Showing papers presented at "Symposium on Computational Geometry in 2010"

Optimal partition trees

Abstract: We revisit one of the most fundamental classes of data structure problems in computational geometry: range searching. Back in SoCG'92, Matousek gave a partition tree method for d-dimensional simplex range searching achieving O(n) space and O(n1-1/d) query time. Although this method is generally believed to be optimal, it is complicated and requires O(n1+e) preprocessing time for any fixed e>0. An earlier method by Matousek (SoCG'91) requires O(n log n) preprocessing time but O(n1-1/d logO(1)n) query time. We give a new method that achieves simultaneously O(n log n) preprocessing time, O(n) space, and O(n1-1/d) query time with high probability. Our method has several advantages: It is conceptually simpler than Matousek's SoCG'92 method. Our partition trees satisfy many ideal properties (e.g., constant degree, optimal crossing number at almost all layers, and disjointness of the children's cells at each node). It leads to more efficient multilevel partition trees, which are important in many data structural applications (each level adds at most one logarithmic factor to the space and query bounds, better than in all previous methods). A similar improvement applies to a shallow version of partition trees, yielding O(n log n) time, O(n) space, and O(n1-1=⌊d=2⌋) query time for halfspace range emptiness in even dimensions d ≥ 4. Numerous consequences follow (e.g., improved results for computing spanning trees with low crossing number, ray shooting among line segments, intersection searching, exact nearest neighbor search, linear programming queries, finding extreme points, ... ).

The tidy set: a minimal simplicial set for computing homology of clique complexes

Abstract: We introduce the tidy set, a minimal simplicial set that captures the topology of a simplicial complex. The tidy set is particularly effective for computing the homology of clique complexes. This family of complexes include the Vietoris-Rips complex and the weak witness complex, methods that are popular in topological data analysis. The key feature of our approach is that it skips constructing the clique complex. We give algorithms for constructing tidy sets, implement them, and present experiments. Our preliminary results show that tidy sets are orders of magnitude smaller than clique complexes, giving us a homology engine with small memory requirements.

Manifold reconstruction using tangential Delaunay complexes

Abstract: We give a provably correct algorithm to reconstruct a k-dimensional manifold embedded in d-dimensional Euclidean space. Input to our algorithm is a point sample coming from an unknown manifold. Our approach is based on two main ideas : the notion of tangential Delaunay complex defined in [6,19,20], and the technique of sliver removal by weighting the sample points [13]. Differently from previous methods, we do not construct any subdivision of the embedding d-dimensional space. As a result, the running time of our algorithm depends only linearly on the extrinsic dimension d while it depends quadratically on the size of the input sample, and exponentially on the intrinsic dimension k. To the best of our knowledge, this is the first certified algorithm for manifold reconstruction whose complexity depends linearly on the ambient dimension. We also prove that for a dense enough sample the output of our algorithm is isotopic to the manifold and a close geometric approximation of the manifold.

A randomized O(m log m) time algorithm for computing Reeb graphs of arbitrary simplicial complexes

Abstract: Given a continuous scalar field ƒ: X → where X is a topological space, a level set of ƒ is a set {x ∈ X : ƒ (x) = α} for some value α ∈ IR. The level sets of ƒ can be subdivided into connected components. As α changes continuously, the connected components in the level sets appear, disappear, split and merge. The Reeb graph of ƒ encodes these changes in connected components of level sets. It provides a simple yet meaningful abstraction of the input domain. As such, it has been used in a range of applications in fields such as graphics and scientific visualization. In this paper, we present the first sub-quadratic algorithm to compute the Reeb graph for a function on an arbitrary simplicial complex K. Our algorithm is randomized with an expected running time O(m log n), where m is the size of the 2-skeleton of K (i.e, total number of vertices, edges and triangles), and n is the number of vertices. This presents a significant improvement over the previous Θ(mn) time complexity for arbitrary complex, matches (although in expectation only) the best known result for the special case of 2-manifolds, and is faster than current algorithms for any other special cases (e.g, 3-manifolds). Our algorithm is also very simple to implement. Preliminary experimental results show that it performs well in practice.

