Showing papers by "Herbert Edelsbrunner published in 1993"

The union of balls and its dual shape

Abstract: Efficient algorithms are described for computing topological, combinatorial, and metric properties of the union of finitely many balls in ℝd. These algorithms are based on a simplicial complex dual to a certain decomposition of the union of balls, and on short inclusion-exclusion formulas derived from this complex. The algorithms are most relevant in ℝ3 where unions of finitely many balls are commonly used as models of molecules.

On the zone theorem for hyperplane arrangements

Abstract: The zone theorem for an arrangement of n hyperplanes in d-dimensional real space says that the total number of faces bounding the cells intersected by another hyperplane is O(nd−1). This result is the basis of a time-optimal incremental algorithm that constructs a hyperplane arrangement and has a host of other algorithmic and combinatorial applications. Unfortunately, the original proof of the zone theorem, for d ≥ 3, turned out to contain a serious and irreparable error. This paper presents a new proof of the theorem. Our proof is based on an inductive argument, which also applies in the case of pseudo-hyperplane arrangements. We also briefly discuss the fallacies of the old proof along with some ways of partially saving that approach.

An incremental algorithm for Betti numbers of simplicial complexes

Abstract: A general and direct method for computing the betti numbers of the homology groups of a finite simplicial complex is given. For subcomplexes of a triangulation of S3 this method has implementations that run in time O(nα(n)) and O(n), where n is the number of simplices in the triangulation. If applied to the family of α-shapes of a finite point set in ℝ3 it takes time O(nℝ(n)) to compute the betti numbers of all α-shapes.

An upper bound for conforming Delaunay triangulations

Abstract: A plane geometric graphC in ?2conforms to another such graphG if each edge ofG is the union of some edges ofC. It is proved that, for everyG withn vertices andm edges, there is a completion of a Delaunay triangulation ofO(m2n) points that conforms toG. The algorithm that constructs the points is also described.

Diameter, width, closest line pair, and parametric searching

Abstract: We apply Megiddo's parametric searching technique to several geometric optimization problems and derive significantly improved solutions for them. We obtain, for any fixed ?>0, anO(n1+?) algorithm for computing the diameter of a point set in 3-space, anO(8/5+?) algorithm for computing the width of such a set, and onO(n8/5+?) algorithm for computing the closest pair in a set ofn lines in space. All these algorithms are deterministic.

Edge insertion for optimal triangulations

Abstract: Edge insertion iteratively improves a triangulation of a finite point set in ?2 by adding a new edge, deleting old edges crossing the new edge, and retriangulating the polygonal regions on either side of the new edge. This paper presents an abstract view of the edge insertion paradigm, and then shows that it gives polynomial-time algorithms for several types of optimal triangulations, including minimizing the maximum slope of a piecewise-linear interpolating surface.

Computing a face in an arrangement of line segments and related problems

Abstract: This paper presents a randomized incremental algorithm for computing a single face in an arrangement of n line segments in the plane that is fairly simple to implement. The expected running time of the algorithm is $O(n\alpha (n)\log n)$. The analysis of the algorithm uses a novel approach that generalizes and extends the Clarkson–Shor analysis technique [in Discrete Comput. Geom., 4 (1989), pp. 387–421]. A few extensions of the technique, obtaining efficient randomized incremental algorithms for constructing the entire arrangement of a collection of line segments and for computing a single face in an arrangement of Jordan arcs are also presented.

Counting triangle crossings and halving planes

Abstract: Every collection of t ≥ 2n2 triangles with a total of n vertices in R3 has Ω(t4/n6) crossing pairs. This implies that one of their edges meets Ω(t3/n6) of the triangles. From this it follows that n points in R3 have only O(8/3) halving planes.

Improved bounds on weak e-nets for convex sets

Abstract: LetS be a set ofn points in ℝ d . A setW is aweak e-net for (convex ranges of)S if, for anyT⊆S containing en points, the convex hull ofT intersectsW. We show the existence of weak e-nets of size\(O((1/\varepsilon ^d )\log ^{\beta _d } (1/\varepsilon ))\), whereβ2=0,β3=1, andβ d ≈0.149·2d-1(d-1)!, improving a previous bound of Alonet al. Such a net can be computed effectively. We also consider two special cases: whenS is a planar point set in convex position, we prove the existence of a net of sizeO((1/e) log1.6(1/e)). In the case whereS consists of the vertices of a regular polygon, we use an argument from hyperbolic geometry to exhibit an optimal net of sizeO(1/e), which improves a previous bound of Capoyleas.

A quadratic time algorithm for the minmax length triangulation

Abstract: It is shown that a triangulation of a set of n points in the plane that minimizes the maximum edge length can be computed in time $O(n^2 )$. The algorithm is reasonably easy to implement and is based on the theorem that there is a triangulation with minmax edge length that contains the relative neighborhood graph of the points as a subgraph. With minor modifications the algorithm works for arbitrary normed metrics.