Showing papers presented at "Symposium on Computational Geometry in 2002"

A local search approximation algorithm for k-means clustering

Abstract: In k-means clustering we are given a set of n data points in d-dimensional space ℜd and an integer k, and the problem is to determine a set of k points in ℜd, called centers, to minimize the mean squared distance from each data point to its nearest center. No exact polynomial-time algorithms are known for this problem. Although asymptotically efficient approximation algorithms exist, these algorithms are not practical due to the extremely high constant factors involved. There are many heuristics that are used in practice, but we know of no bounds on their performance.We consider the question of whether there exists a simple and practical approximation algorithm for k-means clustering. We present a local improvement heuristic based on swapping centers in and out. We prove that this yields a (9+e)-approximation algorithm. We show that the approximation factor is almost tight, by giving an example for which the algorithm achieves an approximation factor of (9-e). To establish the practical value of the heuristic, we present an empirical study that shows that, when combined with Lloyd's algorithm, this heuristic performs quite well in practice.

Finite metric spaces: combinatorics, geometry and algorithms

Abstract: In the last several years a number of very interesting results were proved about finite metric spaces. Some of this work is motivated by practical considerations: Large data sets (coming e.g. from computational molecular biology, brain research or data mining) can be viewed as large metric spaces that should be analyzed (e.g. correctly clustered).On the other hand, these investigations connect to some classical areas of geometry - the asymptotic theory of finite-dimensional normed spaces and differential geometry. Finally, the metric theory of finite graphs has proved very useful in the study of graphs per se and the design of approximation algorithms for hard computational problems. In this talk I will try to explain some of the results and review some of the emerging new connections and the many fascinating open problems in this area.

A global approach to automatic solution of jigsaw puzzles

Abstract: We present a new algorithm for automatically solving jigsaw puzzles by shape alone. The algorithm can solve more difficult puzzles than could be solved before, without the use of backtracking or branch-and-bound. The algorithm can handle puzzles in which pieces border more than four neighbors, and puzzles with as many as 200 pieces. Our overall strategy follows that of previous algorithms but applies a number of new ideas, such as robust fiducial points, "highest- confidence-first" search, and frequent global reoptimization of partial solutions.

Conforming Delaunay triangulations in 3D

Abstract: We describe an algorithm which, for any piecewise linear complex (PLC) in 3D, builds a Delaunay triangulation conforming to this PLC.The algorithm has been implemented, and yields in practice a relatively small number of Steiner points due to the fact that it adapts to the local geometry of the PLC. It is, to our knowledge, the first practical algorithm devoted to this problem.

Approximate nearest neighbor algorithms for Frechet distance via product metrics

Abstract: The Nearest Neighbor Search (NNS) problem is: Given a set P of n points in a metric space X, preprocess P so as to eÆciently answer queries for nding the point in P closest to a query point q. NNS and its approximate versions are among the most extensively studied problems in the elds of Computational Geometry and Algorithms, resulting in discovery of many eÆcient algorithms. In particular, for the case when the metric spaceX is a low-dimensional Euclidean space l 2 , it is known how to construct data structure for exact [5, 15] or approximate [4, 13, 12, 9] NNS with query time (d+log n). Unfortunately, those data structures require space exponential in d. More recently, several data structures using space (quasi)-polynomial in n and d, and query time sublinear in n, have been discovered for approximate NNS under l1 and l2 [14, 12, 11] and l1 [10] norms. While many metrics of interest for nearest neighbor search are norms, quite a few of them are not. A prominent example of the latter is the Frechet metric. Unlike the norms, Frechet metric is de ned for sequences of points (possibly in nite and of di erent length), not vectors. More speci cally, the Frechet distance between two sequences of points is de ned as a minimum, over all alignments between the two sequences, of the maximum distance between two corresponding points (see Preliminaries for formal de nition). Frechet metric is a natural measure of similarity between sequences of points (e.g., handwritten signatures). It also has been studied in the Computational Geometry literature, e.g., in [2] (see also [3]). Unfortunately, the optimum alignment between any two sequences is not xed (i.e., it could be

