Solving Connected Dominating Set Faster than 2<Superscript><Emphasis Type="Italic">n</Emphasis></Superscript>

In the connected dominating set problem we are given an n-node undirected graph, and we are asked to find a minimum cardinality connected subset S of nodes such that each node not in S is adjacent to some node in S. This problem is also equivalent to finding a spanning tree with maximum number of leaves. Despite its relevance in applications, the best known exact algorithm for the problem is the trivial Ω(2 ⁿ ) algorithm that enumerates all the subsets of nodes. This is not the case for the general (unconnected) version of the problem, for which much faster algorithms are available. Such a difference is not surprising, since connectivity is a global property, and non-local problems are typically much harder to solve exactly. In this paper we break the 2 ⁿ barrier, by presenting a simple O(1.9407 ⁿ ) algorithm for the connected dominating set problem. The algorithm makes use of new domination rules, and its analysis is based on the Measure and Conquer technique. An extended abstract of this paper appeared in the proceedings of FSTTCS’06. Fedor V. Fomin was additionally supported by the Research Council of Norway.     

Perspective silhouette of a general swept volume

We present an efficient and robust algorithm for computing the perspective silhouette of the boundary of a general swept volume. We also construct the topology of connected components of the silhouette. At each instant t, a three-dimensional object moving along a trajectory touches the envelope surface of its swept volume along a characteristic curve K^t. The same instance of the moving object has a silhouette curve L^t on its own boundary. The intersection K^t∩L^t contributes to the silhouette of the general swept volume. We reformulate this problem as a system of two polynomial equations in three variables. The connected components of the resulting silhouette curves are constructed by detecting the instances where the two curves K^t and L^t intersect each other tangentially on the surface of the moving object. We also consider a general case where the eye position changes while moving along a predefined path. The problem is reformulated as a system of two polynomial equations in four variables, where the zero-set is a two-manifold. By analyzing the topology of the zero-set, we achieve an efficient algorithm for generating a continuous animation of perspective silhouettes of a general swept volume.     

Wang tilings and distributed verification on anonymous torus networks

Considern ² processors arranged in ann × n torus network in which each processor is connected by direct communication channels with its four neighbours. This paper studies the followingverification problem on anonymousn × n torus networks: verify whether the network is oriented; that is, verify whether there is an agreement, among all processors, on a consistent channel labelling. The problem is to be solved by a distributed algorithm executed by the processors themselves. If processors can label their channels arbitrarily, then there are network labellings that are not oriented but, to the processors, are indistinguishable from ones that are oriented. Hence there is no deterministic distributed verification algorithm. However, a verification algorithm does exist if the initial labellings are suitably restricted. We describe the restrictions placed on the initial labellings by subsets of the permutation groupS ₄. We show that the existence of an algorithm for verification is equivalent to the existence of certain tilings of the torus with Wang tiles. Using this equivalence, we have determined the existence of a distributed algorithm for the verification problem for alln × n torus networks for an important class of restrictions, the subgroups ofS ₄.     

Complete subdivision algorithms, II: Isotopic meshing of singular algebraic curves

Given a real valued function f(X,Y), a box region B₀⊆R² and ε>0, we want to compute an ε-isotopic polygonal approximation to the restriction of the curve S=f⁻¹(0)={p∈R²:f(p)=0} to B₀. We focus on subdivision algorithms because of their adaptive complexity and ease of implementation. Plantinga & Vegter gave a numerical subdivision algorithm that is exact when the curve S is bounded and non-singular. They used a computational model that relied only on function evaluation and interval arithmetic. We generalize their algorithm to any bounded (but possibly non-simply connected) region that does not contain singularities of S. With this generalization as a subroutine, we provide a method to detect isolated algebraic singularities and their branching degree. This appears to be the first complete purely numerical method to compute isotopic approximations of algebraic curves with isolated singularities.     

The Perspective Silhouette of a Canal Surface

We present an efficient and robust algorithm for parameterizing the perspective silhouette of a canal surface and detecting each connected component of the silhouette. A canal surface is the envelope of a moving sphere with varying radius, defined by the trajectory C(t) of its center and a radius function r(t) . This moving sphere, S(t) , touches the canal surface at a characteristic circle K(t) . We decompose the canal surface into a set of characteristic circles, compute the silhouette points on each characteristic circle, and then parameterize the silhouette curve. The perspective silhouette of the sphere S(t) from a given viewpoint consists of a circle Q(t) ; by identifying the values of t at which K(t) and Q(t) touch, we can find all the connected components of the silhouette curve of the canal surface. ACM CSS: I.3.7 Computer Graphics–Three Dimensional Graphics and Realism     

