for Constructing Fault-Tolerant Spanners

Let S be a set of n points in a metric space, and let k be a positive integer. Algorithms are given that construct k -fault-tolerant spanners for S . If in such a spanner at most k vertices and/ or edges are removed, then each pair of points in the remaining graph is still connected by a ``short'' path. First, an algorithm is given that transforms an arbitrary spanner into a k -fault-tolerant spanner. For the Euclidean metric in R ^d , this leads to an O(n log n + c ^k n) -time algorithm that constructs a k -fault-tolerant spanner of degree O(c ^k ) , whose total edge length is O(c ^k ) times the weight of a minimum spanning tree of S , for some constant c . For constant values of k , this result is optimal. In the second part of the paper, algorithms are presented for the Euclidean metric in R ^d . These algorithms construct (i) in O(n log n + k ² n) time, a k -fault-tolerant spanner with O(k ² n) edges, and (ii) in O(k n log n) time, such a spanner with O(k n log n) edges.     

Constructing Plane Spanners of Bounded Degree and Low Weight

Given a set S of n points in the plane, we give an O(n log n)-time algorithm that constructs a plane t-spanner for S, with t ≈ 10, such that the degree of each point of S is bounded from above by 27, and the total edge length is proportional to the weight of a minimum spanning tree of S. Previously, no algorithms were known for constructing plane t-spanners of bounded degree.     

Realistic Input Models for Geometric Algorithms

Abstract. The traditional worst-case analysis often fails to predict the actual behavior of the running time of geometric algorithms in practical situations. One reason is that worst-case scenarios are often very contrived and do not occur in practice. To avoid this, models are needed that describe the properties that realistic inputs have, so that the analysis can take these properties into account. We try to bring some structure to this emerging research direction. In particular, we present the following results: • We show the relations between various models that have been proposed in the literature. • For several of these models, we give algorithms to compute the model parameter(s) for a given (planar) scene; these algorithms can be used to verify whether a model is appropriate for typical scenes in some application area. • As a case study, we give some experimental results on the appropriateness of some of the models for one particular type of scene often encountered in geographic information systems, namely certain triangulated irregular networks.     

Fast Algorithms for Approximating Distances

In this paper we present efficient approximation algorithms for the distance selection problem. Our technique is based on the well-separated pair decomposition proposed in [8]. Received May 16, 1999; revised June 5, 2001.     

Efficient construction of a bounded-degree spanner with low weight

LetS be a set ofn points in ℝ ^d and lett>1 be a real number. At-spanner forS is a graph having the points ofS as its vertices such that for any pairp, q of points there is a path between them of length at mostt times the Euclidean distance betweenp andq. An efficient implementation of a greedy algorithm is given that constructs at-spanner having bounded degree such that the total length of all its edges is bounded byO (logn) times the length of a minimum spanning tree forS. The algorithm has running timeO (n log ^d n). Applying recent results of Das, Narasimhan, and Salowe to thist-spanner gives anO(n log ^d n)-time algorithm for constructing at-spanner having bounded degree and whose total edge length is proportional to the length of a minimum spanning tree forS. Previously, noo(n ²)-time algorithms were known for constructing at-spanner of bounded degree. In the final part of the paper, an application to the problem of distance enumeration is given. This work was supported by the ESPRIT Basic Research Actions Program, under Contract No. 7141 (Project ALCOM II).     

A Tutorial for Designing Flexible Geometric Algorithms

The implementation of an algorithm is faced with the issues of efficiency, flexibility, and ease-of-use. In this paper we suggest a design concept that greatly increases the flexibility of an implementation without sacrificing ease-of-use and efficiency. We demonstrate the advantages of our concept through a C++ implementation of a simple rectangle-intersection algorithm, which follows the well-known sweep-line paradigm. We lead the reader, who should be familiar with C++ and its template mechanism, from a naive interface in a step-by-step guide to an interface offering full flexibility. The gain in flexibility can reduce the overall implementation effort by facilitating code reuse. Reusability in turn helps to achieve correctness simply because more users mean more testing. Though most of the ingredients of our concept have already been suggested elsewhere, to our knowledge this is the first time that they are applied vigorously in a geometric setting. We include a thorough experimental analysis on random and real-world data. Received October 30, 1998; revised June 11, 2001.     

