Distance Queries Research Articles

GPU-Accelerated Minimum Distance and Clearance Queries

We present practical algorithms for accelerating distance queries on models made of trimmed NURBS surfaces using programmable Graphics Processing Units (GPUs). We provide a generalized framework for using GPUs as coprocessors in accelerating CAD operations. By supplementing surface data with a surface bounding-box hierarchy on the GPU, we answer distance queries such as finding the closest point on a curved NURBS surface given any point in space and evaluating the clearance between two solid models constructed using multiple NURBS surfaces. We simultaneously output the parameter values corresponding to the solution of these queries along with the model space values. Though our algorithms make use of the programmable fragment processor, the accuracy is based on the model space precision, unlike earlier graphics algorithms that were based only on image space precision. In addition, we provide theoretical bounds for both the computed minimum distance values as well as the location of the closest point. Our algorithms are at least an order of magnitude faster and about two orders of magnitude more accurate than the commercial solid modeling kernel ACIS.

Guest Editor's Introduction: Special Section on the Joint Conference on Geometric Design and Solid and Physical Modeling (GDSPM)

THE 2009 SIAM/ACM Joint Conference on Geometric Design and Solid and Physical Modeling (GDSPM), which was a federation of the 2009 SIAM Conference on Geometric Design and the 2009 ACM Symposium on Solid and Physical Modeling, was held in San Francisco, CA, from 5 October to 8 October 2009. The goal of the conference was to present theoretically well-founded new methods for geometric and physical modeling that have useful practical applications. In response to the call for papers of the conference, 85 papers were submitted on many aspects of geometric and physical modeling, and their application in design, analysis, manufacturing, biomedicine, digital entertainment, and other areas. All papers were assessed by five reviewers, usually a member from the international program committee. A total of 24 papers were selected for plenary presentation and publication as full papers, and an additional 18 papers for poster presentation and publication as short papers. All papers appeared in the proceedings of the conference, published by ACM. From the full papers, we selected the four papers here for extended publication in this special section. The papers were chosen on the basis of their high quality and their suitability for the readership of IEEE Transactions on Visualization and Computer Graphics (TVCG). All the papers were revised and expanded from the version that appeared in the conference proceedings, and all papers went through further review and revision. We are very pleased with the quality of the resulting papers, and we believe they will be of interest to a wide audience. “Model Synthesis: A General Procedural Modeling Algorithm,” by Paul Merrell and Dinesh Manocha, presents a method for automatically generating geometric models that resemble those provided by an input example. The method examines input models to determine a variety of geometric constraints (such as connectivity and spacing between various features) that help describe the given shape. These constraints are then used in a procedural method to generate new models that follow the same constraints. As a result, a wide variety of new models that resemble the input model can be created very easily, which should be beneficial for numerous graphics applications that require extensive geometry creation with limited user input. “GPU-Accelerated Minimum Distance and Clearance Queries,” by Adarsh Krishnamurthy, Sara McMains, and Kirk Haller, presents a number of fundamental algorithms for computing distance queries for NURBS surfaces. Bounded distance measures are computed for axis-aligned bounding boxes rapidly using the GPU. This allows one to quickly compute the nearest point on a NURBS surface, given a point in space, or to compute clearance between two NURBS models. The paper provides a significant performance increase for fundamental geometric queries that form the basis for several operations in CAD and other applications. “Voronoi-Based Curvature and Feature Estimation from Point Clouds,” by Quentin Merigot, Maks Ovsjanikov, and Leonidas Guibas, an extension of the best paper award winner at the conference, presents a method for estimating certain surface properties directly from point cloud data. In particular, normals, principal curvatures, and sharp features can be computed on point clouds. Perhaps more importantly, theoretical guarantees on many of the computations are provided, including robustness to noise in the point sampling. This paper provides a significant advance in making use of point cloud data sets. “Ball-Morph: Definition, Implementation, and Comparative Evaluation,” by Brian Whited and Jaroslaw (Jarek) Rossignac, describes three methods in detail for computing morphs between pairs of certain planar curves. These methods are compared to several standard morphing methods, and it is shown that the proposed morphs consistently provide better performance on particular metrics. This paper offers a significant improvement to 2D morphing, and should be of particular interest in animation applications. We believe that these four papers are strong representatives of the papers presented at the conference, and hope that you will enjoy reading them. We thank all the authors of the papers and the reviewers for their work in ensuring the high quality of these final papers, along with Dr. Tom Ertl and the TVCG staff for their help in seeing this special section published.