Enhancing 3D mesh topological skeletons with discrete contour constrictions

This paper describes a unified and fully automatic algorithm for Reeb graph construction and simplification as well as constriction approximation on triangulated surfaces. The key idea of the algorithm is that discrete contours – curves carried by the edges of the mesh and approximating the continuous contours of a mapping function – encode both topological and geometrical shape characteristics. Therefore, a new concise shape representation, enhanced topological skeletons, is proposed, encoding the contours’ topological and geometrical evolution. First, mesh feature points are computed. Then they are used as geodesic origins for the computation of an invariant mapping function that reveals the shape most significant features. Next, for each vertex in the mesh, its discrete contour is computed. As the set of discrete contours recovers the whole surface, each of them can be analyzed, both to detect topological changes and constrictions. Constriction approximations enable Reeb graphs refinement into more visually meaningful skeletons, which we refer to as enhanced topological skeletons. Extensive experiments showed that, without any preprocessing stage, proposed algorithms are fast in practice, affine-invariant and robust to a variety of surface degradations (surface noise, mesh sampling and model pose variations). These properties make enhanced topological skeletons interesting shape abstractions for many computer graphics applications.     

Improved Algorithms for Constructing Fault-Tolerant Spanners

Abstract. 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.     

Surface reconstruction by layer peeling

Given an input point cloud P in ?³, this paper proposes a novel algorithm to identify surface neighbors of each point p∈P respecting the underlying surface S and then to construct a piecewise linear surface for P. The algorithm utilizes the simple k-nearest neighborhood in constructing local surfaces. It makes use of two concepts: a local convexity criterion to extract a set of surface neighbors for each point, and a global projection test to determine an order for the reconstruction. Our algorithm not only produces a topologically correct surface for well-sampled point sets, but also adapts well to handle under-sampled point sets. Furthermore, the computational cost of the algorithm increases almost linearly in the size of the point cloud. It, thus, scales well to deal with large input point sets.     

Parallel breadth-first search algorithms for trees and graphs

Parallel Breadth-First Search (BFS) algorithms for ordered trees and graphs on a shared memory model of a Single Instruction-stream Multiple Data-stream computer are proposed. The parallel BFS algorithm for trees computes the BFS rank of eachnode of an ordered tree consisting of n nodes in time of 0(β log n) when 0(n ^1+1/β) processors are used, β being an integer greater than or equal to 2. The parallel BFS algorithm for graphs produces Breadth-First Spanning Trees (BFSTs) of a directedgraph G having n nodes in time 0(log d.log n) using 0(n ³) processors, where d is the diameter of G If G is a strongly connected graph or a connected undirected graph the BFS algorithm produces n BFSTs, each BFST having a different start node.     

Isotopic meshing of implicit surfaces

Implicit surfaces are given as the zero set of a function F:ℝ³→ℝ. Although several algorithms exist for generating piecewise linear approximations, most of these are based on a user-defined stepsize or bounds to indicate the precision, and therefore cannot guarantee topological correctness. Interval arithmetic provides a mechanism to determine global properties of the implicit function. In this paper we present an algorithm that uses these properties to generate a piecewise linear approximation of implicit curves and surfaces, that is isotopic to the curve or surface itself. The algorithm is simple and fast, and is among the first to guarantee isotopy for implicit surface meshing.     

Efficient Algorithms for Computing a Complete Visibility Region in Three-Dimensional Space

Given a set P of polygons in three-dimensional space, two points p and q are said to be visible from each other with respect to P if the line segment joining them does not intersect any polygon in P . A point p is said to be completely visible from an area source S if p is visible from every point in S . The completely visible region CV(S, P) from S with respect to P is defined as the set of all points in three-dimensional space that are completely visible from S . We present two algorithms for computing CV(S, P) for P with a total of n vertices and a convex polygonal source S with m vertices. Our first result is a divide-and-conquer algorithm which runs in O(m ² n ² α(mn)) time and space, where α(mn) is the inverse of Ackermann's function. We next give an incremental algorithm for computing CV(S,P) in O(m ² n+mn ² α(n)) time and O(mn+n ² ) space. We also prove that CV(S,P) consists of Θ(mn+n ² ) surface elements such as vertices, edges, and faces. Received November 16, 1995; revised November 11, 1996.     

An Approximation Algorithm for the Minimum Co-Path Set Problem

We present an approximation algorithm for the problem of finding a minimum set of edges in a given graph G whose removal from G leaves a graph in which each connected component is a path. It achieves a ratio of \frac 107\frac {10}{7} and runs in O(n ^1.5) time, where n is the number of vertices in the input graph. The previously best approximation algorithm for this problem achieves a ratio of 2 and runs in O(n ²) time.     