Projective clustering in high dimensions using core-sets

Abstract: (MATH) Let P be a set of n points in $\Red, and for any integer 0 ≤ k ≤ d--1, let $\RDk(P) denote the minimum over all k-flats $\FLAT$ of maxpeP Dist(p,\FLAT). We present an algorithm that computes, for any 0 k-flat that is within a distance of (1 + $egr;) \RDk(P) from each point of P. The running time of the algorithm is dnO(k/e5log(1/e)). The crucial step in obtaining this algorithm is a structural result that says that there is a near-optimal flat that lies in an affine subspace spanned by a small subset of points in P. The size of this "core-set" depends on k and e but is independent of the dimension.This approach also extends to the case where we want to find a k-flat that is close to a prescribed fraction of the entire point set, and to the case where we want to find j flats, each of dimension k, that are close to the point set. No efficient approximation schemes were known for these problems in high-dimensions, when k>1 or j>1.

Collision detection for deforming necklaces

Abstract: In this paper, we propose to study deformable necklaces --- flexible chains of balls, called beads, in which only adjacent balls may intersect. Such objects can be used to model macro-molecules, muscles, rope, and other 'linear' objects in the physical world. In this paper, we exploit this linearity to develop geometric structures associated with necklaces that are useful in physical simulations. We show how these structures can be implemented efficiently and maintained under necklace deformation. In particular, we study a bounding volume hierarchy based on spheres built on a necklace. Such a hierarchy is easy to compute and is suitable for maintenance when the necklace deforms, as our theoretical and experimental results show. This hierarchy can be used for collision and self-collision detection. In particular, we achieve an upper bound of O(nlog n) in two dimensions and O(n 2-2/d) in d-dimensions, d 3, for collision checking. To our knowledge, this is the first sub-quadratic bound proved for a collision detection algorithm using predefined hierarchies. In addition, we show that the power diagram, with the help of some additional mechanisms, can be also used to detect self-collisions of a necklace in certain ways complementary to the sphere hierarchy.

Guaranteed: quality parallel delaunay refinement for restricted polyhedral domains

Abstract: We describe a distributed memory parallel Delaunay refinement algorithm for polyhedral domains which can generate meshes containing tetrahedra with circumradius to shortest edge ratio less than 2, as long as the angle separating any two incident segments and/or facets is between 90° and 270° degrees. Input to our implementation is an element--wise partitioned, conforming Delaunay mesh of a restricted polyhedral domain which has been distributed to the processors of a parallel system. The submeshes of the distributed mesh are then independently refined by concurrently inserting new mesh vertices.Our algorithm allows a new mesh vertex to affect both the submesh tetrahedralizations and the submesh interfaces induced by the partitioning. This flexibility is crucial to ensure mesh quality, but it introduces unpredictable and variable latencies due to long delays in gathering remote data required for updating mesh data structures. In our experiments, more than 80% of this latency was masked with computation due to the fine--grained concurrency of our algorithm.Our experiments also show that the algorithm is efficient in practice, even for certain domains whose boundaries do not conform to the theoretical limits imposed by the algorithm. The algorithm we describe is the first step in the development of much more sophisticated guaranteed--quality parallel mesh generation algorithms.

Hybrid meshes: multiresolution using regular and irregular refinement

Abstract: A hybrid mesh is a multiresolution surface representation that combines advantages from regular and irregular meshes. Irregular operations allow a hybrid mesh to change topology throughout the hierarchy and approximate detailed features at multiple scales. A preponderance of regular refinements allows for efficient data-structures and processing algorithms. We provide a user driven procedure for creating a hybrid mesh from scanned geometry and present a progressive hybrid mesh compression algorithm.

Parametric search made practical

Abstract: In this paper we show that in sorting-based applications of parametric search, Quicksort can replace the parallel sorting algorithms that are usually advocated, and we argue that Cole's optimization of certain parametric-search algorithms may be unnecessary under realistic assumptions about the input. Furthermore, we present a generic, flexible, and easy-to-use framework that greatly simplifies the implementation of algorithms based on parametric search. We use our framework to implement an algorithm that solves the Frechet-distance problem. The implementation based on parametric search is faster than the binary-search approach that is often suggested as a practical replacement for the parametric-search technique.