GProximity: Hierarchical GPU‐based Operations for Collision and Distance Queries

AbstractWe present novel parallel algorithms for collision detection and separation distance computation for rigid and deformable models that exploit the computational capabilities of many‐core GPUs. Our approach uses thread and data parallelism to perform fast hierarchy construction, updating, and traversal using tight‐fitting bounding volumes such as oriented bounding boxes (OBB) and rectangular swept spheres (RSS). We also describe efficient algorithms to compute a linear bounding volume hierarchy (LBVH) and update them using refitting methods. Moreover, we show that tight‐fitting bounding volume hierarchies offer improved performance on GPU‐like throughput architectures. We use our algorithms to perform discrete and continuous collision detection including self‐collisions, as well as separation distance computation between non‐overlapping models. In practice, our approach (gProximity) can perform these queries in a few milliseconds on a PC with NVIDIA GTX 285 card on models composed of tens or hundreds of thousands of triangles used in cloth simulation, surgical simulation, virtual prototyping and N‐body simulation. Moreover, we observe more than an order of magnitude performance improvement over prior GPU‐based algorithms.

Simple and parallel proximity algorithms for general polygonal models

AbstractWe present simple and fast parallel proximity algorithms for rigid polygonal models. Given two polygon‐soup models in space, if they overlap, our algorithm can find all the intersected primitives between them; otherwise, it reports their Euclidean minimum distance. Our algorithm is performed in a parallel fashion and shows scalable performance in terms of the number of available computing cores. The key ingredient of our algorithm is a simple load‐balancing metric based on the penetration depth (PD) (for collision detection) and approximate Euclidean distance (for Euclidean distance computation) between bounding volumes. To compute the PD between oriented bounding boxes (OBBs), we present a novel algorithm based on the well‐known separating axis theorem (SAT) and also shows that the PD can be trivially obtained as a byproduct of SAT. We have implemented these algorithms on a commodity PC with eight cores and benchmarked their performance on complicated geometric models. In practice, the performance of our algorithm shows up to 5 and 9.7 times improvement for collision and distance queries, respectively, compared to single core computation. Copyright © 2010 John Wiley & Sons, Ltd.

A method for mapping the boundaries of collision-free reachable workspaces

This paper proposes a method for mapping the boundaries of manipulators' collision-free reachable workspaces. Knowledge of manipulators' actual available workspaces is important in applications and design, e.g., the pre-surgery planning for medical robots, the design parameter optimization for manipulators, etc. Algorithms for mapping the boundaries of reachable workspaces have been well developed in previous works, but we have not seen any method that takes into account the obstacles' effects on the manipulators' available workspaces. In this paper, the concept of collision-free reachable workspace is delivered. The obstacle and the end-effecter are modeled as intersections of convex implicit surfaces, which in this paper are defined as convex superquadrics. The minimum distance query between each two superquadrics is collapsed to a convex optimization problem. By employing an interior-point method, the collision-free conditions are obtained and integrated with the continuation method, creating an algorithm which enables the boundary mapping of collision-free reachable workspaces. To demonstrate and validate our approach, a planar three-bar manipulator is tested with the proposed algorithm; the boundaries of the manipulator's collision-free reachable workspaces are calculated and compared with the results of the manipulator's reachable workspaces.

Dynamic Wavelet Synopses Management over Sliding Windows in Sensor Networks

Due to the dynamic nature of data streams, a sliding window is used to generate synopses that approximate the most recent data within the retrospective horizon to answer queries or discover patterns. In this paper, we propose a dynamic scheme for wavelet synopses management in sensor networks. We define a data structure sliding dual tree, abbreviated as SDT, to generate dynamic synopses that adapts to the insertions and deletions in the most recent sliding window. By exploiting the properties of Haar wavelet transform, we develop several operations to incrementally maintain SDT over consecutive time windows in a time- and space-efficient manner. These operations directly operate on the transformed time-frequency domain without the need of storing/reconstructing the original data. As shown in our thorough analysis, our SDT-based approach greatly reduces the required resources for synopses generation and maximizes the storage utilization of wavelet synopses in terms of the window length and quality measures. We also show that the approximation error of the dynamic wavelet synopses, i.e., L <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> -norm error, can be incrementally updated. We also derive the bound of the overestimation of the approximation error due to the incremental thresholding scheme. Furthermore, the synopses can be used to answer various kinds of numerical queries such as point and distance queries. In addition, we show that our SDT can adapt to resource allocation to further enhance the overall storage utilization over time. As demonstrated by our experimental results, our proposed framework can outperform current techniques in both real and synthetic data.

