Empty pseudo-triangles in point sets

We study empty pseudo-triangles in a set P of n points in the plane, where an empty pseudo-triangle has its vertices at the points of P, and no points of P lie inside. We give bounds on the minimum and maximum number of empty pseudo-triangles. If P lies inside a triangle whose corners must be the convex vertices of the pseudo-triangle, then there can be between Θ(n²) and Θ(n³) empty pseudo-triangles. If the convex vertices of the pseudo-triangle are also chosen from P, this number lies between Θ(n³) and Θ(n⁶). If we count only star-shaped pseudo-triangles, the bounds are Θ(n²) and Θ(n⁵). We also study optimization problems: minimizing or maximizing the perimeter or the area over all empty pseudo-triangles defined by P. If P lies inside a triangle whose corners must be used, we can solve these problems in O(n³) time. In the general case, the running times are O(n⁶) for the maximization problems and O(nlogn) for the minimization problems.     

Centroids,Representations, and Submodular Flows

The notion of centroid of a tree is generalized to apply to an arbitrary intersecting family of sets. Centroids are used to construct a compact representation for any intersecting family of sets, as well as any crossing family. The size of the representation for a family on n elements is O(n²), compared to size O(n³) for previous representations. Efficient algorithms to construct the representation are given. For example on a network of n vertices and m edges, the representation of all minimum cuts uses O(m log(n²/m)) space; it is constructed in O(nm log(n²/m)) time (this is the best-known time for finding one minimum cut). The representation is used to improve several submodular flow algorithms. For example a minimum-cost dijoin is found in time O(n²m); as a result a minimum-cost planar feedback are set is found in time O(n³). The previous best-known time bounds for these two problems are both a factor n larger.     

Incidences between Points and Circles in Three and Higher Dimensions

We show that the number of incidences between m distinct points and n distinct circles in ^d, for any d 3, is O(m^6/11n^9/11(m³/n)+m^2/3n^2/3+m+n), where (n)=(log n)^O((n))2 and where (n) is the inverse Ackermann function. The bound coincides with the recent bound of Aronov and Sharir, or rather with its slight improvement by Agarwal et al., for the planar case. We also show that the number of incidences between m points and n unrestricted convex (or bounded-degree algebraic) plane curves, no two in a common plane, is O(m^4/7n^17/21+m^2/3n^2/3+m+n), in any dimension d 3. Our results improve the upper bound on the number of congruent copies of a fixed tetrahedron in a set of n points in 4-space and the lower bound for the number of distinct distances in a set of n points in 3-space. Another application is an improved bound for the number of incidences (or, rather, containments) between lines and reguli in three dimensions. The latter result has already been applied by Feldman and Sharir to obtain a new bound on the number of joints in an arrangement of lines in three dimensions.     

Applications of random sampling in computational geometry,II

We use random sampling for several new geometric algorithms. The algorithms are Las Vegas, and their expected bounds are with respect to the random behavior of the algorithms. These algorithms follow from new general results giving sharp bounds for the use of random subsets in geometric algorithms. These bounds show that random subsets can be used optimally for divide-and-conquer, and also give bounds for a simple, general technique for building geometric structures incrementally. One new algorithm reports all the intersecting pairs of a set of line segments in the plane, and requiresO(A+n logn) expected time, whereA is the number of intersecting pairs reported. The algorithm requiresO(n) space in the worst case. Another algorithm computes the convex hull ofn points inE ^d inO(n logn) expected time ford=3, andO(n ^[d/2]) expected time ford>3. The algorithm also gives fast expected times for random input points. Another algorithm computes the diameter of a set ofn points inE ³ inO(n logn) expected time, and on the way computes the intersection ofn unit balls inE ³. We show thatO(n logA) expected time suffices to compute the convex hull ofn points inE ³, whereA is the number of input points on the surface of the hull. Algorithms for halfspace range reporting are also given. In addition, we give asymptotically tight bounds for (k)-sets, which are certain halfspace partitions of point sets, and give a simple proof of Lee's bounds for high-order Voronoi diagrams.     