Incidences in three dimensions and distinct distances in the plane

Abstract: We first describe a reduction from the problem of lower-bounding the number of distinct distances determined by a set S of s points in the plane to an incidence problem between points and a certain class of helices (or parabolas) in three dimensions. We offer conjectures involving the new setup, but are still unable to fully resolve them. Instead, we adapt the recent new algebraic analysis technique of Guth and Katz [9], as further developed by Elekes et al. [6], to obtain sharp bounds on the number of incidences between these helices or parabolas and points in R3. Applying these bounds, we obtain, among several other results, the upper bound O(s3) on the number of rotations (rigid motions) which map (at least) three points of S to three other points of S. In fact, we show that the number of such rotations which map at least k ≥ 3 points of S to k other points of S is close to O(S3/k12/7). One of our unresolved conjectures is that this number is O(s3/k2), for k ≥ 2. If true, it would imply the lower bound Ω(s/ log s) on the number of distinct distances in the plane.

Approximating loops in a shortest homology basis from point data

Abstract: Inference of topological and geometric attributes of a hidden manifold from its point data is a fundamental problem arising in many scientific studies and engineering applications. In this paper we present an algorithm to compute a set of loops from a point data that presumably sample a smooth manifold M ⊂ Rd. These loops approximate a shortest basis of the one dimensional homology group H1(M) over coefficients in finite field Z2. Previous results addressed the issue of computing the rank of the homology groups from point data, but there is no result on approximating the shortest basis of a manifold from its point sample. In arriving our result, we also present a polynomial time algorithm for computing a shortest basis of H1 (Κ) for any finite simplicial complex Κ whose edges have non-negative weights.

A constant-factor approximation algorithm for TSP with pairwise-disjoint connected neighborhoods in the plane

Abstract: In the Euclidean TSP with neighborhoods (TSPN) problem we seek. shortest tour that visits a given set of n neighborhoods. The Euclidean TSPN generalizes the standard TSP on points. We present the first constant-factor approximation algorithm for planar TSPN with pairwise-disjoint connected neighborhoods of any size or shape. Prior approximation bounds were O(log n), except in special cases.

Orthogonal range reporting: query lower bounds, optimal structures in 3-d, and higher-dimensional improvements

Abstract: Orthogonal range reporting is the problem of storing a set of n points in d-dimensional space, such that the k points in an axis-orthogonal query box can be reported efficiently. While the 2-d version of the problem was completely characterized in the pointer machine model more than two decades ago, this is not the case in higher dimensions. In this paper we provide a space optimal pointer machine data structure for 3-d orthogonal range reporting that answers queries in O(log n + k) time. Thus we settle the complexity of the problem in 3-d. We use this result to obtain improved structures in higher dimensions, namely structures with a log n/ log log n factor increase in space and query time per dimension. Thus for d e 3 we obtain a structure that both uses optimal O(n(log n/ log log n)d--1) space and answers queries in the best known query bound O(log n(log n/ log log n)d--3 + k). Furthermore, we show that any data structure for the d-dimensional orthogonal range reporting problem in the pointer machine model of computation that uses S(n) space must spend Ω((log n/ log(S(n)/n))⌊d/2⌋--1) time to answer queries. Thus, if S(n)/n is poly-logarithmic, then the query time is at least Ω((log n/ log log n)⌊d/2⌋--1). This is the first known non-trivial higher dimensional orthogonal range reporting query lower bound and it has two important implications. First, it shows that the query bound increases with dimension. Second, in combination with our upper bounds it shows that the optimal query bound increases from Θ(log n + k) to Ω((log n/ log log n)2 + k) somewhere between three and six dimensions. Finally, we show that our techniques also lead to improved structures for point location in rectilinear subdivisions, that is, the problem of storing a set of n disjoint d-dimensional axis-orthogonal rectangles, such that the rectangle containing a query point q can be found efficiently.

