On the performance of peeling algorithms

The peeling of a d-dimensional set of points is usually performed with successive calls to a convex hull algorithm; the optimal worst-case convex hull algorithm, known to have an O(n^˙ Log (n)) execution time, may give an O(n^˙n^˙ Log (n)) to peel all the set; an O(n^˙n) convex hull algorithm, m being the number of extremal points, is shown to peel every set with an O(n-n) time, and proved to be optimal; an implementation of this algorithm is given for planar sets and spatial sets, but the latter give only an approximate O(n^˙n) performance.     

On the spectrum of critical sets in latin squares of order 2n

Suppose that L is a latin square of order m and P ? L is a partial latin square. If L is the only latin square of order m which contains P, and no proper subset of P has this property, then P is a critical set of L. The critical set spectrum problem is to determine, for a given m, the set of integers t for which there exists a latin square of order m with a critical set of size t. We outline a partial solution to the critical set spectrum problem for latin squares of order 2ⁿ. The back circulant latin square of even order m has a well‐known critical set of size m²/4, and this is the smallest known critical set for a latin square of order m. The abelian 2‐group of order 2ⁿ has a critical set of size 4ⁿ‐3ⁿ, and this is the largest known critical set for a latin square of order 2ⁿ. We construct a set of latin squares with associated critical sets which are intermediate between the back circulant latin square of order 2ⁿ and the abelian 2‐group of order 2ⁿ. © 2007 Wiley Periodicals, Inc. J Combin Designs 16: 25–43, 2008     

Regular reals

Say that α is an n‐strongly c. e. (n‐strongly computably enumerable) real if α is a sum of n many strongly c. e. reals, and that α is regular if α is n‐strongly c. e. for some n. Let Sⁿ be the set of all n‐strongly c. e. reals, Reg be the set of regular reals and CE be the set of c. e. reals. Then we have: S¹ ? S² ? · · · ? Sⁿ ? · · · ? ? Reg ? CE . This gives a hierarchy of the c. e. reals. We also study the regularity of the d. c. e. reals. (© 2005 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Polyhedral line transversals in space

Algorithms are developed for determining if a set of polyhedral objects inR ³ can be intersected by a common transversal (stabbing) line. It can be determined inO(n logn) time if a set ofn line segments in space has a line transversal, and such a transversal can be found in the same time bound. For a set of polyhedra with a total ofn vertices, we give anO(n ⁴ logn) algorithm for determining the existence of, and computing, a line transversal. Helly-type theorems for lines and segments are also given. In particular, it is shown that if every six of a set of lines in space are intersected by a common transversal, then the entire set has a common transversal.     

Polygonizations of point sets in the plane

We examine the different ways a set ofn points in the plane can be connected to form a simple polygon. Such a connection is called apolygonization of the points. For some point sets the number of polygonizations is exponential in the number of points. For this reason we restrict our attention to star-shaped polygons whose kernels have nonempty interiors; these are callednondegenerate star-shaped polygons.We develop an algorithm and data structure for determining the nondegenerate star-shaped polygonizations of a set ofn points in the plane. We do this by first constructing an arrangement of line segments from the point set. The regions in the arrangement correspond to the kernels of the nondegenerate star-shaped polygons whose vertices are the originaln points. To obtain the data structure representing this arrangement, we show how to modify data structures for arrangements of lines in the plane. This data structure can be computed inO(n ⁴) time and space. By visiting the regions in this data structure in a carefully chosen order, we can compute the polygon associated with each region inO(n) time, yielding a total computation time ofO(n ⁵) to compute a complete list ofO(n ⁴) nondegenerate star-shaped polygonizations of the set ofn points.     

A model of the coNP-complete non-Hamilton tour decision problem for directed graphs

A truncated permutation matrix polytope is defined as the convex hull of a proper subset of n-permutations represented as 0/1 matrices. We present a linear system that models the coNP-complete non-Hamilton tour decision problem based upon constructing the convex hull of a set of truncated permutation matrix polytopes. Define polytope P^n–1 as the convex hull of all n-1 by n-1 permutation matrices. Each extreme point of P^n–1 is placed in correspondence (a bijection) with each Hamilton tour of a complete directed graph on n vertices. Given any n vertex graph G_n, a polynomial sized linear system F⁽ⁿ⁾ is constructed for which the image of its solution set, under an orthogonal projection, is the convex hull of the complete set of extrema of a subset of truncated permutation matrix polytopes, where each extreme point is in correspondence with each Hamilton tour not in G_n. The non-Hamilton tour decision problem is modeled by F⁽ⁿ⁾ such that G_n is non-Hamiltonian if and only if, under an orthogonal projection, the image of the solution set of F⁽ⁿ⁾ is P^n–1. The decision problem Is the projection of the solution set of F⁽ⁿ⁾=P^n–1? is therefore coNP-complete, and this particular model of the non-Hamilton tour problem appears to be new.Dedicated to the 250+ families in Kelowna BC, who lost their homes due to forest fires in 2003.I visited Ted at his home in Kelowna during this time - his family opened their home to evacuees and we shared happy and sad times with many wonderful people.     