A Bichromatic Incidence Bound and an Application

We prove a new, tight upper bound on the number of incidences between points and hyperplanes in Euclidean d-space. Given n points, of which k are colored red, there are O _d (m ^2/3 k ^2/3 n ^(d−2)/3+kn ^d−2+m) incidences between the k red points and m hyperplanes spanned by all n points provided that m=Ω(n ^d−2). For the monochromatic case k=n, this was proved by Agarwal and Aronov (Discrete Comput. Geom. 7(4):359–369, 1992).     

Eppstein's bound on intersecting triangles revisited

Let S be a set of n points in the plane, and let T be a set of m triangles with vertices in S. Then there exists a point in the plane contained in Ω(m³/(n⁶log²n)) triangles of T. Eppstein [D. Eppstein, Improved bounds for intersecting triangles and halving planes, J. Combin. Theory Ser. A 62 (1993) 176-182] gave a proof of this claim, but there is a problem with his proof. Here we provide a correct proof by slightly modifying Eppstein's argument.     

Simple Proofs of Classical Theorems in Discrete Geometry via the Guth–Katz Polynomial Partitioning Technique

Recently Guth and Katz (arXiv:1011.4105, 2010) invented, as a step in their nearly complete solution of Erd?s??s distinct distances problem, a new method for partitioning finite point sets in ? ^d , based on the Stone?CTukey polynomial ham-sandwich theorem. We apply this method to obtain new and simple proofs of two well known results: the Szemerédi?CTrotter theorem on incidences of points and lines, and the existence of spanning trees with low crossing numbers. Since we consider these proofs particularly suitable for teaching, we aim at self-contained, expository treatment. We also mention some generalizations and extensions, such as the Pach?CSharir bound on the number of incidences with algebraic curves of bounded degree.     

Combinatorial complexity bounds for arrangements of curves and spheres

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 (m ^2/3 n ^2/3 +n), and that it isO(m ^2/3 n ^2/3 (n) +n) forn unit-circles, where(n) (and later(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(m ^3/5 n ^4/5 (n) +n). The same bounds (without the(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(m ^4/7 n ^9/7 (m, n) +n ²), in general, andO(m ^3/4 n ^3/4 (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(m ^3/2 (m)) which improves the best previous upper bound on this problem. Applications of our results to other distance problems are also given.The research of the second author was supported by the National Science Foundation under Grant CCR-8714565. Work by the fourth author has been supported by Office of Naval Research Grant N00014-87-K-0129, by National Science Foundation Grant No. 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. A preliminary version of this paper has appeared in theProceedings of the 29th IEEE Symposium on Foundations of Computer Science, 1988.     

Counting facets and incidences

We show thatm distinct cells in an arrangement ofn planes in ³ are bounded byO(m ^2/3 n+n ²) faces, which in turn yields a tight bound on the maximum number of facets boundingm cells in an arrangement ofn hyperplanes in ^d , for everyd3. In addition, the method is extended to obtain tight bounds on the maximum number of faces on the boundary of all nonconvex cells in an arrangement of triangles in ³. We also present a simpler proof of theO(m ^2/3 n ^d/3+n ^d–1) bound on the number of incidences betweenn hyperplanes in ^d andm vertices of their arrangement.Work on this paper was supported by DIMACS (Center for Discrete Mathematics and Theoretical Computer Science), a National Science Foundation Science and Technology Center Grant NSF-STC88-09648, and by NSA Grant MDA 904-89-H-2030.     

Bound reduction using pairs of linear inequalities

We describe a procedure to reduce variable bounds in mixed integer nonlinear programming (MINLP) as well as mixed integer linear programming (MILP) problems. The procedure works by combining pairs of inequalities of a linear programming (LP) relaxation of the problem. This bound reduction procedure extends the feasibility based bound reduction technique on linear functions, used in MINLP and MILP. However, it can also be seen as a special case of optimality based bound reduction, a method to infer variable bounds from an LP relaxation of the problem. For an LP relaxation with m constraints and n variables, there are O(m ²) pairs of constraints, and a naïve implementation of our bound reduction scheme has complexity O(n ³) for each pair. Therefore, its overall complexity O(m ² n ³) can be prohibitive for relatively large problems. We have developed a more efficient procedure that has complexity O(m ² n ²), and embedded it in two Open-Source solvers: one for MINLP and one for MILP. We provide computational results which substantiate the usefulness of this bound reduction technique for several instances.     