Algorithms for Approximate Shortest Path Queries on Weighted Polyhedral Surfaces

We consider the well-known geometric problem of determining shortest paths between pairs of points on a polyhedral surface P, where P consists of triangular faces with positive weights assigned to them. The cost of a path in P is defined to be the weighted sum of Euclidean lengths of the sub-paths within each face of P. We present query algorithms that compute approximate distances and/or approximate shortest paths on P. Our all-pairs query algorithms take as input an approximation parameter e∈(0,1) and a query time parameter $\mathfrak{q}$, in a certain range, and build a data structure $\mathrm{APQ}(P,\varepsilon;\mathfrak{q})$, which is then used for answering e-approximate distance queries in $O(\mathfrak{q})$time. As a building block of the $\mathrm{APQ}(P,\varepsilon;\mathfrak{q})$data structure, we develop a single-source query data structure SSQ(a;P,e) that can answer e-approximate distance queries from a fixed point a to any query point on P in logarithmic time. Our algorithms answer shortest path queries in weighted surfaces, which is an important extension, both theoretically and practically, to the extensively studied Euclidean distance case. In addition, our algorithms improve upon previously known query algorithms for shortest paths on surfaces. The algorithms are based on a novel graph separator algorithm introduced and analyzed here, which extends and generalizes previously known separator algorithms.

On Bounded Leg Shortest Paths Problems

Let V be a set of points in a d-dimensional l p-metric space. Let s,t∈V and let L be a real number. An L-bounded leg path from s to t is an ordered set of points which connects s to t such that the leg between any two consecutive points in the set has length of at most L. The minimal path among all these paths is the L-bounded leg shortest path from s to t. In the s–t Bounded Leg Shortest Path (stBLSP) problem we are given two points s and t and a real number L, and are required to compute an L-bounded leg shortest path from s to t. In the All-Pairs Bounded Leg Shortest Path (apBLSP) problem we are required to build a data structure that, given any two query points from V and a real number L, outputs the length of the L-bounded leg shortest path (a distance query) or the path itself (a path query). In this paper we obtain the following results: An algorithm for the apBLSP problem in any l p-metric which, for any fixed e>0, computes in O(n3(log 3n+log 2n⋅e−d)) time a data structure which approximates any bounded leg shortest path within a multiplicative error of (1+e). It requires O(n2log n) space and distance queries are answered in O(log log n) time. An algorithm for the stBLSP problem that, given s,t∈V and a real number L, computes in O(n⋅polylog(n)) the exact L-bounded shortest path from s to t. This algorithm works in l1 and l∞ metrics. In the Euclidean metric we also obtain an exact algorithm but with a running time of O(n4/3+e), for any e>0. For any weighted directed graph we give a data structure of size O(n2.5log n) which is capable of answering path queries with a multiplicative error of (1+e) in O(log log n+l) time, where l is the length of the reported path. Our results improve upon the results given by Bose et al. (Comput. Geom. Theory Appl. 29:233–249, 2004). Our algorithms incorporate several new ideas along with an interesting observation made on geometric spanners, which is of independent interest.

Localized and compact data-structure for comparability graphs

We show that every comparability graph of any two-dimensional poset over n elements (a.k.a. permutation graph) can be preprocessed in O ( n ) time, if two linear extensions of the poset are given, to produce an O ( n ) space data-structure supporting distance queries in constant time. The data-structure is localized and given as a distance labeling, that is each vertex receives a label of O ( log n ) bits so that distance queries between any two vertices are answered by inspecting their labels only. This result improves the previous scheme due to Katz, Katz and Peleg [M. Katz, N.A. Katz, D. Peleg, Distance labeling schemes for well-separated graph classes, Discrete Applied Mathematics 145 (2005) 384–402] by a log n factor. As a byproduct, our data-structure supports all-pair shortest-path queries in O ( d ) time for distance- d pairs, and so identifies in constant time the first edge along a shortest path between any source and destination. More fundamentally, we show that this optimal space and time data-structure cannot be extended for higher dimension posets. More precisely, we prove that for comparability graphs of three-dimensional posets, every distance labeling scheme requires Ω ( n 1 / 3 ) bit labels.