Steinitz theorems for orthogonal polyhedra

Abstract: We define a simple orthogonal polyhedron to be a three-dimensional polyhedron with the topology of a sphere in which three mutually-perpendicular edges meet at each vertex. By analogy to Steinitz's theorem characterizing the graphs of convex polyhedra, we characterize the graphs of simple orthogonal polyhedra: they are exactly the 3-regular bipartite planar graphs in which the removal of any two vertices produces at most two connected components. We also characterize two subclasses of these polyhedra: corner polyhedra, which can be drawn by isometric projection in the plane with only one hidden vertex, and xyz polyhedra, in which each axis-parallel line through a vertex contains exactly one other vertex. Based on our characterizations we find efficient algorithms for constructing orthogonal polyhedra from their graphs

Approximating the Fréchet distance for realistic curves in near linear time

Abstract: We present a simple and practical (1+e)-approximation algorithm for the Frechet distance between polygonal curves. To analyze this algorithm we introduce a new realistic family of curves, c-packed curves, that is closed under simplification. We believe the notion of c-packed curves to be of independent interest. We show that our algorithm has near linear running time for c-packed polygonal curves, and show similar results for other input models, such as low density.

Finding shortest non-trivial cycles in directed graphs on surfaces

Abstract: Let D be a weighted directed graph cellularly embedded in a surface of genus g, orientable or not, possibly with boundary. We describe algorithms to compute a shortest non-contractible and a shortest surface non-separating cycle in D. This generalizes previous results that only dealt with undirected graphs. Our first algorithm computes such cycles in O(n2 log n) time, where n is the total number of vertices and edges of D, thus matching the complexity of the best known algorithm in the undirected case. It revisits and extends Thomassen's 3-path condition; the technique applies to other families of cycles as well. We also give an algorithm with subquadratic complexity in the complexity of the input graph, if g is fixed. Specifically, we can solve the problem in O(√g n3/2 log n) time, using a divide-and-conquer technique that simplifies the graph while preserving the topological properties of its cycles. A variant runs in O(ng log g + n log2 n) for graphs of bounded treewidth.

3D Euler spirals for 3D curve completion

Abstract: Shape completion is an intriguing problem in geometry processing with applications in CAD and graphics. This paper defines a new type of 3D curves, which can be utilized for curve completion. It can be considered as the extension to three dimensions of the 2D Euler spiral. We prove several properties of these curves - properties that have been shown to be important for the appeal of curves. We illustrate their utility in two applications. The first is "fixing" curves detected by algorithms for edge detection on surfaces. The second is shape illustration in archaeology, where the user would like to draw curves that are missing due to the incompleteness of the input model.

Kinetic stable Delaunay graphs

Abstract: The best known upper bound on the number of topological changes in the Delaunay triangulation of a set of moving points in ℜ2 is (nearly) cubic, even if each point is moving with a fixed velocity. We introduce the notion of a stable Delaunay graph (SDG in short), a dynamic subgraph of the Delaunay triangulation, that is less volatile in the sense that it undergoes fewer topological changes and yet retains many useful properties of the full Delaunay triangulation. SDG is defined in terms of a parameter ± > 0, and consists of Delaunay edges pq for which the (equal) angles at which p and q see the corresponding Voronoi edge epq are at least ±. We prove several interesting properties of SDG and describe two kinetic data structures for maintaining it. Both structures use O*(n) storage. They process O*(n2) events during the motion, each in O*(1) time, provided that the points of P move along algebraic trajectories of bounded degree; the O*(·) notation hides multiplicative factors that are polynomial in 1/± and polylogarithmic in n. The first structure is simpler but the dependency on 1/± in its performance is higher.