New lower bounds for Hopcroft's problem

We establish new lower bounds on the complexity of the following basic geometric problem, attributed to John Hopcroft: Given a set ofn points andm hyperplanes in $\mathbb{R}^d $ , is any point contained in any hyperplane? We define a general class ofpartitioning algorithms, and show that in the worst case, for allm andn, any such algorithm requires time Ω(n logm + n ^2/3m^2/3 + m logn) in two dimensions, or Ω(n logm + n ^5/6m^1/2 + n^1/2m^5/6 + m logn) in three or more dimensions. We obtain slightly higher bounds for the counting version of Hopcroft's problem in four or more dimensions. Our planar lower bound is within a factor of 2O(log^*(n+m)) of the best known upper bound, due to Matou?ek. Previously, the best known lower bound, in any dimension, was Ω(n logm + m logn). We develop our lower bounds in two stages. First we define a combinatorial representation of the relative order type of a set of points and hyperplanes, called amonochromatic cover, and derive lower bounds on its size in the worst case. We then show that the running time of any partitioning algorithm is bounded below by the size of some monochromatic cover. As a related result, using a straightforward adversary argument, we derive aquadratic lower bound on the complexity of Hopcroft's problem in a surprisingly powerful decision tree model of computation.     

Localized and compact data-structure for comparability graphs

We show that every comparability graph of any two-dimensional poset over n elements (a.k.a. permutation graph) can be preprocessed in O(n) time, if two linear extensions of the poset are given, to produce an O(n) space data-structure supporting distance queries in constant time. The data-structure is localized and given as a distance labeling, that is each vertex receives a label of O(logn) bits so that distance queries between any two vertices are answered by inspecting their labels only. This result improves the previous scheme due to Katz, Katz and Peleg [M. Katz, N.A. Katz, D. Peleg, Distance labeling schemes for well-separated graph classes, Discrete Applied Mathematics 145 (2005) 384-402] by a log n factor.As a byproduct, our data-structure supports all-pair shortest-path queries in O(d) time for distance-d pairs, and so identifies in constant time the first edge along a shortest path between any source and destination.More fundamentally, we show that this optimal space and time data-structure cannot be extended for higher dimension posets. More precisely, we prove that for comparability graphs of three-dimensional posets, every distance labeling scheme requires Ω(n^1/3) bit labels.     

Space-optimal, backtracking algorithms to list the minimal vertex separators of a graph

For a graph G in read-only memory on n vertices and m edges and a write-only output buffer, we give two algorithms using only O(n) rewritable space. The first algorithm lists all minimal a−b separators of G with a polynomial delay of O(nm). The second lists all minimal vertex separators of G with a cumulative polynomial delay of O(n³m).One consequence is that the algorithms can list the minimal a−b separators (and minimal vertex separators) spending O(nm) time (respectively, O(n³m) time) per object output.     

Some remarks on lengths of propositional proofs

We survey the best known lower bounds on symbols and lines in Frege and extended Frege proofs. We prove that in minimum length sequent calculus proofs, no formula is generated twice or used twice on any single branch of the proof. We prove that the number of distinct subformulas in a minimum length Frege proof is linearly bounded by the number of lines. Depthd Frege proofs ofm lines can be transformed into depthd proofs ofO(m ^d+1) symbols. We show that renaming Frege proof systems are p-equivalent to extended Frege systems. Some open problems in propositional proof length and in logical flow graphs are discussed. Supported in part by NSF grant DMS-9205181     

Sorting with networks of data structures

We consider the problem of sorting a permutation using a network of data structures as introduced by Knuth and Tarjan. In general the model as considered previously was restricted to networks that are directed acyclic graphs (DAGs) of stacks and/or queues. In this paper we study the question of which are the smallest general graphs that can sort an arbitrary permutation and what is their efficiency. We show that certain two-node graphs can sort in time Θ(n²) and no simpler graph can sort all permutations. We then show that certain three-node graphs sort in time Ω(n^3/2), and that there exist graphs of k nodes which can sort in time Θ(nlog_kn), which is optimal.     