Approximating average parameters of graphs

AbstractInspired by Feige (36th STOC, 2004), we initiate a study of sublinear randomized algorithms for approximating average parameters of a graph. Specifically, we consider the average degree of a graph and the average distance between pairs of vertices in a graph. Since our focus is on sublinear algorithms, these algorithms access the input graph via queries to an adequate oracle.We consider two types of queries. The first type is standard neighborhood queries (i.e., what is the ith neighbor of vertex v?), whereas the second type are queries regarding the quantities that we need to find the average of (i.e., what is the degree of vertex v? and what is the distance between u and v?, respectively).Loosely speaking, our results indicate a difference between the two problems: For approximating the average degree, the standard neighbor queries suffice and in fact are preferable to degree queries. In contrast, for approximating average distances, the standard neighbor queries are of little help whereas distance queries are crucial. © 2008 Wiley Periodicals, Inc. Random Struct. Alg., 2008

Optimal Distance Labeling for Interval Graphs and Related Graph Families

A distance labeling scheme is a distributed graph representation that assigns labels to the vertices and enables answering distance queries between any pair $(x,y)$ of vertices by using only the labels of $x$ and $y$. This paper presents an optimal distance labeling scheme with labels of $\mathcal{O}(\log n)$ bits for the $n$-vertex interval graphs family. It improves by $\log n$ factor the best known upper bound of [M. Katz, N. A. Katz, and D. Peleg, Distance labeling schemes for well-separated graph classes, in Proceedings of the 17th Annual Symposium on Theoretical Aspects of Computer Science, Lecture Notes in Comput. Sci. 1770, Springer-Verlag, Berlin, 2000, pp. 516-528]. Moreover, the scheme supports constant time distance queries, and if the interval representation of the input graph is given and the intervals are sorted, then the set of labels can be computed in $\mathcal{O}(n)$ time. Our result is tight as we show that the length of any label is at least $3\log n-\mathcal{O}(\log\log n)$ bits. This lower bound derives from a new estimator of the number of unlabeled $n$-vertex interval graphs, that is, $2^{\Omega(n \log n)}$. To our knowledge, interval graphs are thereby the first known nontrivial hereditary family with $2^{\Omega(n f(n))}$ unlabeled elements and with a distance labeling scheme with $f(n)$ bit labels.

Approximate distance oracles for unweighted graphs in expected O ( n 2 ) time

Let G = ( V , E ) be an undirected graph on n vertices, and let δ( u , v ) denote the distance in G between two vertices u and v . Thorup and Zwick showed that for any positive integer t , the graph G can be preprocessed to build a data structure that can efficiently report t -approximate distance between any pair of vertices. That is, for any u , v ∈ V , the distance reported is at least δ( u , v ) and at most t δ( u , v ). The remarkable feature of this data structure is that, for t ≥3, it occupies subquadratic space, that is, it does not store all-pairs distances explicitly, and still it can answer any t -approximate distance query in constant time. They named the data structure “approximate distance oracle” because of this feature. Furthermore, the trade-off between the stretch t and the size of the data structure is essentially optimal.In this article, we show that we can actually construct approximate distance oracles in expected O ( n 2 ) time if the graph is unweighted. One of the new ideas used in the improved algorithm also leads to the first expected linear-time algorithm for computing an optimal size (2, 1)-spanner of an unweighted graph. A (2, 1) spanner of an undirected unweighted graph G = ( V , E ) is a subgraph ( V , Ê), Ê ⊆ E , such that for any two vertices u and v in the graph, their distance in the subgraph is at most 2δ( u , v ) + 1.

Formula omitted] query time algorithm for all pairs shortest distances on permutation graphs

We present an algorithm for the all pairs shortest distance problem on permutation graphs. Given a permutation model for the graph on n vertices, after O ( n ) preprocessing the algorithm will deliver answers to distance queries in O ( 1 ) time. In the EREW PRAM model, preprocessing can be accomplished in O ( log n ) time with O ( n ) work. Where the distance between query vertices is k, a path can be delivered in O ( k ) time. The method is based on reduction to bipartite permutation graphs, a further reduction to unit interval graphs, and a coordinatization of unit interval graphs.

Fast proximity computation among deformable models using discrete Voronoi diagrams

We present novel algorithms to perform collision and distance queries among multiple deformable models in dynamic environments. These include inter-object queries between different objects as well as intra-object queries. We describe a unified approach to compute these queries based on N-body distance computation and use properties of the 2 nd order discrete Voronoi diagram to perform N-body culling. Our algorithms involve no preprocessing and also work well on models with changing topologies. We can perform all proximity queries among complex deformable models consisting of thousands of triangles in a fraction of a second on a high-end PC. Moreover, our Voronoi-based culling algorithm can improve the performance of separation distance and penetration queries by an order of magnitude.

