Input Vertices Research Articles

Fast segment insertion and incremental construction of constrained Delaunay triangulations

The most commonly implemented method of constructing a constrained Delaunay triangulation (CDT) in the plane is to first construct a Delaunay triangulation, then incrementally insert the input segments one by one. For typical implementations of segment insertion, this method has a Θ(kn2) worst-case running time, where n is the number of input vertices and k is the number of input segments.We give a randomized algorithm for inserting a segment into a CDT in expected time linear in the number of edges the segment crosses. We demonstrate with a performance comparison that for segments that cross many edges, our algorithm is faster than gift-wrapping. We also show that a simple algorithm for segment location, which precedes segment insertion, is fast enough never to be a bottleneck in CDT construction. A result of Agarwal, Arge, and Yi implies that randomized incremental construction of CDTs by our segment insertion algorithm takes expected O(nlog⁡n+nlog2⁡k) time. We show that this bound is tight by deriving a matching lower bound. Although there are CDT construction algorithms guaranteed to run in O(nlog⁡n) time, incremental CDT construction is easier to program and competitive in practice.Lastly, we partly extend the analysis (albeit not the linear-time insertion algorithm) to randomized incremental CDT construction in three dimensions.

Feasibility Study on the Implementation of Cubic Motion Curve for Vehicle Trajectory Planning

Cubic Motion curve has been introduced to integrate the information of path and motion. Since this is a new approach, a feasibility study has been carried out to analyse the practicality of Cubic Motion curves used as the input for the study of vehicle dynamic response. This study requires the development of three modules, namely, the Cubic Motion module, the vehicle dynamic model, the driver model (VDM-DM) module, and, lastly, the integrated module. The Cubic Motion module will generate Cubic Motion curve. The VDM-DM module applies the 2 DOF bicycle model and links it with Reński's driver model. Finally, an integrated module will merge both previous modules as a complete system for trajectory and vehicle response as the output. The double lane change and the Slalom test were the case studies used to develop an understanding of the applicability of the three types of Cubic Motion input, namely, natural, adjusted tangent, and additional vertex cubic motion. The finding of this study is that Cubic Motion is in fact a compact representation of curve with motion attributes and can be directly associated with a vehicle dynamic response study.

Supercriticality of an annealed approximation of Boolean networks

We consider a model recently proposed by Chatterjee and Durrett as an "annealed approximation'' of boolean networks, which are a class of cellular automata on a random graph, as defined by S. Kauffman. The starting point is a random directed graph on $n$ vertices; each vertex has $r$ input vertices pointing to it. For the model of Chatterjee and Durrett, a discrete time threshold contact process is then considered on this graph: at each instant, each vertex has probability $q$ of choosing to receive input; if it does, and if at least one of its input vertices were in state 1 at the previous instant, then it is labelled with a 1; in all other cases, it is labelled with a 0. $r$ and $q$ are kept fixed and $n$ is taken to infinity. Improving a result of Chatterjee and Durrett, we show that if $qr > 1$, then the time of persistence of activity of the dynamics is exponential in $n$

Complexity of Computing Convex Subgraphs in Custom Instruction Synthesis

Synthesis of custom instruction processors from high-level application descriptions involves automated evaluation of data-flow subgraphs as custom instruction candidates. A subgraph S of a graph D , is convex if no two vertices of S are connected by a path in D that is not also in S. An algorithm for enumerating all convex subgraphs of a directed acyclic graph (DAG) under input, output, and forbidden vertex constraints was given by Pozzi, Atasu, and Ienne. We show that this algorithm makes no more than O(|V(D)|Nin+Nout+1) recursive calls, where |V(D)| is the number of vertices in D, and Nin and Nout are input and output constraints, respectively. Therefore, when Nin and Nout are constants, the algorithm is of polynomial complexity. Furthermore, a convex subgraph S is a maximal convex subgraph if it is not a proper subgraph of some other convex subgraph, assuming that both are valid under forbidden vertex constraints. The largest maximal convex subgraph is called the maximum convex subgraph. There exist popular algorithms that enumerate maximal convex subgraphs, which all have exponential worst-case time complexity. This work shows that although no polynomial-time maximal convex subgraph enumeration algorithm can exist, the related maximum convex subgraph problem can be solved in polynomial time.