Extremal problems on triangle areas in two and three dimensions

The study of extremal problems on triangle areas was initiated in a series of papers by Erd?s and Purdy in the early 1970s. In this paper we present new results on such problems, concerning the number of triangles of the same area that are spanned by finite point sets in the plane and in 3-space, and the number of distinct areas determined by the triangles.In the plane, our main result is an O(n44/19)=O(n^2.3158) upper bound on the number of unit-area triangles spanned by n points, which is the first breakthrough improving the classical bound of O(n7/3) from 1992. We also make progress in a number of important special cases. We show that: (i) For points in convex position, there exist n-element point sets that span Ω(nlogn) triangles of unit area. (ii) The number of triangles of minimum (nonzero) area determined by n points is at most ; there exist n-element point sets (for arbitrarily large n) that span (6/π²−o(1))n² minimum-area triangles. (iii) The number of acute triangles of minimum area determined by n points is O(n); this is asymptotically tight. (iv) For n points in convex position, the number of triangles of minimum area is O(n); this is asymptotically tight. (v) If no three points are allowed to be collinear, there are n-element point sets that span Ω(nlogn) minimum-area triangles (in contrast to (ii), where collinearities are allowed and a quadratic lower bound holds).In 3-space we prove an O(n17/7β(n))=O(n^2.4286) upper bound on the number of unit-area triangles spanned by n points, where β(n) is an extremely slowly growing function related to the inverse Ackermann function. The best previous bound, O(n8/3), is an old result of Erd?s and Purdy from 1971. We further show, for point sets in 3-space: (i) The number of minimum nonzero area triangles is at most n²+O(n), and this is worst-case optimal, up to a constant factor. (ii) There are n-element point sets that span Ω(n4/3) triangles of maximum area, all incident to a common point. In any n-element point set, the maximum number of maximum-area triangles incident to a common point is O(n4/3+ε), for any ε>0. (iii) Every set of n points, not all on a line, determines at least Ω(n2/3/β(n)) triangles of distinct areas, which share a common side.     

Caps in PG(n,q),q even,n⩾3

Letm′₂(3,q) be the largest value ofk(k<q ²+1) for which there exists a completek-cap in PG(3,q),q even. In this paper, the known upper bound onm′₂(3,q) is improved. We also describe a number of intervals, fork, for which there does not exist a completek-cap in PG(3,q),q even. These results are then used to improve the known upper bounds on the number of points of a cap in PG(n, q),q even,n?4.     

The complexity of many cells in arrangements of planes and related problems

We consider several problems involving points and planes in three dimensions. Our main results are: (i) The maximum number of faces boundingm distinct cells in an arrangement ofn planes isO(m ^2/3 n logn +n ²); we can calculatem such cells specified by a point in each, in worst-case timeO(m ^2/3 n log³ n+n ² logn). (ii) The maximum number of incidences betweenn planes andm vertices of their arrangement isO(m ^2/3 n logn+n ²), but this number is onlyO(m ^3/5– n ^4/5+2 +m+n logm), for any>0, for any collection of points no three of which are collinear. (iii) For an arbitrary collection ofm points, we can calculate the number of incidences between them andn planes by a randomized algorithm whose expected time complexity isO((m ^3/4– n ^3/4+3 +m) log² n+n logn logm) for any>0. (iv) Givenm points andn planes, we can find the plane lying immediately below each point in randomized expected timeO([m ^3/4– n ^3/4+3 +m] log² n+n logn logm) for any>0. (v) The maximum number of facets (i.e., (d–1)-dimensional faces) boundingm distinct cells in an arrangement ofn hyperplanes ind dimensions,d>3, isO(m ^2/3 n ^d/3 logn+n ^d–1). This is also an upper bound for the number of incidences betweenn hyperplanes ind dimensions andm vertices of their arrangement. The combinatorial bounds in (i) and (v) and the general bound in (ii) are almost tight.Work on this paper by the first author has been supported by Amoco Fnd. Fac. Dev. Comput. Sci. 1-6-44862 and by NSF Grant CCR-8714565. Work by the third author has been supported by Office of Naval Research Grant N00014-87-K-0129, by National Science Foundation Grant DCR-82-20085, 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. An abstract of this paper has appeared in theProceedings of the 13th International Mathematical Programming Symposium, Tokyo, 1988, p. 147.     

A Faster Katz Status Score Algorithm

A new graph theoretical algorithm to calculate Katz status scores reduces computational complexity from time O(n ³) to O(n + m). Randomly-generated graphs as well as data from a large empiric study are used to test the performance of two commercial network analysis packages (GRADAP and UCINET V), compared to the performance achieved by the authors' algorithm, implemented in Visual Basic.