Optimally cutting a surface into a disk

Abstract: We consider the problem of cutting a set of edges on a polyhedral manifold surface, possibly with boundary, to obtain a single topological disk, minimizing either the total number of cut edges or their total length. We show that this problem is NP-hard, even for manifolds without boundary and for punctured spheres. We also describe an algorithm with running time n O(g+k), where n is the combinatorial complexity, g is the genus, and k is the number of boundary components of the input surface. Finally, we describe a greedy algorithm that outputs a O(log2 g)-approximation of the minimum cut graph in O(g 2 n log n) time.

Efficient maintenance and self-collision testing for Kinematic Chains

Abstract: The kinematic chain is a ubiquitous and extensively studied representation in robotics as well as a useful model for studying the motion of biological macro-molecules. Both fields stand to benefit from algorithms for efficient maintenance and collision detection in such chains. This paper introduces a novel hierarchical representation of a kinematic chain allowing for efficient incremental updates and relative position calculation. A hierarchy of oriented bounding boxes is superimposed on this representation, enabling high performance collision detection, self-collision testing, and distance computation. This representation has immediate applications in the field of molecular biology, for speeding up molecular simulations and studies of folding paths of proteins. It could be instrumental in path planning applications for robots with many degrees of freedom, also known as hyper-redundant robots. A comparison of the performance of the algorithm with the current state of the art in collision detection is presented for a number of benchmarks.

On the number of embeddings of minimally rigid graphs

Abstract: (MATH) Rigid frameworks in some Euclidian space are embedded graphs having a unique local realization (up to Euclidian motions) for the given edge lengths, although globally they may have several. We study first the number of distinct planar embeddings of rigid graphs with n vertices. We show that, modulo planar rigid motions, this number is at most $2n-4\choose n-2 \approx 4n. We also exhibit several families which realize lower bounds of the order of 2n, 2.21n and 2.88n.(MATH) For the upper bound we use techniques from complex algebraic geometry, based on the (projective) Cayley-Menger variety CM 2,n(C)\subset P_n\choose 2-1(C)$ over the complex numbers C. In this context, point configurations are represented by coordinates given by squared distances between all pairs of points. Sectioning the variety with 2n-4 hyperplanes yields at most deg(CM 2,n) zero-dimensional components, and one finds this degree to be D 2,n =\frac122n-4\choose n-2$. The lower bounds are related to inductive constructions of minimally rigid graphs via Henneberg sequences.(MATH) The same approach works in higher dimensions. In particular we show that it leads to an upper bound of 2 D^3,n= \frac2^n-3n-2n-6\choosen-3$ for the number of spatial embeddings with generic edge lengths of the $1$-skeleton of a simplicial polyhedron, up to rigid motions.

Quickest paths, straight skeletons, and the city Voronoi diagram

Abstract: The city Voronoi diagram is induced by quickest paths, in the L 1 plane speeded up by an isothetic transportation network. We investigate the rich geometric and algorithmic properties of city Voronoi diagrams, and report on their use in processing quickest-path queries.In doing so, we revisit the fact that not every Voronoi-type diagram has interpretations in both the distance model and the wavefront model. Especially, straight skeletons are a relevant example where an interpretation in the former model is lacking. We clarify the relation between these models, and further draw a connection to the bisector-defined abstract Voronoi diagram model, with the particular goal of computing the city Voronoi diagram efficiently.

The one-round Voronoi game

Abstract: (MATH) In the one-round Voronoi game, the first player chooses an n-point set $\PFRST$ in a square $Q$, and then the second player places another n-point set $\PSCND$ into $Q$. The payoff for the second player is the fraction of the area of $Q$ occupied by the regions of the points of $\PSCND$ in the Voronoi diagram of $\PFRST\cup\PSCND$. We give a strategy for the second player that always guarantees him a payoff of at least $\frac12+\alpha$, for a constant $\alpha>0$ independent of n. This contrasts with the one-dimensional situation, with $Q=[0,1]$, where the first player can always win more than 1/2.

