Showing papers presented at "Symposium on Computational Geometry in 2005"

Stability of persistence diagrams

Abstract: The persistence diagram of a real-valued function on a topological space is a multiset of points in the extended plane. We prove that under mild assumptions on the function, the persistence diagram is stable: small changes in the function imply only small changes in the diagram. We apply this result to estimating the homology of sets in a metric space and to comparing and classifying geometric shapes.

Fast construction of nets in low dimensional metrics, and their applications

Abstract: We present a near linear time algorithm for constructing hierarchical nets in finite metric spaces with constant doubling dimension. This data-structure is then applied to obtain improved algorithms for the following problems: Approximate nearest neighbor search, well-separated pair decomposition, spanner construction, compact representation scheme, doubling measure, and computation of the (approximate) Lipschitz constant of a function. In all cases, the running (preprocessing) time is near-linear and the space being used is linear.

Smaller coresets for k-median and k-means clustering

Abstract: In this paper, we show that there exists a (k, e)-coreset for k-median and k-means clustering of n points in Rd, which is of size independent of n. In particular, we construct a (k, e)-coreset of size O(k2/ed) for k-median clustering, and of size O(k3/ed+1) for k-means clustering.

Sampling in dynamic data streams and applications

Abstract: A dynamic geometric data stream is a sequence of m Add/Remove operations of points from a discrete geometric space (1,...,Δ)d [21]. Add(p) inserts a point p from (1,...,Δ)d into the current point set, Remove(p) deletes p from P. We develop low-storage data structures to (i) maintain e-approximations of range spaces of P with constant VC-dimension and (ii) maintain an e-approximation of the weight of the Euclidean minimum spanning tree of P. Our data structures use O(log3e • log3(1/e) • log(1/e)/e2) and O(log (1/δ) • (log Δ/e)O(d)) bits of memory, respectively (we assume that the dimension d is a constant), and they are correct with probability 1-δ. These results are based on a new data structure that maintains a set of elements chosen (almost) uniformly at random from P.

The skip quadtree: a simple dynamic data structure for multidimensional data

Abstract: We present a new multi-dimensional data structure, which we call the skip quadtree (for point data in R2) or the skip octree (for point data in Rd, with constant d > 2). Our data structure combines the best features of two well-known data structures, in that it has the well-defined "box"-shaped regions of region quadtrees and the logarithmic-height search and update hierarchical structure of skip lists. Indeed, the bottom level of our structure is exactly a region quadtree (or octree for higher dimensional data). We describe efficient algorithms for inserting and deleting points in a skip quadtree, as well as fast methods for performing point location, approximate range, and approximate nearest neighbor queries.

Weak feature size and persistent homology: computing homology of solids in Rn from noisy data samples

Abstract: In this work, one proves that under quite general assumptions one can deduce the topology of a bounded open set in Rn from a Hausdorff distance approximation of it. For this, one introduces the weak feature size (wfs) that generalizes the notion of local feature size. Our results apply to open sets with positive wfs, which include many sets whose boundaries are not smooth and even nowhere smooth. This class includes also the piecewise analytic open sets which cover many cases encountered in practical applications. The proofs are based on the study of distance functions to closed sets and their critical points. As an application, one gives an algorithmic way, thanks to persistent homology techniques, to compute the homology groups of open sets from noisy samples of points on their boundary.

On the exact computation of the topology of real algebraic curves

Abstract: We consider the problem of computing a representation of the plane graph induced by one (or more) algebraic curves in the real plane. We make no assumptions about the curves, in particular we allow arbitrary singularities and arbitrary intersection. This problem has been well studied for the case of a single curve. All proposed approaches to this problem so far require finding and counting real roots of polynomials over an algebraic extension of Q, i.e. the coefficients of those polynomials are algebraic numbers. Various algebraic approaches for this real root finding and counting problem have been developed, but they tend to be costly unless speedups via floating point approximations are introduced, which without additional checks in some cases can render the approach incorrect for some inputs.We propose a method that is always correct and that avoids finding and counting real roots of polynomials with non-rational coefficients. We achieve this using two simple geometric approaches: a triple projections method and a curve avoidance method. We have implemented our approach for the case of computing the topology of a single real algebraic curve. Even this prototypical implementation without optimizations appears to be competitive with other implementations.

