Partitioning arrangements of lines I: An efficient deterministic algorithm

In this paper we consider the following problem: Given a set ofn lines in the plane, partition the plane intoO(r ²) triangles so that no triangle meets more thanO(n/r) lines of . We present a deterministic algorithm for this problem withO(nr logn/r) running time, where is a constant <3.33.Work on this paper has been supported by Office of Naval Research Grant N00014-87-K-0129, by National Science Foundation Grant DCR-83-20085, and by grants from the Digital Equipment Corporation and the IBM Corporation. A preliminary version of this paper appears in theProceedings of the 5th Annual Symposium on Computational Geometry, 1989, pp. 11–22.     

Triangles in arrangements of lines

A set of n lines in the projective plane divides the plane into a certain number of polygonal cells. We show that if we insist that all of these cells be triangles, then there are at most $\text{1}\text{3}\text{n(n ? 1) + 4 ?}\text{2}\text{7}\text{n}$ of them. We also observe that if no point of the plane belongs to more than two of the lines and n is at least 4, some of the cells must be either quadrangles or pentagons. We further show that for $\text{n ≧ 4}$, there is a set of n lines which divides the plane into at least $\text{1}\text{3}\text{n(n ? 3) + 4}$ triangles.     

Triangles in arrangements of lines

A set of n nonconcurrent lines in the projective plane (called an arrangment) divides the plane into polygonal cells. It has long been a problem to find a nontrivial upper bound on the number of triangular regions. We show that $\text{5}\text{12}\text{n(n ? 1)}$ is such a bound. We also show that if no three lines are concurrent, then the number of quadrilaterals, pentagons and hexagons is at least cn².     

Triangles in arrangements of lines

Givenn lines in the real projective plane, Grünbaum conjectures that, for n16, the numberp ₃ of triangular regions determined by the lines is at most 1/3n(n–1). We show that ifn7 thenp ₃ 8/21n(n–1)+2/7, we also point out that if no vertex is a point of intersection of exactly three of the lines, thenp ₃1/3n(n–1).Professor Gu died while on a visit to Poland in April 1997     

Quadrilaterals and pentagons in arrangements of lines

Let p _k denote the number of k-sided faces in an arrangement of n5 lines in the real projective plane. B. Grünbaum has shown that p ₄1/2n(n–3) and has conjectured that equality can occur only for simple arrangements. We prove this conjecture here. We also show that 4p ₄+5p ₅3n holds for every simple arrangement of n4 lines. This latter result is a strengthening of a theorem of T. O. Strommer.     

Implicitly representing arrangements of lines or segments

Anarrangement ofn lines (or line segments) in the plane is the partition of the plane defined by these objects. Such an arrangement consists ofO(n ²) regions, calledfaces. In this paper we study the problem of calculating and storing arrangementsimplicitly, using subquadratic space and preprocessing, so that, given any query pointp, we can calculate efficiently the face containingp. First, we consider the case of lines and show that with (n) space¹ and (n ^3/2) preprocessing time, we can answer face queries in (n)+O(K) time, whereK is the output size. (The query time is achieved with high probability.) In the process, we solve three interesting subproblems: (1) given a set ofn points, find a straight-edge spanning tree of these points such that any line intersects only a few edges of the tree, (2) given a simple polygonal path , form a data structure from which we can find the convex hull of any subpath of quickly, and (3) given a set of points, organize them so that the convex hull of their subset lying above a query line can be found quickly. Second, using random sampling, we give a tradeoff between increasing space and decreasing query time. Third, we extend our structure to report faces in an arrangement of line segments in (n ^1/3)+O(K) time, given(n ^4/3) space and (n ^5/3) preprocessing time. Lastly, we note that our techniques allow us to computem faces in an arrangement ofn lines in time (m ^2/3 n ^2/3+n), which is nearly optimal.The first author is pleased to acknowledge the support of Amoco Fnd. Fac. Dev. Comput. Sci. 1-6-44862 and National Science Foundation Grant CCR-8714565. Work on this paper by the fifth author has been supported by Office of Naval Research Grant N00014-87-K-0129, by National Science Foundation Grant NSF-DCR-83-20085, by grants from the Digital Equipment Corporation, and the IBM Corporation, and by a research grant from the NCRD—the Israeli National Council for Research and Development. The sixth author was supported in part by a National Science Foundation Graduate Fellowship. This work was begun while the non-DEC authors were visiting at the DEC Systems Research Center.     

Freeness for 13 lines arrangements is combinatorial

We prove Terao conjecture saying that the freeness is determined by the combinatorics for arrangements of 13 lines in the complex projective plane.     

Partitioning with two lines in the plane

An O(n) algorithm is presented for the problem of partitioning a set of n points in the plane into four equal parts by means of two straight lines.     

A theorem on arrangements of lines in the plane

LetA be an arrangement ofn lines in the plane. IfR ₁, …,R _r arer distinct regions ofA, andR _i is ap _i-gon (i=1, …,r) then we show that . Further we show that for allr this bound is the best possible ifn is sufficiently large. Financial support for this research was provided by the Carnegie Trust for the Universities of Scotland.     

