Extremal point queries with lines and line segments and related problems

We address a number of extremal point query problems when P is a set of n points in , d3 a constant, including the computation of the farthest point from a query line and the computation of the farthest point from each of the lines spanned by the points in P. In , we give a data structure of size O(n¹⁺), that can be constructed in O(n¹⁺) time and can report the farthest point of P from a query line segment in O(n^2/3+) time, where >0 is an arbitrarily small constant. Applications of our results also include: (1) Sub-cubic time algorithms for fitting a polygonal chain through an indexed set of points in , d3 a constant, and (2) A sub-quadratic time and space algorithm that, given P and an anchor point q, computes the minimum (maximum) area triangle defined by q with P{q}.     

An elementary algorithm for reporting intersections of red/blue curve segments

Let E_r and E_b be two sets of x-monotone and non-intersecting curve segments, E=E_rE_b and |E|=n. We give a new sweep-line algorithm that reports the k intersecting pairs of segments of E. Our algorithm uses only three simple predicates that allow to decide if two segments intersect, if a point is left or right to another point, and if a point is above, below or on a segment. These three predicates seem to be the simplest predicates that lead to subquadratic algorithms. Our algorithm is almost optimal in this restricted model of computation. Its time complexity is O(nlogn+kloglogn) and it requires O(n) space.     

Multi-pattern matching with bidirectional indexes

We study multi-pattern matching in a scenario where the pattern set is to be matched to several texts and hence indexing the pattern set is affordable. This kind of scenarios arise, for example, in metagenomics, where pattern set represents DNA of several species and the goal is to find out which species are represented in the sample and in which quantity. We develop a generic search method that exploits bidirectional indexes both for the pattern set and texts, and analyze the best and worst case running time of the method on worst case text. We show that finding the instance of the search method with minimum best case running time on worst case text is NP-hard. The positive result is that an instance with logarithm factor approximation to minimum best case running time can be found in polynomial time using a bidirectional index called affix tree. We further show that affix trees can be simulated, in reduced space, using bidirectional variant of compressed suffix trees. Lastly, we provide a practical implementation of this approach. We show that one can obtain 3-fold speed up against the basic scenario of searching each pattern independently with data sets typical in high-throughput DNA sequencing.     

Foliations with good geometry

The goal of this article is to show that there is a large class of closed hyperbolic 3-manifolds admitting codimension one foliations with good large scale geometric properties. We obtain results in two directions. First there is an internal result: A possibly singular foliation in a manifold is quasi-isometric if, when lifted to the universal cover, distance along leaves is efficient up to a bounded multiplicative distortion in measuring distance in the universal cover. This means that leaves reflect very well the geometry in the large of the universal cover and are geometrically tight-this is the best geometric behavior. We previously proved that nonsingular codimension one foliations in closed hyperbolic 3-manifolds can never be quasi-isometric. In this article we produce a large class of singular quasi-isometric, codimension one foliations in closed hyperbolic 3-manifolds. The foliations are stable and unstable foliations of pseudo-Anosov flows. Our second result is an external result, relating (nonsingular) foliations in hyperbolic 3-manifolds with their limit sets in the universal cover, that is, showing that leaves in the universal cover have good asymptotic behavior. Let be a Reebless, finite depth foliation in a closed hyperbolic 3-manifold. Then is not quasi-isometric, but suppose that is transverse to a quasigeodesic pseudo-Anosov flow with quasi-isometric stable and unstable foliations-which are given by the internal result. We then show that the lifts of leaves of to the universal cover extend continuously to the sphere at infinity and we also produce infinitely many examples satisfying the hypothesis. The main tools used to prove these results are first a link between geometric properties of stable/unstable foliations of pseudo-Anosov flows and the topology of these foliations in the universal cover, and second a topological theory of the joint structure of the pseudo-Anosov foliation in the universal cover.

     

Algorithms for generalized halfspace range searching and other intersection searching problems

In a generalized intersection searching problem, a set S of colored geometric objects is to be preprocessed so that, given a query object q, the distinct colors of the objects of S that are intersected by q can be reported or counted efficiently. These problems generalize the well-studied standard intersection searching problems and have many applications. Unfortunately, the solutions known for the standard problems do not yield efficient solutions to the generalized problems. Recently, efficient solutions have been given for generalized problems where the input and query objects are iso-oriented (i.e., axes-parallel) or where the color classes satisfy additional properties (e.g., connectedness). In this paper, efficient algorithms are given for several generalized problems involving objects that are not necessarily iso-oriented. These problems include: generalized halfspace range searching in , for any fixed d ≥ 2, and segment intersection searching, triangle stabbing, and triangle range searching in for certain classes of line segments and triangles. The techniques used include: computing suitable sparse representations of the input, persistent data structures, and filtering search.     

Computing a flattest, undercut-free parting line for a convex polyhedron, with application to mold design

A parting line for a polyhedron is a closed curve on its surface, which identifies the two halves of the polyhedron for which mold-boxes must be made. A parting line is undercut-free if the two halves that it generates do not contain facets that obstruct the de-molding of the polyhedron. Computing an undercut-free parting line that is as “flat” as possible is an important problem in mold design. In this paper, algorithms are presented to compute such a parting line for a convex polyhedron, based on different flatness criteria.     