VERTEX ISOPERIMETRIC PARAMETER OF A COMPUTATION GRAPH

Let G = (V,E) be a computation graph, which is a directed graph representing a straight line computation and S ⊂ V. We say a vertex v is an input vertex for S if there is an edge (v, u) such that v ∉ S and u ∈ S. We say a vertex u is an output vertex for S if there is an edge (u, v) such that u ∈ S and v ∉ S. A vertex is called a boundary vertex for a set S if it is either an input vertex or an output vertex for S. We consider the problem of determining the minimum value of boundary size of S over all sets of size M in an infinite directed grid. This problem is related to the vertex isoperimetric parameter of a graph, and is motivated by the need for deriving a lower bound for memory traffic for a computation graph representing a financial application. We first extend the notion of vertex isoperimetric parameter for undirected graphs to computation graphs, and then provide a complete solution for this problem for all M. In particular, we show that a set S of size M = 3k2 + 3k + 1 vertices of an infinite directed grid, the boundary size must be at least 6k + 3, and this is obtained when the vertices in S are arranged in a regular hexagonal shape with side k + 1.

Advice Complexity of the Online Vertex Coloring Problem

We study online algorithms with advice for the problem of coloring graphs which come as input vertex by vertex. We consider the class of all 3-colorable graphs and its sub-classes of chordal and maximal outerplanar graphs, respectively. We show that, in the case of the first two classes, for coloring optimally, essentially log2 3 advice bits per vertex (bpv) are necessary and sufficient. In the case of maximal outerplanar graphs, we show a lower bound of 1.0424 bpv and an upper bound of 1.2932 bpv. Finally, we develop algorithms for 4-coloring in these graph classes. The algorithm for 3-colorable chordal and outerplanar graphs uses 0.9865 bpv, and in case of general 3-colorable graphs, we obtain an algorithm using < 1.1583 bpv.

A local spectral method for graphs: with applications to improving graph partitions and exploring data graphs locally

The second eigenvalue of the Laplacian matrix and its associated eigenvector are fundamental features of an undirected graph, and as such they have found widespread use in scientific computing, machine learning, and data analysis. In many applications, however, graphs that arise have several local regions of interest, and the second eigenvector will typically fail to provide information fine-tuned to each local region. In this paper, we introduce a locally-biased analogue of the second eigenvector, and we demonstrate its usefulness at highlighting local properties of data graphs in a semi-supervised manner. To do so, we first view the second eigenvector as the solution to a constrained optimization problem, and we incorporate the local information as an additional constraint; we then characterize the optimal solution to this new problem and show that it can be interpreted as a generalization of a Personalized PageRank vector; and finally, as a consequence, we show that the solution can be computed in nearly-linear time. In addition, we show that this locally-biased vector can be used to compute an approximation to the best partition near an input seed set in a manner analogous to the way in which the second eigenvector of the Laplacian can be used to obtain an approximation to the best partition in the entire input graph. Such a primitive is useful for identifying and refining clusters locally, as it allows us to focus on a local region of interest in a semi-supervised manner. Finally, we provide a detailed empirical evaluation of our method by showing how it can applied to finding locally-biased sparse cuts around an input vertex seed set in social and information networks.

DELAUNAY REFINEMENT ALGORITHMS FOR ESTIMATING LOCAL FEATURE SIZE IN 2D AND 3D

We present Delaunay refinement algorithms for estimating local feature size on the input vertices of a 2D piecewise linear complex and on the input vertices and segments of a 3D piecewise linear complex. These algorithms are designed to eliminate the need for a local feature size oracle during quality mesh generation of domains containing acute input angles. In keeping with Ruppert's algorithm, encroachment in these algorithms can be determined through only local information in the current Delaunay triangulation. The algorithms are practical to implement and several examples are given.