Incidences of not-too-degenerate hyperplanes

Abstract: We present a multi-dimensional generalization of the Szemeredi-Trotter Theorem, and give a sharp bound on the number of incidences of points and not-too-degenerate hyperplanes in three- or higher-dimensional Euclidean spaces. We call a hyperplane not-too-degenerate if at most a constant portion of its incident points lie in a lower dimensional affine subspace.

An exact, complete and efficient implementation for computing planar maps of quadric intersection curves

Abstract: We present the first exact, complete and efficient implementation that computes for a given set P=p1,...,pn of quadric surfaces the planar map induced by all intersection curves p1∩ pi, 2 ≤ i ≤ n, running on the surface of p1. The vertices in this graph are the singular and x-extreme points of the curves as well as all intersection points of pairs of curves. Two vertices are connected by an edge if the underlying points are connected by a branch of one of the curves. Our work is based on and extends ideas developed in [20] and [9].Our implementation is complete in the sense that it can handle all kind of inputs including all degenerate ones where intersection curves have singularities or pairs of curves intersect with high multiplicity. It is exact in that it always computes the mathematical correct result. It is efficient measured in running times.

Improved approximation algorithms for geometric set cover

Abstract: Given a collection S of subsets of some set U, and M ⊂ U, the set cover problem is to find the smallest subcollection C ⊂ S such that M is a subset of the union of the sets in C. While the general problem is NP-hard to solve, even approximately, here we consider some geometric special cases, where usually U = Rd. Combining previously known techniques [3, 4], we show that polynomial time approximation algorithms with provable performance exist, under a certain general condition: that for a random subset R ⊂ S and function f(), there is a decomposition of the complement U ∖ ∪Y ∈ R Y into an expected f(|R|) regions, each region of a particular simple form. Under this condition, a cover of size O(f(|C|)) can be found in polynomial time. Using this result, and combinatorial geometry results implying bounding functions f(c) that are nearly linear, we obtain o(log c) approximation algorithms for covering by fat triangles, by pseudodisks, by a family of fat objects, and others. Similarly, constant-factor approximations follow for similar-sized fat triangles and fat objects, and for fat wedges. With more work, we obtain constant-factor approximation algorithms for covering by unit cubes in R3, and for guarding an x-monotone polygonal chain.

Star splaying: an algorithm for repairing delaunay triangulations and convex hulls

Abstract: Star splaying is a general-dimensional algorithm that takes as input a triangulation or an approximation of a convex hull, and produces the Delaunay triangulation, weighted Delaunay triangulation, or convex hull of the vertices in the input. If the input is "nearly Delaunay" or "nearly convex" in a certain sense quantified herein, and it is sparse (i.e. each input vertex adjoins only a constant number of edges), star splaying runs in time linear in the number of vertices. Thus, star splaying can be a fast first step in repairing a high-quality finite element mesh that has lost the Delaunay property after its vertices have moved in response to simulated physical forces. Star splaying is akin to Lawson's edge flip algorithm for converting a triangulation to a Delaunay triangulation, but it works in any dimensionality.

Curvature-bounded traversals of narrow corridors

Abstract: We consider the existence and efficient construction of bounded curvature paths traversing constant-width regions of the plane, called corridors. We make explicit a width threshold τ with the property that (a) all corridors of width at least τ admit a unit-curvature traversal and (b) for any width w

Multi-pass geometric algorithms

Abstract: We initiate the study of exact geometric algorithms that require limited storage and make only a small number of passes over the input. Fundamental problems such as low-dimensional linear programming and convex hulls are considered.