Topological inference via meshing

Abstract: We apply ideas from mesh generation to improve the time and space complexities of computing the full persistent homological information associated with a point cloud P in Euclidean space ℜd. Classical approaches rely on the Cech, Rips, ±-complex, or witness complex filtrations of P, whose complexities scale up very badly with d. For instance, the ±-complex filtration incurs the n Ω(d) size of the Delaunay triangulation, where n is the size of P. The common alternative is to truncate the filtrations when the sizes of the complexes become prohibitive, possibly before discovering the most relevant topological features. In this paper we propose a new collection of filtrations, based on the Delaunay triangulation of a carefully-chosen superset of P, whose sizes are reduced to 2O(d2)n. Our filtrations interleave multiplicatively with the family of offsets of P, so that the persistence diagram of P can be approximated in 2O(d2)n3 time in theory, with a near-linear observed running time in practice. Thus, our approach remains tractable in medium dimensions, say 4 to 10.

Dynamic well-spaced point sets

Abstract: In a well-spaced point set, when there is a bounding hypercube, the Voronoi cells all have bounded aspect ratio, i.e., the distance from the Voronoi site to the farthest point in the Voronoi cell divided by the distance to the nearest neighbor in the set is bounded by a small constant. Well-spaced point sets satisfy some important geometric properties and yield quality Voronoi or simplicial meshes that can be important in scientific computations. In this paper, we consider the dynamic well-spaced point sets problem, which requires computing the well-spaced superset of a dynamically changing input set, e.g., as input points are inserted or deleted. We present a dynamic algorithm that allows inserting/deleting points into/from the input in worst-case O(log Δ) time, where Δ is the geometric spread, a natural measure that is bounded by O(log n) when input points are represented by log-size words. We show that the runtime of the dynamic update algorithm is optimal in the worst case. Our algorithm generates size-optimal outputs: the resulting output sets are never more than a constant factor larger than the minimum size necessary. A preliminary implementation indicates that the algorithm is indeed fast in practice. To the best of our knowledge, this is the first time- and size-optimal dynamic algorithm for well-spaced point sets.

Tight lower bounds for halfspace range searching

Abstract: We establish two new lower bounds for the halfspace range searching problem: Given a set of n points in ℜd, where each point is associated with a weight from a commutative semigroup, compute the semigroup sum of the weights of the points lying within any query halfspace. Letting $m$ denote the space requirements, we prove a lower bound for general semigroups of Ω(n1-1/(d+1)/m1/(d+1)) and for integral semigroups of Ω(n/m1/d). Our lower bounds are proved in the semigroup arithmetic model. Neglecting logarithmic factors, our result for integral semigroups matches the best known upper bound due to Matousek. Our result for general semigroups improves upon the best known lower bound due to Bronnimann, Chazelle, and Pach. Moreover, Fonseca and Mount have recently shown that, given uniformly distributed points, halfspace range queries over idempotent semigroups can be answered in O(n1-1/(d+1)/m1/(d+1)) time in the semigroup arithmetic model. As our lower bounds are established for uniformly distributed point sets, it follows that they also resolve the computational complexity of halfspace range searching over idempotent semigroups in this important special case.

The complexity of the normal surface solution space

Abstract: Normal surface theory is a central tool in algorithmic three-dimensional topology, and the enumeration of vertex normal surfaces is the computational bottleneck in many important algorithms. However, it is not well understood how the number of such surfaces grows in relation to the size of the underlying triangulation. Here we address this problem in both theory and practice. In theory, we tighten the exponential upper bound substantially; furthermore, we construct pathological triangulations that prove an exponential bound to be unavoidable. In practice, we undertake a comprehensive analysis of millions of triangulations and find that in general the number of vertex normal surfaces is remarkably small, with strong evidence that our pathological triangulations may in fact be the worst case scenarios. This analysis is the first of its kind, and the striking behaviour that we observe has important implications for the feasibility of topological algorithms in three dimensions.