Cover geometry design using multiple convex hulls

We present a design method to create close-fitting customized covers for given three-dimensional (3D) objects such as cameras, toys and figurines. The system first computes clustering of the input vertices using multiple convex hulls, then generates multiple convex hulls using the results. It then outputs a cover geometry to set union operation of these hulls, and the resulting intersection curves are set as seam lines. However, as some of the regions created are not necessarily suitable for flattening, the user can design seam lines by drawing and erasing. The system flattens the patches of the target cover geometry after segmentation, allowing the user to obtain a corresponding 2D pattern and sew the shapes in actual fabric. This paper’s contribution lies in its proposal of a clustering method to generate multiple convex hulls, i.e., a set of convex hulls that individually cover part of the input mesh and together cover all of it. The method is based on vertex clustering to allow handling of mesh models with poor vertex connectivity such as those obtained by 3D scanning, and accommodates conventional meshes with multiple connected components and point-based models with no connectivity information. Use of the system to design actual covers confirmed that it functions as intended.

Multi‐Scale Geometry Interpolation

AbstractInterpolating vertex positions among triangle meshes with identical vertex‐edge graphs is a fundamental part of many geometric modelling systems. Linear vertex interpolation is robust but fails to preserve local shape. Most recent approaches identify local affine transformations for parts of the mesh, model desired interpolations of the affine transformations, and then optimize vertex positions to conform with the desired transformations. However, the local interpolation of the rotational part is non‐trivial for more than two input configurations and ambiguous if the meshes are deformed significantly. We propose a solution to the vertex interpolation problem that starts from interpolating the local metric (edge lengths) and mean curvature (dihedral angles) and makes consistent choices of local affine transformations using shape matching applied to successively larger parts of the mesh. The local interpolation can be applied to any number of input vertex configurations and due to the hierarchical scheme for generating consolidated vertex positions, the approach is fast and can be applied to very large meshes.

Fast 3D Mesh Compression Using Shared Vertex Analysis

A trend in 3D mesh compression is codec design with low computational complexity which preserves the input vertex and face order. However, this added information increases the complexity. We present a fast 3D mesh compression method that compresses the redundant shared vertex information between neighboring faces using simple first-order differential coding followed by fast entropy coding with a fixed length prefix. Our algorithm is feasible for low complexity designs and maintains the order, which is now part of the MPEG-4 scalable complexity 3D mesh compression standard. The proposed algorithm is 30 times faster than MPEG-4 3D mesh coding extension.

SLIVER REMOVAL BY LATTICE REFINEMENT

I present an algorithm that can create a mesh that is free of slivers away from the boundary, or that can eliminate such slivers from a pre-existing mesh by refining it. The resulting tetrahedra have dihedral angles between 30 and 135 degrees and radius-edge ratios of at most 1.368, except near the boundary. In comparison, previous bounds on dihedral angles were microscopic. The final mesh can respect specified input vertices and a user-defined sizing function. The algorithm comes with a bound on the sizes of the features it creates, and can provably grade from small to large tetrahedra.

An extremal result for geometries in the one-way measurement model

We present an extremal result for the class of graphs $G$ which (together with some specified sets of input and output vertices, $I$ and $O$) have a certain ``flow'' property introduced by Danos and Kashefi for the one-way measurement model of quantum computation. The existence of a flow for a triple $(G,I,O)$ allows a unitary embedding to be derived from any choice of measurement bases allowed in the one-way measurement model. We prove an upper bound on the number of edges that a graph $G$ may have, in order for a triple $(G,I,O)$ to have a flow for some $I, O \subseteq V(G)$, in terms of the number of vertices in $G$ and $O$. This implies that finding a flow for a triple $(G,I,O)$ when $\lvert I \rvert = \lvert O \rvert = k$ (corresponding to unitary transformations in the measurement model) and $\lvert V(G) \rvert = n$ can be performed in time $O(k^2 n)$, improving the earlier known bound of $O(km)$ given in \cite{B06a}, where $m = \lvert E(G) \rvert$.