Efficient computation of query point visibility in polygons with holes

Abstract: In this paper, we consider the problem of computing the visibility of a query point inside polygons with holes. The goal is to perform this computation efficiently per query with more cost in the preprocessing phase. Our algorithm is based on solutions in [13] and [2] proposed for simple polygons. In our solution, the preprocessing is done in time O(n3 log(n)) to construct a data structure of size O(n3). It is then possible to report the visibility polygon of any query point q in time O((1+h′) log n+|V(q)|), in which n and h are the number of the vertices and holes of the polygon respectively, |V(q)| is the size of the visibility polygon of q, and h′ is an output and preprocessing sensitive parameter of at most min(h,|V(q)|). This is claimed to be the best query-time result on this problem so far.

Critical points of the distance to an epsilon-sampling of a surface and flow-complex-based surface reconstruction

Abstract: The distance function to surfaces in three dimensions plays a key role in many geometric modeling applications such as medial axis approximations, surface reconstructions, offset computations, feature extractions and others. In most cases, the distance function induced by the surface is approximated by a discrete distance function induced by a discrete sample of the surface. The critical points of the distance function determine the topology of the set inducing the function. However, no earlier theoretical result has linked the critical points of the distance to a sampling of geometric structures to their topological properties. We provide this link by showing that the critical points of the distance function induced by a discrete sample of a surface either lie very close to the surface or near its medial axis and this closeness is quantified with the sampling density. Based on this result, we provide a new flow-complex-based surface reconstruction algorithm that, given a tight e-sampling of a surface, approximates the surface geometrically, both in Hausdorff distance and normals, and captures its topology.

The visibility--voronoi complex and its applications

Abstract: We introduce a new type of diagram called the VV(c)-diagram (the Visibility--Voronoi diagram for clearance c), which is a hybrid between the visibility graph and the Voronoi diagram of polygons in the plane. It evolves from the visibility graph to the Voronoi diagram as the parameter c grows from 0 to ∞. This diagram can be used for planning natural-looking paths for a robot translating amidst polygonal obstacles in the plane. A natural-looking path is short, smooth, and keeps --- where possible --- an amount of clearance c from the obstacles. The VV(c)-diagram contains such paths. We also propose an algorithm that is capable of preprocessing a scene of configuration-space polygonal obstacles and constructs a data structure called the VV(c)-complex. The VV(c)-complex can be used to efficiently plan motion paths for any start and goal configuration and any clearance value c, without having to explicitly construct the VV(c)-diagram for that c-value. The preprocessing time is O(n2 log n), where n is the total number of obstacle vertices, and the data structure can be queried directly for any c-value by merely performing a Dijkstra search. We have implemented a Cgal-based software package for computing the VV(c)-diagram in an exact manner for a given clearance value, and used it to plan natural-looking paths in various applications.

Clustered planarity

Abstract: A cluster of a graph is a non empty subset of vertices. A clustered graph C(G, T ) is a graph G plus a rooted tree T such that the leaves of T are the vertices of G (see Fig. 1 for an example). Each internal node ν of T corresponds to the cluster V (ν) of G whose vertices are the leaves of the subtree rooted at ν. The subgraph of G induced by V (ν) is denoted as G(ν). An edge e between a vertex of V (ν) and a vertex of V − V (ν) is said to be incident on ν. Graph G and tree T are called underlying graph and inclusion tree, respectively. Drawing a clustered graph is requided in many applications. To give a few examples:

A time-optimal delaunay refinement algorithm in two dimensions

Abstract: We propose a new refinement algorithm to generate size-optimal quality-guaranteed Delaunay triangulations in the plane. The algorithm takes O(n log n + m) time, where n is the input size and m is the output size. This is the first time-optimal Delaunay refinement algorithm.