Some (n − 1)-dimensional rigid sets inR n

We prove that a rigid set inR ⁿ remains rigid if we remove a countable subset of its interior. This gives us a method of obtaining (n – 1)-dimensional rigid sets inR ⁿ .Recently V. A. Aleksandrov announced that he had found a 1-dimensional rigid set inR ². Our method is quite different and more general (for arbitrary dimensionn).     

A lower bound for the number of triangular embeddings of some complete graphs and complete regular tripartite graphs

We prove that, for a certain positive constant a and for an infinite set of values of n, the number of nonisomorphic triangular embeddings of the complete graph K_n is at least n^an2. A similar lower bound is also given, for an infinite set of values of n, on the number of nonisomorphic triangular embeddings of the complete regular tripartite graph K_n,n,n.     

Better lower bounds on detecting affine and spherical degeneracies

We show that in the worst case, Ω(n ^d ) sidedness queries are required to determine whether a set ofn points in ℝ ^d is affinely degenerate, i.e., whether it containsd+1 points on a common hyperplane. This matches known upper bounds. We give a straightforward adversary argument, based on the explicit construction of a point set containing Ω(n ^d ) “collapsible” simplices, any one of which can be made degenerate without changing the orientation of any other simplex. As an immediate corollary, we have an Ω(n ^d ) lower bound on the number of sidedness queries required to determine the order type of a set ofn points in ℝ ^d . Using similar techniques, we also show that Ω(n ^d+1) in-sphere queries are required to decide the existence of spherical degeneracies in a set ofn points in ℝ ^d . An earlier version of this paper was presented at the 34th Annual IEEE Symposium on Foundations of Computer Science [8]. This research has been supported by NSF Presidential Young Investigator Grant CCR-9058440.     

Diameter of a direct power of a finite group

We present two conjectures concerning the diameter of a direct power of a finite group. The first conjecture states that the diameter of Gⁿ with respect to each generating set is at most n(|G|?rank(G)); and the second one states that there exists a generating set 𝒜, of minimum size, for Gⁿ such that the diameter of Gⁿ with respect to 𝒜 is at most n(|G|?rank(G)). We will establish evidence for each of the above mentioned conjectures.     

An Improvement to “A Note on Euclidean Ramsey Theory”

We characterize those subsets Y⊆ℝ ⁿ such that for every infinite X⊆ℝ ⁿ , there is a red/blue coloring of ℝ ⁿ having no monochromatic red set similar to X and no monochramtic blue set similar to Y.     

A pre-hilbert space consisting of classes of convex sets

An equivalence relation is defined in the set of all bounded closed convex sets in Euclidean spaceE ⁿ. The equivalence classes are shown to be elements of a pre-Hilbert spaceA ⁿ, and geometrical relationships betweenA ⁿ andE ⁿ are investigated.     

Off-line dynamic maintenance of the width of a planar point set

Agarwal, P.K. and M. Sharir, Off-line dynamic maintenance of the width of a planar point set, Computational Geometry: Theory and Applications 1 (1990) 65-78. In this paper we present an efficient algorithm for the off-line dynamic maintenance of the width of a planar point set in the following restricted case: We are given a real parameter W and a sequence Σ=(σ₁,...,σ_n) of n insert and delete operations on a set S of points in ², initially consisting of n points, and we want to determine whether there is an i such that the width of S the ith operation is less than or equal to W. Our algorithm runs in time O(nlog³n) and uses O(n) space.     