A dynamic data structure for approximate range searching

Abstract: In this paper, we introduce a simple, randomized dynamic data structure for storing multidimensional point sets, called a quadtreap. This data structure is a randomized, balanced variant of a quadtree data structure. In particular, it defines a hierarchical decomposition of space into cells, which are based on hyperrectangles of bounded aspect ratio, each of constant combinatorial complexity. It can be viewed as a multidimensional generalization of the treap data structure of Seidel and Aragon. When inserted, points are assigned random priorities, and the tree is restructured through rotations as if the points had been inserted in priority order. In any fixed dimension d, we show it is possible to store a set of n points in a quadtreap of space O(n). The height h of the tree is O(log n) with high probability. It supports point insertion in time O(h). It supports point deletion in worst-case time O(h2) and expected-case time O(h), averaged over the points of the tree. It can answer e-approximate spherical range counting queries over groups and approximate nearest neighbor queries in time O(h + (1/e)d-1).

I/O-efficient computation of water flow across a terrain

Abstract: Consider rain falling at a uniform rate onto a terrain T represented as a triangular irregular network. Over time, water collects in the basins of T, forming lakes that spill into adjacent basins. Our goal is to compute, for each terrain vertex, the time this vertex is flooded (covered by water). We present an I/O-efficient algorithm that solves this problem using O(sort(X) log (X/M) + sort(N)) I/Os, where N is the number of terrain vertices, X is the number of pits of the terrain, sort(N) is the cost of sorting N data items, and M is the size of the computer's main memory. Our algorithm assumes that the volumes and watersheds of the basins of T have been precomputed using existing methods.

Reconstructing shapes with guarantees by unions of convex sets

Abstract: A simple way to reconstruct a shape A from a sample P is to output an r-offset P + r B, where B = {x ∈ RN x ≤ 1} designates the unit Euclidean ball centered at the origin. Recently, it has been proved that the output P + r B is homotopy equivalent to the shape A, for a dense enough sample P of A and for a suitable value of the parameter r. In this paper, we extend this result and find convex sets C ⊂ RN, besides the unit Euclidean ball B, for which P + rC reconstructs the topology of A. This class of convex sets includes in particular N-dimensional cubes in RN. We proceed in two steps. First, we establish the result when P is an e-offset of A. Building on this first result, we then consider the case when P is a finite noisy sample of A.

Adding one edge to planar graphs makes crossing number hard

Abstract: A graph is near-planar if it can be obtained from a planar graph by adding an edge. We show that it is NP-hard to compute the crossing number of near-planar graphs. The main idea in the reduction is to consider the problem of simultaneously drawing two planar graphs inside a disk, with some of its vertices fixed at the boundary of the disk. This approach can be used to prove hardness of some other geometric problems. As an interesting consequence we obtain a new, geometric proof of NP-completeness of the crossing number problem, even when restricted to cubic graphs. This resolves a question of Hlinený.

On degrees in random triangulations of point sets

Abstract: We study the expected number of interior vertices of degree i in a triangulation of a point set S, drawn uniformly at random from the set of all triangulations of S, and derive various bounds and inequalities for these expected values. One of our main results is: For any set S of N points in general position, and for any fixed i, the expected number of vertices of degree i in a random triangulation is at least γiN, for some fixed positive constant γi (assuming that N > i and that at least some fixed fraction of the points are interior). We also present a new application for these expected values, using upper bounds on the expected number of interior vertices of degree 3 to get a new lower bound, Ω(2.4317N), for the minimal number of triangulations any N-element planar point set in general position must have. This improves the previously best known lower bound of Ω(2.33N).

Better bounds on the union complexity of locally fat objects

Abstract: We prove that the union complexity of a set of n constant-complexity locally fat objects (which can be curved and/or non-convex) in the plane is O(λt+2(n) log n), where t is the maximum number of times the boundaries of any two objects intersect. This improves the previously best known bound by a logarithmic factor.

An improved algorithm for computing the volume of the union of cubes

Abstract: Let C be a set of n axis-aligned cubes in ℜ3, and let U(C) denote the union of C. We present an algorithm that computes the volume of U(C) in time O(n polylog(n)). The previously best known algorithm takes O(n4/3 log2 n) time.