Certifying and constructing minimally rigid graphs in the plane

Abstract: We study minimally rigid graphs in the plane or plane isostatic graphs. These graphs (also called Laman graphs) admit characterizations based on decomposition into trees (Crapo's theorem and Recski's theorem). Tree partitions can be viewed as certificates of plane isostatic graphs. Unfortunately, they require Ω(n2) time to verify their validity where n is the number of vertices in the graph. We present a new construction (which can be viewed as a hierarchical decomposition of the graph) called red-black hierarchy that (i) is a certificate for plane isostatic graphs, and (ii) can be verified in linear time. We also show that it can be computed in O(n2) time.A classical result in Rigidity Theory by Henneberg [9] states that the plane isostatic graphs can be constructed incrementally by special vertex insertions. We study the following computational problem: given a Laman graph G, compute a sequence of Henneberg insertions that yields G. We show that the red-bl ack hierarchy can be used to compute a Henneberg construction in O(n2) time. Applied to planar graphs our algorithm can speed up a recent algorithm by Haas et al. [8] for embedding a planar Laman graph as a pointed pseudo-triangulation by a factor of O(n).

Hadwiger and Helly-type theorems for disjoint unit spheres in R3

Abstract: Let S be an ordered set of disjoint unit spheres in R3 We show that if every subset of at most six spheres from S admits a line transversal respecting the ordering, then the entire family has a line transversal. Without the order condition, we show that the existence of a line transversal for every subset of at most 11 spheres from S implies the existence of a line transversal forS.

Cache-oblivious planar orthogonal range searching and counting

Abstract: We present the first cache-oblivious data structure for planar orthogonal range counting, and improve on previous results for cache-oblivious planar orthogonal range searching.Our range counting structure uses O(N log2 N) space and answers queries using O(logB N) memory transfers, where B is the block size of any memory level in a multilevel memory hierarchy. Using bit manipulation techniques, the space can be further reduced to O(N). The structure can also be modified to support more general semigroup range sum queries in O(logB N) memory transfers, using O(N log2 N) space for three-sided queries and O(N log22 N/log2 log2 N) space for four-sided queries.Based on the O(N log N) space range counting structure, we develop a data structure that uses O(N log2 N) space and answers three-sided range queries in O(logB N+T/B) memory transfers, where T is the number of reported points. Based on this structure, we present a general four-sided range searching structure that uses O(N log22 N/log2 log2 N) space and answers queries in O(logB N + T/B) memory transfers.

Finding the best shortcut in a geometric network

Abstract: Given a Euclidean graph G in Rd with n vertices and m edges we consider the problem of adding a shortcut such that the stretch factor of the resulting graph is minimized. Currently, the fastest algorithm for computing the stretch factor of a Euclidean graph runs in O(mn+n2 log n) time, resulting in a trivial O(mn3+n4 log n) time algorithm for computing the optimal shortcut. First, we show that a simple modification yields the optimal solution in O(n4) time using O(n2) space. To reduce the running times we consider several approximation algorithms. Our main result is a (2+e)-approximation algorithm with running time O(nm+n2(log n+1/e3d)) using O(n2) space.

Shortest path amidst disc obstacles is computable

Abstract: An open question in Exact Geometric Computation is whether there re transcendental computations that can be made "geometrically exact".Perhaps the simplest such problem in computational geometry is that of computing the shortest obstacle-avoiding path between two points p, q in the plane, where the obstacles re collection of n discs.This problem can be solved in O (n 2 log n)time in the Real RAM model, but nothing was known about its computability in the standard (Turing) model of computation. We first show the Turing-computability of this problem,provided the radii of the discs are rationally related. We make the usual assumption that the numerical input data are real algebraic numbers. By appealing to effective bounds from transcendental number theory, we further show single-exponential time upper bound when the input numbers are rational.Our result ppears to be the first example of non-algebraic combinatorial problem which is shown computable. It is also rare example of transcendental number theory yielding positive computational results.