Addressing, distances and routing in triangular systems with applications in cellular networks

Triangular systems are the subgraphs of the regular triangular grid which are formed by a simple circuit of the grid and the region bounded by this circuit. They are used to model cellular networks where nodes are base stations. In this paper, we propose an addressing scheme for triangular systems by employing their isometric embeddings into the Cartesian product of three trees. This embedding provides a simple representation of any triangular system with only three small integers per vertex, and allows to employ the compact labeling schemes for trees for distance queries and routing. We show that each such system with n vertices admits a labeling that assigns O(log2 n) bit labels to vertices of the system such that the distance between any two vertices u and v can be determined in constant time by merely inspecting the labels of u and v, without using any other information about the system. Furthermore, there is a labeling, assigning labels of size O(log n) bits to vertices, which allows, given the label of a source vertex and the label of a destination, to compute in constant time the port number of the edge from the source that heads in the direction of the destination. These results are used in solving some problems in cellular networks. Our addressing and distance labeling schemes allow efficient implementation of distance and movement based tracking protocols in cellular networks, by providing information, generally not available to the user, and means for accurate cell distance determination. Our routing and distance labeling schemes provide elegant and efficient routing and connection rerouting protocols for cellular networks.

Randomized vs. deterministic distance query strategies for point location on the line

Suppose that n points are located at n mutually distinct but unknown positions on the line, and we can measure their pairwise distances. How many measurements are needed to determine their relative positions uniquely? The problem is motivated by DNA mapping techniques based on pairwise distance measures. It is also interesting by itself for its own and surprisingly deep. Continuing our earlier work on this problem, we give a simple randomized two-round strategy that needs, with high probability, only ( 1 + o ( 1 ) ) n measurements. We show that deterministic strategies cannot manage the task in two rounds with ( 1 + o ( 1 ) ) n measurements in the worst case. We improve an earlier deterministic bound to roughly 4 n / 3 measurements.

Distance Labeling for Permutation Graphs

Abstract We show that every permutation graph with n elements can be preprocessed in O ( n ) time, if two linear extensions of the corresponding poset are given, to produce an O ( n ) space data-structure supporting distance queries in constant time. The data-structure is localized and given as a distance labeling, that is each vertex receives a label of O (log n ) bits so that distance queries between any two vertices are answered by inspecting on their labels only. This result improves the previous scheme due to Katz, Katz and Peleg [M. Katz , N.A. Katz , and D. Peleg , Distance labeling schemes for well-separated graph classes , in 17 th Annual Symposium on Theoretical Aspects of Computer Science (STACS), H. Reichel and S. Tison, eds., vol. 1770 of Lecture Notes in Computer Science, Springer Verlag, Feb. 2000, pp. 516–528] in the STACS '00 by a log n factor.

A simple GPU-based approach for 3D Voronoi diagram construction and visualization

In this paper we propose a simple GPU-based approach for discrete incremental approximation of 3D Voronoi diagram. By constructing region maps via GPU. Nearest sites, space clustering, and shortest distance query can be quickly answered by looking up the region map. In addition, we propose another representation of the 3D Voronoi diagram for visualization.

A new approach to dynamic all pairs shortest paths

We study novel combinatorial properties of graphs that allow us to devise a completely new approach to dynamic all pairs shortest paths problems. Our approach yields a fully dynamic algorithm for general directed graphs with non-negative real-valued edge weights that supports any sequence of operations in O ( n 2 log 3 n ) amortized time per update and unit worst-case time per distance query, where n is the number of vertices. We can also report shortest paths in optimal worst-case time. These bounds improve substantially over previous results and solve a long-standing open problem. Our algorithm is deterministic, uses simple data structures, and appears to be very fast in practice.

Minimum distance queries for time series data

In this paper, we propose an indexing scheme for time sequences which supports the minimum distance of arbitrary L p norms as a similarity measurement. In many applications where the shape of the time sequence is a major consideration, the minimum distance is a more suitable similarity measurement than the simple L p norm. To support minimum distance queries, most of the previous work has the preprocessing step for vertical shifting which normalizes each sequence by its mean. The vertical shifting, however, has the additional overhead to get the mean of a sequence and to subtract it from each element of the sequence. The proposed method can match time series of similar shape without vertical shifting and guarantees no false dismissals. In addition, the proposed method needs only one index structure to support minimum distance queries in any arbitrary L p norm. The experiments are performed on real data (stock price movement) to verify the performance of the proposed method.