Showing papers by "Leonidas J. Guibas published in 1990"

Combinatorial complexity bounds for arrangements of curves and spheres

Abstract: We present upper and lower bounds for extremal problems defined for arrangements of lines, circles, spheres, and alike. For example, we prove that the maximum number of edges boundingm cells in an arrangement ofn lines is ?(m2/3n2/3 +n), and that it isO(m2/3n2/3s(n) +n) forn unit-circles, wheres(n) (and laters(m, n)) is a function that depends on the inverse of Ackermann's function and grows extremely slowly. If we replace unit-circles by circles of arbitrary radii the upper bound goes up toO(m3/5n4/5s(n) +n). The same bounds (without thes(n)-terms) hold for the maximum sum of degrees ofm vertices. In the case of vertex degrees in arrangements of lines and of unit-circles our bounds match previous results, but our proofs are considerably simpler than the previous ones. The maximum sum of degrees ofm vertices in an arrangement ofn spheres in three dimensions isO(m4/7n9/7s(m, n) +n2), in general, andO(m3/4n3/4s(m, n) +n) if no three spheres intersect in a common circle. The latter bound implies that the maximum number of unit-distances amongm points in three dimensions isO(m3/2s(m)) which improves the best previous upper bound on this problem. Applications of our results to other distance problems are also given.

The complexity and construction of many faces in arrangements of lines and of segments

Abstract: We show that the total number of edges ofm faces of an arrangement ofn lines in the plane isO(m2/3??n2/3+2?+n) for any?>0. The proof takes an algorithmic approach, that is, we describe an algorithm for the calculation of thesem faces and derive the upper bound from the analysis of the algorithm. The algorithm uses randomization and its expected time complexity isO(m2/3??n2/3+2? logn+n logn logm). If instead of lines we have an arrangement ofn line segments, then the maximum number of edges ofm faces isO(m2/3??n2/3+2?+n? (n) logm) for any?>0, where?(n) is the functional inverse of Ackermann's function. We give a (randomized) algorithm that produces these faces and takes expected timeO(m2/3??n2/3+2? log+n?(n) log2n logm).

Randomized Incremental Construction of Delaunay and Voronoi Diagrams

Abstract: In this paper we give a new randomized incremental algorithm for the construction of planar Voronoi diagrams and Delaunay triangulations. The new algorithm is more “online” than earlier similar methods, takes expected time O(n log n) and space O(n), and is eminently practical to implement. The analysis of the algorithm is also interesting in its own right and can serve as a model for many similar questions in both two and three dimensions. Finally we demonstrate how this approach for constructing Voronoi diagrams obviates the need for building a separate point-location structure for nearest-neighbor queries.

Points and triangles in the plane and halving planes in space

Abstract: We prove that for any set S of n points in the plane and n3-α triangles spanned by the points of S there exists a point (not necessarily of S) contained in at least n3-3α/(512 log5 n) of the triangles. This implies that any set of n points in three-dimensional space defines at most 6.4n8/3 log5/3 n halving planes.

Slimming down by adding; selecting heavily covered points

Abstract: We show that for any set Π of n points in three-dimensional space there is a set Q of 𝒪(n1/2 log3 n) points so that the Delaunay triangulation of Π ∪ Q has at most 𝒪(n3/2 log3 n) edges — even though the Delaunay triangulation of Π may have Ω(n2) edges. The main tool of our construction is the following geometric covering result: For any set Π of n points in three-dimensional space and any set S of m spheres, where each sphere passes through a distinct point pair in Π, there exists a point x, not necessarily in Π, that is enclosed by Ω(m2/n2 log3 n2/m) of the spheres in S.Our results generalize to arbitrary fixed dimensions, to geometric bodies other than spheres, and to geometric structures other than Delaunay triangulations.

Counting and cutting cycles of lines and rods in space

Abstract: A number of rendering algorithms in computer graphics sort three-dimensional objects by depth and assume that there is no cycle that makes the sorting impossible. One way to resolve the problem caused by cycles is to cut the objects into smaller pieces. The problem of estimating how many such cuts are always sufficient is addressed. A few related algorithmic and combinatorial geometry problems are considered. >

Constructing strongly convex approximate hulls with inaccurate primitives

Abstract: The first half of this paper introducesEpsilon Geometry, a framework for the development of robust geometric algorithms using inaccurate primitives. Epsilon Geometry is based on a very general model of imprecise computations, which includes floating-point and rounded-integer arithmetic as special cases. The second half of the paper introduces the notion of a (−ɛ)-convex polygon, a polygon that remains convex even if its vertices are all arbitrarily displaced by a distance ofɛ of less, and proves some interesting properties of such polygons. In particular, we prove that for every point set there exists a (−ɛ)-convex polygonH such that every point is at most 4ɛ away fromH. Using the tools of Epsilon Geometry, we develop robust algorithms for testing whether a polygon is (−ɛ)-convex, for testing whether a point is inside a (−ɛ)-convex polygon, and for computing a (−ɛ)-convex approximate hull for a set of points.

Compact interval trees: a data structure for convex hulls

Abstract: In this paper, we investigate the problem of finding the common tangents of two convex polygons that intersect in two (unknown) points. First, we give a Θ(log2n) bound for algorithms that store the polygons in independent arrays. Second, we show how to beat the lower bound if the vertices of the convex polygons are drawn from a fixed set of n points. We introduce a data structure called a compact interval tree that supports common tangent computations, as well as the standard binary-search-based queries, in O(logn) time apiece. Third, we apply compact interval trees to solve the subpath hull query problem: given a simple path, preprocess it so that we can find the convex hull of a query subpath quickly. With O(nlogn) preprocessing, we can assemble a compact interval tree that represents the convex hull of a query subpath in O(logn) time. In order to represent arrangements of Lines implicitly, Edelsbrunner et al. used a less efficient structure, called bridge trees, to solve the subpath hull query problem. Our compact interval trees improve their results by a factor of O(logn). Thus, the present paper replaces the paper on bridge trees referred to by Edelsbrunner et al.