Maximizing the overlap of two planar convex sets under rigid motions

Abstract: Given two compact convex sets P and Q in the plane, we compute an image of P under a rigid motion that approximately maximizes the overlap with Q. More precisely, for any e > 0, we compute a rigid motion such that the area of overlap is at least 1 - e times the maximum possible overlap. Our algorithm uses O(1/e) extreme point and line intersection queries on P and Q, plus O((1/e2) log(1/e)) running time. If only translations are allowed, the extra running time reduces to O((1/e) log(1/e)). If P and Q are convex polygons with n vertices in total, the total running time is O((1/e) log n + (1/e2) log(1/e)) for rigid motions and O((1/e) log n + (1/e) log(1/e)) for translations.

Minimum dilation stars

Abstract: The dilation of a Euclidean graph is defined as the ratio of distance in the graph divided by distance in Rd. In this paper we consider the problem of positioning the root of a star such that the dilation of the resulting star is minimal. We present a deterministic O(n log n)-time algorithm for evaluating the dilation of a given star; a randomized O(n log n) expected-time algorithm for finding an optimal center in Rd; and for the case d = 2, a randomized O(n2α(n) log2n) expected-time algorithm for finding an optimal center among the input points.

Dynamic maintenance of molecular surfaces under conformational changes

Abstract: We present an efficient algorithm for maintaining the boundary and surface area of protein molecules as they undergo conformational changes. We also describe a robust implementation of the algorithm and report on experimental results with our implementation on proteins with hundreds of residues. Our work extends and combines two previous results: (i) controlled perturbation for static molecular surfaces [18], and (ii) data structures for self-collision testing and energy maintenance of proteins that change conformation [26]. As our method keeps a highly accurate representation of the boundary surface and of the voids in the molecule, it can be useful in various applications such as Monte Carlo Simulation or Molecular Dynamics Simulation. In addition we propose and analyze an alternative method for efficiently recalculating the surface area under conformational (and hence topological) changes based on techniques for efficient dynamic maintenance of graph connectivity; initial results of the implementation of this method show great promise.

Provable dimension detection using principal component analysis

Abstract: We present simple algorithms for detecting the dimension k of a smooth manifold M ⊂ Rd from a set P of point samples, provided that P satisfies a standard sampling condition as in previous results. The best running time so far is O(d2O(k7 log k)) worst-case by Giesen and Wagner after the adaptive neighborhood graph is constructed in O(d|P|2) worst-case time. Given the adaptive neighborhood graph, for any l ≥ 1, our algorithm outputs the true dimension with probability at least 1-2-l in O(2O(k)kd(k + l log d)) expected time. Our experimental results validate the effectiveness of our approach in computing the dimension. A further advantage is that both the algorithm and its analysis can be generalized to the noisy case, in which outliers and a small perturbation of the samples are allowed.

Cache-oblivious R-trees

Abstract: We develop a cache-oblivious data structure for storing a set S of N axis-aligned rectangles in the plane, such that all rectangles in S intersecting a query rectangle or point can be found efficiently. Our structure is an axis-aligned bounding-box hierarchy and as such it is the first cache-oblivious R-tree with provable performance guarantees. If no point in the plane is contained in B or more rectangles in S, the structure answers a rectangle query using O(√N/B + T/B) memory transfers and a point query using O((N/B)e) memory transfers for any e > 0, where B is the block size of memory transfers between any two levels of a multilevel memory hierarchy. We also develop a variant of our structure that achieves the same performance on input sets with arbitrary overlap among the rectangles. The rectangle query bound matches the bound of the best known linear-space cache-aware structure.

Vertical ray shooting for fat objects

Abstract: We describe a data structure for vertical ray shooting in a set of n convex fat polyhedra of constant complexity in 3-space. The structure has O(log2 n) query time, and it uses O(n log3 n(log log n)2) storage. It can also be used for fat objects with curved boundaries, at the cost of a small increase in storage.

Inequalities for the curvature of curves and surfaces

Abstract: In this paper, we bound the difference between the total mean curvatures of two closed surfaces in R3 in terms of their total absolute curvatures and the Frechet distance between the volumes they enclose. The proof relies on a combination of methods from algebraic topology and integral geometry. We also bound the difference between the lengths of two curves using the same methods.