Testing Homotopy for paths in the plane

Abstract: In this paper we present an efficient algorithm to test if two given paths are homotopic; that is, whether they wind around obstacles in the plane in the same way. For simple paths specified by n line segments with obstacles described by n points, our algorithm runs in O(n log n) time, which we show is tight. For self-intersecting paths the problem is related to Hopcroft's problem.

Kinetic maintenance of context-sensitive hierarchical representations for disjoint simple polygons

Abstract: We describe how to construct and kinetically maintain a tessellation of the free space between a collection of k disjoint simple polygonal objects with a total of N vertices, R of which are reflex. Our linear size tessellation consists of pseudo-triangles and has the following properties: (i) it contains disjoint outer hierarchical representations of all objects where the size of the outer boundary of these representations is proportional to a minimum link separator for the objects, and (ii) any line segment in the free space intersects at most O((k + log R) log N) pseudo-triangles (each of constant size).We maintain our tessellation by using the Kinetic Data Structure (KDS) framework. Our structure is compact, maintaining an active set of certificates whose number is linear in the size of a minimum link subdivision for the objects. It is also responsive; on the failure of a certificate invariants can be restored in time logarithmic in the total number of vertices. While its efficiency is difficult to establish precisely, it is shown that at most O(k + κmaxlog R)log N events happen during straight line motion of one object A in the context of k (fixed) others, where κmax denotes the maximum size of the minimum link polygon separating object A from the rest, during the motion.Furthermore, ray shooting queries (that use point location) can be answered in O((k + log R) log N) time for rays with arbitrary direction.

On the crossing number of complete graphs

Abstract: (MATH) Let $\overlinecr(G)$ denote the rectilinear crossing number of a graph $G. We determine $\overlinecr(K 11)=102 and $\overlinecr(K 12)=153. Despite the remarkable hunt for crossing numbers of the complete graph .K n -- initiated by R. Guy in the 1960s -- these quantities have been unknown for n>10 to date. Our solution mainly relies on a tailor-made method for enumerating all inequivalent sets of points (order types) of size 11.(MATH) Based on these findings, we establish new upper and lower bounds on $\overlinecr(K n), for general n. Specific values are given for n, ≤ 45. The new asymptotic lower bound is immediate from the result $\overlinecr(K 11)=102, whereas the upper bound stems from a novel construction of drawings with few crossings. The tantalizing question of determining $\overlinecr(K 13) is left open. The latest ra(n)ge is 221,223,225,227,229; our conjecture is $\overlinecr(K 13) = 229.

Visibility preserving terrain simplification: an experimental study

Abstract: The terrain surface simplification problem has been studied extensively, as it has important applications in geographic information systems and computer graphics. The goal is to obtain a new surface that is combinatorially as simple as possible, while maintaining a prescribed degree of similarity with the original input surface. Generally, the approximation error is measured with respect to distance (e.g., Hausdorff) from the original or with respect to visual similarity. In this paper, we propose a new method of simplifying terrain surfaces, designed specifically to maximize a new measure of quality based on preserving inter-point visibility relationships. Our work is motivated by various problems of terrain analysis that rely on inter-point visibility relationships, such as optimal antenna placement.We have implemented our new method and give experimental evidence of its effectiveness in simplifying terrains according to our quality measure. We experimentally compare its performance with that of other leading simplification methods.

Three dimensional euclidean Voronoi diagrams of lines with a fixed number of orientations

Abstract: (MATH) We show that the combinatorial complexity of the Euclidean Voronoi diagram of n lines in $\reals3 that have at most c distinct orientations, is O(c 4 n 2+e), for any e>0. This result is a step towards proving the long-standing conjecture that the Euclidean Voronoi diagram of lines in three dimensions has near-quadratic complexity. It provides the first natural instance in which this conjecture is shown to hold. In a broader context, our result adds a natural instance to the (rather small) pool of instances of general 3-dimensional Voronoi diagrams for which near-quadratic complexity bounds are known.