Visualizing Whole-Brain DTI Tractography with GPU-based Tuboids and LoD Management

Diffusion Tensor Imaging (DTI) of the human brain, coupled with tractography techniques, enable the extraction of large-collections of three-dimensional tract pathways per subject. These pathways and pathway bundles represent the connectivity between different brain regions and are critical for the understanding of brain related diseases. A flexible and efficient GPU-based rendering technique for DTI tractography data is presented that addresses common performance bottlenecks and image-quality issues, allowing interactive render rates to be achieved on commodity hardware. An occlusion query-based pathway LoD management system for streamlines/streamtubes/tuboids is introduced that optimizes input geometry, vertex processing, and fragment processing loads, and helps reduce overdraw. The tuboid, a fully-shaded streamtube impostor constructed entirely on the GPU from streamline vertices, is also introduced. Unlike full streamtubes and other impostor constructs, tuboids require little to no preprocessing or extra space over the original streamline data. The supported fragment processing levels of detail range from texture-based draft shading to full raycast normal computation, Phong shading, environment mapping, and curvature-correct text labeling. The presented text labeling technique for tuboids provides adaptive, aesthetically pleasing labels that appear attached to the surface of the tubes. Furthermore, an occlusion query aggregating and scheduling scheme for tuboids is described that reduces the query overhead. Results for a tractography dataset are presented, and demonstrate that LoD-managed tuboids offer benefits over traditional streamtubes both in performance and appearance.

Total energies from variational functionals of the Green function and the renormalized four-point vertex

We derive variational expressions for the grand potential or action in terms of the many-body Green function $G$ which describes the propagation of particles and the renormalized four-point vertex $\ensuremath{\Gamma}$ which describes the scattering of two particles in many-body systems. The main ingredient of the variational functionals is a term we denote as the $\ensuremath{\Xi}$-functional which plays a role analogously to the usual $\ensuremath{\Phi}$-functional studied by Baym [G. Baym, Phys. Rev. 127, 1391 (1962)] in connection with the conservation laws in many-body systems. We show that any $\ensuremath{\Xi}$-derivable theory is also $\ensuremath{\Phi}$-derivable and therefore respects the conservation laws. We further set up a computational scheme to obtain accurate total energies from our variational functionals without having to solve computationally expensive sets of self-consistent equations. The input of the functional is an approximate Green function $\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{G}$ and an approximate four-point vertex $\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{\ensuremath{\Gamma}}$ obtained at a relatively low computational cost. The variational property of the functional guarantees that the error in the total energy is only of second order in deviations of the input Green function and vertex from the self-consistent ones that make the functional stationary. The functionals that we will consider for practical applications correspond to infinite order summations of ladder and exchange diagrams and are therefore particularly suited for applications to highly correlated systems. Their practical evaluation is discussed in detail.

A Fast Algorithm for Constructing Multi-resolution Representations of Linear Features to Support the Progressive Transmission of Vector Map Data

This paper presents a fast algorithm designed to generate multi-resolution representations of linear features based on bit interleaving and map tries. The proposed algorithm constructs a hierarchical subdivision of vertices of linear features in order to support the progressive transmission of vector map data over the Internet. Its time complexity is O(n), where n is the number of input vertices of the linear features. The performance of this algorithm is compared quantitatively and qualitatively with both the Ramer-Douglas-Peucker algorithm and the simple and fast n-th point algorithm.