Spanners in sparse graphs

A t-spanner of a graph G is a spanning subgraph S in which the distance between every pair of vertices is at most t times their distance in G. If S is required to be a tree then S is called a tree t-spanner of G. In 1998, Fekete and Kremer showed that on unweighted planar graphs deciding whether G admits a tree t-spanner is polynomial time solvable for t?3 and is NP-complete when t is part of the input. They also left as an open problem if the problem is polynomial time solvable for every fixed t?4. In this work we resolve the open question of Fekete and Kremer by proving much more general results:

• • 

  The problem of finding a t-spanner of treewidth at most k in a given planar graph G is fixed parameter tractable parameterized by k and t. Moreover, for every fixed t and k, the running time of our algorithm is linear.

• • 

  Our technique allows to extend the result from planar graphs to much more general classes of graphs. An apex graph is a graph that can be made planar by the removal of a single vertex. We prove that the problem of finding a t-spanner of treewidth k is fixed parameter tractable on graphs that do not contain some fixed apex graph as a minor, i.e. on apex-minor-free graphs. The class of apex-minor-free graphs contains planar graphs and graphs of bounded genus.

• • 

  Finally, we show that the tractability border of the t-spanner problem cannot be extended beyond the class of apex-minor-free graphs and in this sense our results are tight. In particular, for every t?4, the problem of finding a tree t-spanner is NP-complete on K6-minor-free graphs.

     

Linear-Time Algorithms for Finding the Shadow Volumes from a Convex Area Light Source

An area light source in three-dimensional space shines past a scene polygon to generate two types of shadow volumes for each scene polygon, i.e., one with partial occlusion and the other with complete occlusion. These are called penumbra and umbra, respectively. In this paper we propose linear-time algorithms for computing the penumbra and the umbra of a scene polygon from an area light source, respectively. Received June 27, 1995; revised May 20, 1996.     

Approximation Algorithms for a k-Line Center

Given a set P of n points in ℝd and an integer k ≥ 1, let w^* denote the minimum value so that P can be covered by k congruent cylinders of radius w^*. We describe a randomized algorithm that, given P and an ε > 0, computes k cylinders of radius (1 + ε) w^* that cover P. The expected running time of the algorithm is O(n log n), with the constant of proportionality depending on k, d, and ε. We first show that there exists a small ”certificate” Q ⫅ P, whose size does not depend on n, such that for any k congruent cylinders that cover Q, an expansion of these cylinders by a factor of (1 + ε) covers P. We then use a well-known scheme based on sampling and iterated re-weighting for computing the cylinders.     

Near-Linear Time Approximation Algorithms for Curve Simplification

We consider the problem of approximating a polygonal curve P under a given error criterion by another polygonal curve P’ whose vertices are a subset of the vertices of P. The goal is to minimize the number of vertices of P’ while ensuring that the error between P’ and P is below a certain threshold. We consider two different error measures: Hausdorff and Frechet. For both error criteria, we present near-linear time approximation algorithms that, given a parameter ε > 0, compute a simplified polygonal curve P’ whose error is less than ε and size at most the size of an optimal simplified polygonal curve with error ε/2. We consider monotone curves in ℝ² in the case of the Hausdorff error measure under the uniform distance metric and arbitrary curves in any dimension for the Frechet error measure under L_p metrics. We present experimental results demonstrating that our algorithms are simple and fast, and produce close to optimal simplifications in practice.     

Multiple Translational Containment Part II: Exact Algorithms

We present exact algorithms for finding a solution to the two-dimensional translational containment problem: find translations for k polygons which place them inside a polygonal container without overlapping. The term kCN denotes the version in which the polygons are convex and the container is nonconvex, and the term kNN denotes the version in which the polygons and the container are nonconvex. The notation (r,k)CN, (r,k)NN, and so forth refers to the problem of finding all subsets of size k out of r objects that can be placed in a container. The polygons have up to m vertices, and the container has n vertices, where n is usually much larger than m. We present exact algorithms for the following: 2CN in time, (r,2)CN in time (for ), 3CN in time, kCN in or time, and kNN in time, where LP(a,b) is the time to solve a linear program with a variables and b constraints. All these results are improvements on previously known running times except for the last. The algorithm for kNN is slower asymptotically than the naive algorithm, but is expected to be much faster in practice. The algorithm for 2CN is based on the use of separating line orientations as a means of characterizing the solution. The solution to 3CN also uses a separating line orientation characterization leading to a simple and robust ``carrousel' algorithm. The kCN algorithm uses the idea of disassembling the layout to the left. Finally, the kNN algorithm uses the concept of subdivision trees and linear programming. Received July 11, 1994; revised August 22, 1995, and February 26, 1996.     