Optimal decomposition of polygonal models into triangle strips

Abstract: Motivated by applications in computer graphics, we study the problem of computing an optimal encoding in "triangle strips" of a triangulation of a polygonal surface model. The goal is to facilitate the transmission and rendering of a polygonal model by decomposing its triangulation into a minimum number of "tristrips," each of which has its connectivity stored implicitly in the ordering of the data points. While this optimization problem has been conjectured to be hard, its complexity status has been open. We prove that the tristrip decomposition problem is, in fact, NP-complete. We also propose two methods for solving the problem to optimality, one based on an integer programming formulation, one based on a branch-and-bound scheme that relies on lower bounding techniques for its efficiency. We perform an extensive set of experiments to test the efficiencies of these methods and some of their variants. These methods have been integrated also with the practical system FTSG (Fast Triangle Strip Generator), in order to utilize optimization methods on small subproblems to improve the quality of the heuristic solutions obtained by FTSG. We use experimentation to judge the quality of the improvements.

Finding the consensus shape for a protein family

Abstract: We define and prove properties of the consensus shape for a family of proteins, a protein-like structure that provides a compact summary of the significant structural information for a protein family. If all members of a protein family exhibit a geometric relationship between corresponding alpha carbons then that relationship is preserved in the consensus shape. In particular, distances and angles that are consistent across family members are preserved. For the consensus shape, the spacing between successive alpha carbons is variable, with small distances in regions where the members of the protein family exhibit significant variation and large distances (up to the standard spacing of about 4\AA) in regions where the family members agree. Despite this non-protein-like characteristic, the consensus shape preserves and highlights important structural information. We describe an iterative algorithm for computing the consensus shape and prove that the algorithm converges. We also present the results of experiments in which we build consensus shapes for several known protein families.

Polyhedral Voronoi diagrams of polyhedra in three dimensions

Abstract: We show that that the complexity of the Voronoi diagram of a collection of disjoint polyhedra in 3-space that have n vertices overall, under a convex distance function induced by a polyhedron with O(1) facets, is O(n 2+e), for any e>0. We also show that when the sites are n segments in 3-space, this complexity is O(n 2 α(n) log n). This generalizes previous results by Chew et al. [9] and by Aronov and Sharir [4], and solves an open problem put forward by Agarwal and Sharir [2]. Specific distance functions for which our results hold are the L 1 and the L ∞ metrics. These results imply that we can preprocess a collection of polyhedra as above into a near-quadratic data structure that can answer Δ-approximate Euclidean nearest-neighbor queries amidst the polyhedra in time O(log (n/Δ)), for an arbitrarily small Δ>0.

Box-trees for collision checking in industrial installations

Abstract: A box-tree is a bounding-volume hierarchy that uses axis-aligned boxes as bounding volumes. We describe a new algorithm to construct a box-tree for a 3D scene consisting of n objects, and we analyze its worst-case query time for approximate range queries. If the input scene has certain characteristics that we derived from our application---collision detection in industrial installations---then the query times are polylogarithmic, not only for searching with boxes but also for range searching with other constant-complexity ranges.

Vertex-unfoldings of simplicial manifolds

Abstract: We present an algorithm to unfold any triangulated 2-manifold (in particular, any simplicial polyhedron) into a non-overlap-linebreak ping, connected planar layout in linear time. The manifold is cut only along its edges. The resulting layout is connected, but it may have a disconnected interior; the triangles are connected at vertices, but not necessarily joined along edges. We extend our algorithm to establish a similar result for simplicial manifolds of arbitrary dimension.