A continuous analogue of Sperner's theorem

One of the best-known results of extremal combinatorics is Sperner's theorem, which asserts that the maximum size of an antichain of subsets of an n-element set equals the binomial coefficient (n/(n/2)), that is, the maximum of the binomial coefficients. In the last twenty years, Sperner's theorem has been generalized to wide classes of partially ordered sets. It is the purpose of the present paper to propose yet another generalization that strikes in a different direction. We consider the lattice Mod(n) of linear subspaces (through the origin) of the vector space Rⁿ. Because this lattice is infinite, the usual methods of extremal set theory do not apply to it. It turns out, however, that the set of elements of rank k of the lattice Mod(n), that is, the set of all subspaces of dimension k of Rⁿ, or Grassmannian, possesses an invariant measure that is unique up to a multiplicative constant. Can this multiplicative constant be chosen in such a way that an analogue of Sperner's theorem holds for Mod(n), with measures on Grassmannians replacing binomial coefficients? We show that there is a way of choosing such constants for each level of the lattice Mod(n) that is natural and unique in the sense defined below and for which an analogue of Sperner's theorem can be proven. The methods of the present note indicate that other results of extremal set theory may be generalized to the lattice Mod(n) by similar reasoning. © 1997 John Wiley & Sons, Inc.     

Toral algebraic sets and function theory on polydisks

A toral algebraic set A is an algebraic set in ℂ ⁿ whose intersection with T ⁿ is sufficiently large to determine the holomorphic functions on A. We develop the theory of these sets, and give a number of applications to function theory in several variables and operator theoretic model theory. In particular, we show that the uniqueness set for an extremal Pick problem on the bidisk is a toral algebraic set, that rational inner functions have zero sets whose irreducible components are not toral, and that the model theory for a commuting pair of contractions with finite defect lives naturally on a toral algebraic set.     

Some remarks on Legendrian rectifiable currents

A rectifiable current of dimension n−1 in the sphere bundle Sℝⁿ≃ℝⁿ×S ^{n −1} for euclidean space is Legendrian if it annihilates the contact 1-form α (i.e. T(α∧φ)=0 for all forms φ of degree n−2). Such a current may be naturally associated to any convex set or to any singular real analytic variety, and induces the curvature measures of such a set. We prove that the projection to ℝⁿ of a carrier of a general such T is C ²-rectifiable in the sense of Anzellotti–Serapioni. We deduce that the boundary of a set with positive reach, as well as its singular skeleta, are C ²-rectifiable. In case ∂T= 0 we prove also that the curvature measures associated to T satisfy the analogues of the classical variational formulas for curvature integrals. It follows that such formulas are valid for the curvature measures of subsets of space forms. Received: 3 December 1997/ Revised version: 25 May 1998     

Finding intersection of rectangles by range search

Three related rectangle intersection problems in k-dimensional space are considered: (1) find the intersections of a rectangle with a given set of rectangles, (2) find the intersecting pairs of rectangles as they are inserted into or deleted from an existing set of rectangles, and (3) find the intersecting pairs of a given set of rectangles. By transforming these problems into range search problems, one need not divide the intersection problem into two subproblems, namely, the edge-intersecting problem and the containment problem, as done by many previous studies. Furthermore, this approach can also solve these subproblems separately, if required. For the first problem the running time is O((log n)^2k−1 + s), where s is the number of intersecting pairs of rectangles. For the second problem the time needed to generate and maintain n rectangles is O(n(log n)^2k) and the time for each query is O((log n)^2k−1 + s). For the third problem the total time is O(n log n + n(log n)^2(k−1) + s) for k ≥ 1.     

Triangles of Extremal Area or Perimeter in a Finite Planar Point Set

We show the following two results on a set of n points in the plane, thus answering questions posed by Erdos and Purdy [11]: 1. The maximum number of triangles of maximum area (or of maximum perimeter) in a set of n points in the plane is exactly n . 2. The maximum possible number of triangles of minimum positive area in a set of n points in the plane is Θ(n ² ) . Received April 19, 1999, and in revised form February 29, 2000, and September 12, 2000. Online publication April 6, 2001.     

Dynamic Euclidean minimum spanning trees and extrema of binary functions

We maintain the minimum spanning tree of a point set in the plane subject to point insertions and deletions, in amortized timeO(n ^1/2 log² n) per update operation. We reduce the problem to maintaining bichromatic closest pairs, which we solve in timeO(n ^e ) per update. Our algorithm uses a novel construction, theordered nearest neighbor path of a set of points. Our results generalize to higher dimensions, and to fully dynamic algorithms for maintaining minima of binary functions, including the diameter of a point set and the bichromatic farthest pair. This research was supported in part by NSF Grant CCR-9258355     

An algorithm for outerplanar graphs with parameter

For n-vertex outerplanar graphs, it is proven that O(n^2.87) is an upper bound on the number of breakpoints of the function which gives the maximum weight of an independent set, where the vertex weights vary as linear functions of a parameter. An O(n^2.87) algorithm for finding the solution is proposed.