On the Topology and Geometry of Spaces of Affine Shapes

We define the space of affine shapes of k points in R ⁿ to be the topological quotient of (R ⁿ ) ^k modulo the natural action of the affine group of R ⁿ . These spaces arise naturally in many image-processing applications, and despite having poor separation properties, have some topological and geometric properties reminiscent of the more familiar Procrustes shape spaces Σ _n ^k in which one identifies configurations related by an orientation-preserving Euclidean similarity transformation. We examine the topology of the connected, non-Hausdorff spaces Sh _n ^k in detail. Each Sh _n ^k is a disjoint union of naturally ordered strata, each of which is homeomorphic in the relative topology to a Grassmannian, and we show how the strata are attached to each other. The top stratum carries a natural Riemannian metric, which we compute explicitly for k>n, expressing the metric purely in terms of “pre-shape” data, i.e. configurations of k points in R ⁿ .     

Two-Way and Multiway Partitioning of a Set of Intervals for Clique-Width Maximization

For a set S of intervals, the clique-interva I _S is defined as the interval obtained from the intersection of all the intervals in S , and the clique-width quantity w _S is defined as the length of I _S . Given a set S of intervals, it is straightforward to compute its clique-interval and clique-width. In this paper we study the problem of partitioning a set of intervals in order to maximize the sum of the clique-widths of the partitions. We present an O(n log n) time algorithm for the balanced bipartitioning problem, and an O(k n ² ) time algorithm for the k -way unbalanced partitioning problem. Received May 27, 1997; revised October 30, 1997.     

A vertex-first parametric algorithm for polyhedron blending

This paper enhances the conventional parametric algorithms for polyhedron blending, by strategically inverting the edges-first approach to vertex-first, so that matching the vertex blending surface (using a triangular or tensor product Bézier surface, or an S-patch) with the edge blending surfaces (generated by Hartmann method) becomes essentially easier. Based on a study of cross boundary derivatives (those of S-patches are deduced herein), G^g-continuity between all the above surfaces and the primary planar faces is achieved by a novel trick as a first step: assigning the vertex, some edge points and some face points to be the proper control points. This still leaves enough free parameters usable for changing the blending configuration. The new algorithm is illustrated with two practical examples involving miscellaneous vertices up to 6-edge convex–concave.     

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.     

Basic Topological Models for Spatial Entities in 3-Dimensional Space

In recent years, models of spatial relations, especially topological relations, have attracted much attention from the GIS community. In this paper, some basic topologic models for spatial entities in both vector and raster spaces are discussed.It has been suggested that, in vector space, an open set in 1-D space may not be an open set any more in 2-D and 3-D spaces. Similarly, an open set in 2-D vector space may also not be an open set any more in 3-D vector spaces. As a result, fundamental topological concepts such as boundary and interior are not valid any more when a lower dimensional spatial entity is embedded in higher dimensional space. For example, in 2-D, a line has no interior and the line itself (not its two end-points) forms a boundary. Failure to recognize this fundamental topological property will lead to topological paradox. It has also been stated that the topological models for raster entities are different in Z ² and R ². There are different types of possible boundaries depending on the definition of adjacency or connectedness. If connectedness is not carefully defined, topological paradox may also occur. In raster space, the basic topological concept in vector space—connectedness—is implicitly inherited. This is why the topological properties of spatial entities can also be studied in raster space. Study of entities in raster (discrete) space could be a more efficient method than in vector space, as the expression of spatial entities in discrete space is more explicit than that in connected space.     

Approximate conversion of a rational boundary gregory patch to a nonuniform B-spline surface

A rational boundary Gregory patch is characterized by the facts that anyn-sided loop can be smoothly interpolated and that it can be smoothly connected to an adjacent patch. Thus, it is well-suited to interpolate complicated wire frames in shape modeling. Although a rational boundary Gregory patch can be exactly converted to a rational Bézier patch to enable the exchange of data, problems of high degree and singularity tend to arise as a result of conversion. This paper presents an algorithm that can approximately convert a rational boundary Gregory patch to a bicubic nonuniform B-spline surface. The approximating surface hasC ¹ continuity between its inner patches.     

Smooth multiple B-spline surface fitting with Catmull%ndash;Clark subdivision surfaces for extraordinary corner patches

This paper presents an algorithm for simultaneously fitting smoothly connected multiple surfaces from unorganized measured data. A hybrid mathematical model of B-spline surfaces and Catmull–Clark subdivision surfaces is introduced to represent objects with general quadrilateral topology. The interconnected multiple surfaces are G ² continuous across all surface boundaries except at a finite number of extraordinary corner points where G ¹ continuity is obtained. The algorithm is purely a linear least-squares fitting procedure without any constraint for maintaining the required geometric continuity. In case of general uniform knots for all surfaces, the final fitted multiple surfaces can also be exported as a set of Catmull–Clark subdivision surfaces with global C ² continuity and local C ¹ continuity at extraordinary corner points. Published online: 14 May 2002 Correspondence to: W. Ma     

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).