Efficient Randomized Algorithms for the Repeated Median Line Estimator

The problem of fitting a straight line to a finite collection of points in the plane is an important problem in statistical estimation. Recently there has been a great deal of interest is robust estimators, because of their lack of sensitivity to outlying data points. The basic measure of the robustness of an estimator is its breakdown point, that is, the fraction (up to 50%) of outlying data points that can corrupt the estimator. One problem with robust estimators is that achieving high breakdown points (near 50%) has proved to be computationally demanding. In this paper we present the best known theoretical algorithm and a practical subquadratic algorithm for computing a 50% breakdown point line estimator, the Siegel or repeated median line estimator. We first present an O(n\log n) randomized expected-time algorithm, where n is the number of given points. This algorithm relies, however, on sophisticated data structures. We also present a very simple O(n log ² n) randomized algorithm for this problem, which uses no complex data structures. We provide empirical evidence that, for many realistic input distributions, the running time of this second algorithm is actually O(n log n) expected time. Received January 25, 1995; revised May 17, 1996. Communciated by L. J. Guibas.     

Two Algorithms for Decomposing a Polyhedron into Convex Parts

Two algorithms are presented for splitting a polyhedron into convex components: one for the case of a simple polyhedron and one for a more general case, when the polyhedron may have ring-shaped faces and cavities. The time requirement in both cases is O ( DN log N ), where D is the number of concave dihedral angles and N is the number of edges. The algorithm for the simple oasis produces at most D + 1 convex pieces which is the minimal number of the convex components.     

构造二叉树的两个改进算法

在数据结构中,已知一棵二叉树的先序序列和中序序列,可唯一确定此二叉树.本文在分析建立二叉树经典算法的时间复杂度的基础上,给出了两个改进算法:①利用哈希函数,使得改进后的算法在最差情况下,时间复杂度由O(n2)降为O(n);②利用栈和控制输入的结点序列构造二叉树,时间复杂度也由O(n2)降为O(n).     

基于FPGA的容错计算机故障注入研究

为了验证以FPGA为主控制器的容错计算机的可靠性,利用构造双NIOSII系统的方法,设计了模拟量、数字量、通讯量以及单粒子效应的故障注入系统。该系统可以通过软件由用户选择故障参数,对容错计算机进行故障输入。它无需改变原容错计算机的硬件设计,对原操作系统也无特殊的要求。     

Circular Separability of Polygons

Two planar sets are circularly separable if there exists a circle enclosing one of the sets and whose open interior disk does not intersect the other set. This paper studies two problems related to circular separability. A linear-time algorithm is proposed to decide if two polygons are circularly separable. The algorithm outputs the smallest separating circle. The second problem asks for the largest circle included in a preprocessed, convex polygon, under some point and/ or line constraints. The resulting circle must contain the query points and it must lie in the halfplanes delimited by the query lines. Received October 25, 1998; revised April 21, 1999.     

Algorithms for computing the center of area of a convex polygon

The center of area of a convex polygonP is the unique pointp ^* that maximizes the minimum area overlap betweenP and any halfplane that includesp ^*. We show thatp ^* is unique and present two algorithms for its computation. The first is a combinatorial algorithm that runs in timeO (n ⁶ log² n). The second is a numerical algorithm that runs in timeO(GK(n+K)) whereK represents the number of desired bits of precision in the output coordinates andG the number of bits used to represent the coordinates of the input polygon vertices. We conclude with a discussion of implementation issues and related results.Research partially supported by the second author's NSF grant CCR-8351468, at Johns Hopkins University and Smith College.     

Improved algorithm for the widest empty 1-corner corridor

Given a set P of n points on a 2D plane, an empty corridor is an open region bounded by two parallel polygonal chains that does not contain any point of P, and partitions the point-set P into two non-empty parts. An empty corridor is said to be a 1-corner empty corridor if each of the two bounding polygonal chains has exactly one corner point. We present an improved algorithm for computing the widest empty 1-corner corridor. It runs in O(n³log²n) time and O(n²) space. This improves the time complexity of the best known algorithm for the same problem by a factor of [J.M. Diaz-Banez, M.A. Lopez, J.A. Sellares, On finding a widest empty 1-corner corridor, Information Processing Letters 98 (2006) 199-205].     

Constructing the visibility graph for n-line segments in O(n2) time

Given a set S of line segments in the plane, its visibility graph G_S is the undirected graph which has the endpoints of the line segments in S as nodes and in which two nodes (points) are adjacent whenever they ‘see’ each other (the line segments in S are regarded as nontransparent obstacles). It is shown that G_S can be constructed in O(n²) time and space for a set S of n nonintersecting line segments. As an immediate implication, the shortest path between two points in the plane avoiding a set of n nonintersecting line segments can be computed in O(n²) time and space