Construction of asymmetric connectors of depth two

An ( n , N ) -connector of depth d is an acyclic digraph with n inputs and N outputs in which for any injective mapping of input vertices into output vertices there exist n vertex disjoint paths of length at most d joining each input to its corresponding output. In this paper we consider the problem of construction of sparse depth two connectors with n ≪ N . We use posets of star products and their matching properties to construct such connectors. In particular, this gives a simple explicit construction for connectors of size O ( N log n / log log n ) . Thus our earlier idea to use other posets than the family of subsets of a finite set was successful.

Sparse asymmetric connectors in communication networks

Abstract An (n, N, d)–connector is an acyclic digraph with n inputs and N outputs in which for any injective mapping of input vertices into output vertices there exist n vertex disjoint paths of length d joining each input to its corresponding output. We consider the problem of construction of sparse (n, N, 2)–connectors (depth 2 connectors) when n ≪ N . The probabilistic argument in [A. Baltz, G. Jager, A. Srivastav, Elementary constructions of sparse asymmetric connectors, in Proceedings of the 23rd Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS), Mumbai, India, LNCS 2914, 13-22, December 2003] shows the existence of (n, N, 2)–connectors of size (number of edges) O(N) if n ⩽ N 1 / 2 − ɛ , ɛ > 0 . However, the known explicit constructions with n ⩽ N in [F.K. Hwang, G.W. Richards, “A two–stage rearrangeable broadcast switching network”, IEEE Trans. on Communications 33 (1985) 1025-1035],[A. Baltz, G. Jager, A. Srivastav, Elementary constructions of sparse asymmetric connectors, in Proceedings of the 23rd Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS), Mumbai, India, LNCS 2914, 13-22, December 2003],[A. Baltz, G. Jager, A. Srivastav, Constructions of sparse asymmetric connectors with Number Theoretic methods, submitted to Networks] are of size O ( N n ) . Here we present a simple combinatorial construction for (n, N, 2)–connectors of size O(N log 2 n). We also consider depth 2 fault–tolerant connectors under arc or node failures.

Interpolating and approximating implicit surfaces from polygon soup

This paper describes a method for building interpolating or approximating implicit surfaces from polygonal data. The user can choose to generate a surface that exactly interpolates the polygons, or a surface that approximates the input by smoothing away features smaller than some user-specified size. The implicit functions are represented using a moving least-squares formulation with constraints integrated over the polygons. The paper also presents an improved method for enforcing normal constraints and an iterative procedure for ensuring that the implicit surface tightly encloses the input vertices.

Field-theoretical three-body relativistic equations for the multichannel πN↔ γN↔ ππN↔ γπN reactions

A new kind of the relativistic three-body equations for the coupled πN and γN scattering reactions with the ππN and γπN three particle final states are suggested. These equations are derived in the framework of the standard field-theoretical S-matrix approach in the time-ordered three-dimensional form. Therefore, corresponding relativistic covariant equations are three-dimensional from the beginning and the considered formulation is free of the ambiguities which appear due to a three dimensional reduction of the four dimensional Bethe–Salpeter equations. The solutions of the considered equations satisfy the unitarity condition and they are exactly gauge invariant even after the truncation of the multiparticle ( n>3) intermediate states. Moreover, the form of these three-body equations does not depend on the choice of the model Lagrangian and it is the same for the formulations with and without quark degrees of freedom. The effective potential of the suggested equations is defined by the vertex functions with two on-mass shell particles. It is emphasized that these INPUT vertex functions can be constructed from experimental data. Special attention is given to the construction of the intermediate on shell and off shell Δ resonance states. These intermediate Δ states are obtained after separation of the Δ resonance pole contributions in the intermediate πN Green function. The resulting amplitudes for the Δ⇔ Nπ; Δ⇔ Nγ; Δ ′⇔ Δγ transition have the same structure as the vertex functions for transitions between the on-mass shell particle states with spin 1/2 and 3/2. Therefore it is possible to introduce the real value for the magnetic momenta for the Δ ′⇔ Δγ transition amplitudes in the same way as it is done for the N ′⇔ Nγ vertex function.