On some geometric optimization problems in layered manufacturing

Efficient geometric algorithms are given for optimization problems arising in layered manufacturing, where a 3D object is built by slicing its CAD model into layers and manufacturing the layers successively. The problems considered include minimizing the stair-step error on the surfaces of the manufactured object under various formulations, minimizing the volume of the so-called support structures used, and minimizing the contact area between the supports and the manufactured object—all of which are factors that affect the speed and accuracy of the process. The stair-step minimization algorithm is valid for any polyhedron, while the support minimization algorithms are applicable only to convex polyhedra. The techniques used to obtain these results include construction and searching of certain arrangements on the sphere, 3D convex hulls, halfplane range searching, and constrained optimization.     

Enumerating pseudo-triangulations in the plane

In this paper, we study the problem of whether a polyhedron can be obtained from a net by folding along the creases. We show that this problem can be solved in polynomial time if the dihedral angle at each crease is given, and it becomes NP-hard if these angles are unknown. We also study the case when the net has rigid faces that should not intersect during the folding process.     

LOOK: A Lazy Object-Oriented Kernel design for geometric computation

In this paper we describe and discuss a new kernel design for geometric computation in the plane. It combines different kinds of floating-point filter techniques and a lazy evaluation scheme with the exact number types provided by LEDA allowing for efficient and exact computation with rational and algebraic geometric objects.

It is the first kernel design which uses floating-point filter techniques on the level of geometric constructions.

The experiments we present—partly using the CGAL framework—show a great improvement in speed and—maybe even more important for practical applications—memory consumption when dealing with more complex geometric computations.     

Recent researches on multivariate spline and piecewise algebraic variety

The multivariate splines as piecewise polynomials have become useful tools for dealing with Computational Geometry, Computer Graphics, Computer Aided Geometrical Design and Image Processing. It is well known that the classical algebraic variety in algebraic geometry is to study geometrical properties of the common intersection of surfaces represented by multivariate polynomials. Recently the surfaces are mainly represented by multivariate piecewise polynomials (i.e. multivariate splines), so the piecewise algebraic variety defined as the common intersection of surfaces represented by multivariate splines is a new topic in algebraic geometry. Moreover, the piecewise algebraic variety will be also important in computational geometry, computer graphics, computer aided geometrical design and image processing. The purpose of this paper is to introduce some recent researches on multivariate spline, piecewise algebraic variety (curve), and their applications.     

Coarse homotopy groups

In this note on coarse geometry we revisit coarse homotopy. We prove that coarse homotopy indeed is an equivalence relation, and this in the most general context of abstract coarse structures. We introduce (in a geometric way) coarse homotopy groups. The main result is that the coarse homotopy groups of a cone over a compact simplicial complex coincide with the usual homotopy groups of the underlying compact simplicial complex. To prove this we develop geometric triangulation techniques for cones which we expect to be of relevance also in different contexts.     

Fast reconstruction of Delaunay triangulations

We present a new linear time algorithm to compute a good order for the point set of a Delaunay triangulation in the plane. Such a good order makes reconstruction in linear time with a simple algorithm possible. Similarly to the algorithm of Snoeyink and van Kreveld [Proceedings of 5th European Symposium on Algorithms (ESA), 1997, pp. 459–471], our algorithm constructs such orders in O(logn) phases by repeatedly removing a constant fraction of vertices from the current triangulation. Compared to [Proceedings of 5th European Symposium on Algorithms (ESA), 1997, pp. 459–471] we improve the guarantee on the number of removed vertices in each such phase. If we restrict the degree of the points (at the time they are removed) to 6, our algorithm removes at least 1/3 of the points while the algorithm from [Proceedings of 5th European Symposium on Algorithms (ESA), 1997, pp. 459–471] gives a guarantee of 1/10. We achieve this improvement by removing the points sequentially using a breadth first search (BFS) based procedure that—in contrast to [Proceedings of 5th European Symposium on Algorithms (ESA), 1997, pp. 459–471]—does not (necessarily) remove an independent set.

Besides speeding up the algorithm, removing more points in a single phase has the advantage that two consecutive points in the computed order are usually closer to each other. For this reason, we believe that our approach is better suited for vertex coordinate compression.

We implemented prototypes of both algorithms and compared their running time on point sets uniformly distributed in the unit cube. Our algorithm is slightly faster. To compare the vertex coordinate compression capabilities of both algorithms we round the resulting sequences of vertex coordinates to 16-bit integers and compress them with a simple variable length code. Our algorithm achieves about 14% better vertex data compression than the algorithm from [Proceedings of 5th European Symposium on Algorithms (ESA), 1997, pp. 459–471].     

Computing optimal islands

Let S be a bicolored set of n points in the plane. A subset I of S is an island if there is a convex set C such that I=C∩S. We give an O(n³)-time algorithm to compute a monochromatic island of maximum cardinality. Our approach is adapted to optimize similar (decomposable) objective functions. Finally, we use our algorithm to give an O(logn)-approximation for the problem of computing the minimum number of convex polygons that cover a class region.