Incidences between points and circles in three and higher dimensions

Abstract: (MATH) We show that the number of incidences between m distinct points and n distinct circles in $\reals^3$ is O(m 4/7 n 17/21+m 2/3 n 2/3+m+n); the bound is optimal for m n 3/2. This result extends recent work on point-circle incidences in the plane, but its proof requires a different analysis. The bound improves upon a previous bound, noted by Akutsu et al. [2] and by Agarwal and Sharir [1], but it is not as sharp (when m is small) as the recent planar bound of Aronov and Sharir [3]. Our analysis extends to yield the same bound (a) on the number of incidences between m points and n circles in any dimension d≥ 3, and (b) on the number of incidences between m points and n arbitrary convex plane curves in $\reals^d$, for any d≥ 3, provided that no two curves are coplanar. Our results improve the upper bound on the number of congruent copies of a fixed tetrahedron in a set of n points in 4-space, and were already used to obtain a lower bound for the number of distinct distances in a set of n points in 3-space.

Densest translational lattice packing of non-convex polygons

Abstract: A translational lattice packing of k polygons P"1,P"2,P"3,...,P"k is a (non-overlapping) packing of the k polygons which is replicated without overlap at each point of a lattice i"0v"0+i"1v"1, where v"0 and v"1 are vectors generating the lattice and i"0 and i"1 range over all integers. A densest translational lattice packing is one which minimizes the area |v"0xv"1| of the fundamental parallelogram. An algorithm and implementation is given for densest translational lattice packing. This algorithm has useful applications in industry, particularly clothing manufacture.

A lower bound on the distortion of embedding planar metrics into Euclidean space

Abstract: (MATH) We exhibit a simple infinite family of series-parallel graphs that cannot be metrically embedded into Euclidean space with distortion smaller than $\Omega(\sqrt\log n\,)$. This matches Rao's general upper bound for metric embedding of planar graphs into Euclidean space, [14], thus resolving the question of how well do planar metrics embed in Euclidean spaces.

The probabilistic complexity of the Voronoi diagram of points on a polyhedron

Abstract: (MATH) It is well known that the complexity, i.e., the number of vertices, edges and faces, of the 3-dimensional Voronoi diagram of n points can be as bad as Θ(n 2). Interest has recently arisen as to what happens, both in deterministic and probabilistic situations, when the 3-dimensional points are restricted to lie on the surface of a 2-dimensional object. In this paper we consider the situation when the points are drawn from a 2-dimensional Poisson distribution with rate n over a fixed union of triangles in $\myRe^3.$ We show that with high probability the complexity of their Voronoi diagram is $\Otn.(MATH) This implies, for example, that the complexity of the Voronoi diagram of points chosen from the surface of a general fixed polyhedron in $\myRe3 will also be $\Otn with high probability.

Pseudo approximation algorithms, with applications to optimal motion planning

Abstract: (MATH) We introduce a technique for computing approximate solutions to optimization problems. If X is the set of feasible solutions, the standard goal of approximation algorithms is to compute k e X that is an e-approximate solution in the following sense: d(k)≤(1+e)d(k*) where k* E X is an optimal solution, d : X → 0 is the optimization function to be minimized, and $\vareps>0 is an input parameter. Our approach is to first devise algorithms that compute pseudo e-approximate solutions satisfying the bound d(k) ≤ d(kR *) + eR where R>0 is a new input parameter. Here k* R denotes an optimal solution in the space X R of R-constrained feasible solutions. The parameterization provides a stratification of X in the sense that (1) XR ⊆ XR' , for R R' and (2) XR = X for R sufficiently large.We first describe a highly efficient scheme for converting a pseudo e-approximation algorithm into a true e-approximation algorithm. This scheme is useful because pseudo approximation algorithms seem to be easier to construct than e-approximation algorithms.We then apply our technique to two problems in robotics: (A) Euclidean Shortest Path (3ESP), namely the shortest path for a point robot amidst polyhedral obstacles in 3D, and (B) d 1-optimal motion for a rod moving amidst polygonal obstacles in 2D. Previously, no true e-approximation algorithm for (B) was known. For (A), our new solution is not only simpler than two previous solutions but also has a lower complexity (in the algebraic model) measured in terms of the input precision. Note that (A) and (B) are the simplest NP-hard motion planning problems in 3-D and 2-D respectively.