Consistent digital line segments

Abstract: We introduce a novel and general approach for digitalization of line segments in the plane that satisfies a set of axioms naturally arising from Euclidean axioms. In particular, we show how to derive such a system of digital segments from any total order on the integers. As a consequence, using a well-chosen total order, we manage to define a system of digital segments such that all digital segments are, in Hausdorff metric, optimally close to their corresponding Euclidean segments, thus giving an explicit construction that resolves the main question of [1].

Planar visibility: testing and counting

Abstract: In this paper we consider query versions of visibility testing and visibility counting. Let S be a set of n disjoint line segments in ℜ2 and let s be an element of S. Visibility testing is to preprocess S so that we can quickly determine if s is visible from a query point q. Visibility counting involves preprocessing S so that one can quickly estimate the number of segments in S visible from a query point q. We present several data structures for the two query problems. The structures build upon a result by O'Rourke and Suri (1984) who showed that the subset, VS(s), of ℜ2 that is weakly visible from a segment s can be represented as the union of a set, CS(s), of O(n2) triangles, even though the complexity of VS(s) can be Ω(n4). We define a variant of their covering, give efficient output-sensitive algorithms for computing it, and prove additional properties needed to obtain approximation bounds. Some of our bounds rely on a new combinatorial result that relates the number of segments of S visible from a point p to the number of triangles in ∪s∈S CS(s) that contain p.

Zone diagrams in Euclidean spaces and in other normed spaces

Abstract: Zone diagram is a variation on the classical concept of a Voronoi diagram. Given n sites in a metric space that compete for territory, the zone diagram is an equilibrium state in the competition. Formally it is defined as a fixed point of a certain "dominance" map. Asano, Matousek, and Tokuyama proved the existence and uniqueness of a zone diagram for point sites in Euclidean plane, and Reem and Reich showed existence for two arbitrary sites in an arbitrary metric space. We establish existence and uniqueness for n disjoint compact sites in a Euclidean space of arbitrary (finite) dimension, and more generally, in a finite-dimensional normed space with a smooth and rotund norm. The proof is considerably simpler than that of Asano et al. We also provide an example of non-uniqueness for a norm that is rotund but not smooth. Finally, we prove existence and uniqueness for two point sites in the plane with a smooth (but not necessarily rotund) norm.

Geometric methods for multi-agent collision avoidance

Abstract: We present an approach to reciprocal collision avoidance, where multiple mobile agents must avoid collisions with each other while moving in a common workspace. Each agent acts fully independently, and does not communicate with others. Yet our approach guarantees that all agents will be collision-free for at least a fixed amount of time. Our approach provides a sufficient condition for collision-free motion. Given the agent's objective, the optimal collision-free action can be computed very efficiently, as it is the solution to a two-dimensional linear program. We show our approach on dense and complex simulation scenarios involving thousands of agents at fast real-time running times.

New constructions of SSPDs and their applications

Abstract: We present a new optimal construction of semi-separated pair decomposition (SSPD) for a set of n points in IRd. In the new construction each point participates in a few pairs, and it extends easily to spaces with low doubling dimension. This is the first optimal construction with these properties. As an application of the new construction, for a fixed t > 1, we present a new construction of a t-spanner with O(n) edges and maximum degree O(log2 n) that has a separator of size O(n1-1/d).

Output-sensitive algorithm for the edge-width of an embedded graph

Abstract: Let G be an unweighted graph of complexity n cellularly embedded in a surface (orientable or not) of genus g. We describe improved algorithms to compute (the length of) a shortest non-contractible and a shortest non-separating cycle of G. If k is an integer, we can compute such a non-trivial cycle with length at most k in O(gnk) time, or correctly report that no such cycle exists. In particular, on a fixed surface, we can test in linear time whether the edge-width or face-width of a graph is bounded from above by a constant. This also implies an output-sensitive algorithm to compute a shortest non-trivial cycle that runs in O(gnk) time, where k is the length of the cycle.