Moduli spaces of arrangements of 10 projective lines with quadruple points

We classify moduli spaces of arrangements of 10 lines with quadruple points. We show that moduli spaces of arrangements of 10 lines with quadruple points may consist of more than 2 disconnected components, namely 3 or 4 distinct points. We also present defining equations to those arrangements whose moduli spaces are still reducible after taking quotients of complex conjugations.     

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

We show that the total number of edges ofm faces of an arrangement ofn lines in the plane isO(m ^2/3– n ^2/3+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(m ^2/3– n ^2/3+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(m ^2/3– n ^2/3+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(m ^2/3– n ^2/3+2 log+n(n) log² n logm).The first author is pleased to acknowledge partial support by the Amoco Fnd. Fac. Dev. Comput. Sci. 1-6-44862 and the National Science Foundation under Grant CCR-8714565. Work on this paper by the third author has been supported by Office of Naval Research Grant N00014-82-K-0381, by National Science Foundation Grant DCR-83-20085, by grants from the Digital Equipment Corporation, and the IBM Corporation, and by a research grant from the NCRD-the Israeli National Council for Research and Development. A preliminary version of this paper has appeared in theProceedings of the 4th ACM Symposium on Computational Geometry, 1988, pp. 44–55.     

Bounded-Independence Derandomization of Geometric Partitioning with Applications to Parallel Fixed-Dimensional Linear Programming

We give fast and efficient methods for constructing ε-nets and ε-approximations for range spaces with bounded VC-exponent. These combinatorial structures have wide applicability to geometric partitioning problems, which are often used in divide-and-conquer constructions in computational geometry algorithms. In addition, we introduce a new deterministic set approximation for range spaces with bounded VC-exponent, which we call the δ-relative ε-approximation, and we show how such approximations can be efficiently constructed in parallel. To demonstrate the utility of these constructions we show how they can be used to solve the linear programming problem in deterministically in time using linear work in the PRAM model of computation, for any fixed constant d. Our method is developed for the CRCW variant of the PRAM parallel computation model, and can be easily implemented to run in time using linear work on an EREW PRAM. Received August 7, 1995, and in revised form November 11, 1996.     

Noncommutative Burkholder/Rosenthal inequalities II: Applications

We show norm estimates for the sum of independent random variables in noncommutative L _p -spaces for 1 < p < ∞, following our previous work. These estimates generalize the classical Rosenthal inequality in the commutative case. As applications, we derive an equivalence for the p-norm of the singular values of a random matrix with independent entries, and characterize those symmetric subspaces and unitary ideals which can be realized as subspaces of a noncommutative L _p for 2 < p < ∞. The first author is partially supported by the National Science Foundation DMS-0301116. The second author is partially supported by the Agence Nationale de Recherche 06-BLAN-0015.     

The diffeomorphic types of the complements of arrangements in ℂℙ3 II

For any arrangement of hyperplanes in ??³, we introduce the soul of this arrangement. The soul, which is a pseudo-complex, is determined by the combinatorics of the arrangement of hyperplanes. In this paper, we give a sufficient combinatoric condition for two arrangements of hyperplanes to be diffeomorphic to each other. In particular we have found sufficient conditions on combinatorics for the arrangement of hyperplanes whose moduli space is connected. This generalizes our previous result on hyperplane point arrangements in ??³.     

Arrangements on Parametric Surfaces II: Concretizations and Applications

We describe the algorithms and implementation details involved in the concretizations of a generic framework that enables exact construction, maintenance, and manipulation of arrangements embedded on certain two-dimensional orientable parametric surfaces in three-dimensional space. The fundamentals of the framework are described in a companion paper. Our work covers arrangements embedded on elliptic quadrics and cyclides induced by intersections with other algebraic surfaces, and a specialized case of arrangements induced by arcs of great circles embedded on the sphere. We also demonstrate how such arrangements can be used to accomplish various geometric tasks efficiently, such as computing the Minkowski sums of polytopes, the envelope of surfaces, and Voronoi diagrams embedded on parametric surfaces. We do not assume general position. Namely, we handle degenerate input, and produce exact results in all cases. Our implementation is realized using Cgal and, in particular, the package that provides the underlying framework. We have conducted experiments on various data sets, and documented the practical efficiency of our approach.     

Nonsmooth Analysis of Singular Values. Part II: Applications

In this work we continue the nonsmooth analysis of absolutely symmetric functions of the singular values of a real rectangular matrix. Absolutely symmetric functions are invariant under permutations and sign changes of its arguments. We extend previous work on subgradients to analogous formulae for the proximal subdifferential and Clarke subdifferential when the function is either locally Lipschitz or just lower semicontinuous. We illustrate the results by calculating the various subdifferentials of individual singular values. Another application gives a nonsmooth proof of Lidskii’s theorem for weak majorization. Mathematics Subject Classifications (2000) Primary 90C31, 15A18; secondary 49K40, 26B05.Research supported by NSERC.     

Applications of k-covers II

We continue the study of applications of k-covers to some topological constructions, mostly to function spaces and hyperspaces.