A result about the density of iterated line intersections in the plane

Let S be a finite set of points in the plane and let be the set of intersection points between pairs of lines passing through any two points in S. We characterize all configurations of points S such that iteration of the above operation produces a dense set. We also discuss partial results on the characterization of those finite point-sets with rational coordinates that generate all of through iteration of .     

Minimum weight pseudo-triangulations

We consider the problem of computing a minimum weight pseudo-triangulation of a set of n points in the plane. We first present an -time algorithm that produces a pseudo-triangulation of weight which is shown to be asymptotically worst-case optimal, i.e., there exists a point set for which every pseudo-triangulation has weight , where is the weight of a minimum weight spanning tree of . We also present a constant factor approximation algorithm running in cubic time. In the process we give an algorithm that produces a minimum weight pseudo-triangulation of a simple polygon.     

On finding widest empty curved corridors

An -siphon of width w is the locus of points in the plane that are at the same distance w from a 1-corner polygonal chain C such that is the interior angle of C. Given a set P of n points in the plane and a fixed angle , we want to compute the widest empty -siphon that splits P into two non-empty sets. We present an efficient O(nlog³n)-time algorithm for computing the widest oriented -siphon through P such that the orientation of a half-line of C is known. We also propose an O(n³log²n)-time algorithm for the widest arbitrarily-oriented version and an Θ(nlogn)-time algorithm for the widest arbitrarily-oriented -siphon anchored at a given point.     

The minimum Manhattan network problem: Approximations and exact solutions

Given a set of points in the plane and a constant t1, a Euclidean t-spanner is a network in which, for any pair of points, the ratio of the network distance and the Euclidean distance of the two points is at most t. Such networks have applications in transportation or communication network design and have been studied extensively.

In this paper we study 1-spanners under the Manhattan (or L₁-) metric. Such networks are called Manhattan networks. A Manhattan network for a set of points is a set of axis-parallel line segments whose union contains an x- and y-monotone path for each pair of points. It is not known whether it is NP-hard to compute minimum Manhattan networks (MMN), i.e., Manhattan networks of minimum total length. In this paper we present an approximation algorithm for this problem. Given a set P of n points, our algorithm computes in O(nlogn) time and linear space a Manhattan network for P whose length is at most 3 times the length of an MMN of P.

We also establish a mixed-integer programming formulation for the MMN problem. With its help we extensively investigate the performance of our factor-3 approximation algorithm on random point sets.     

On automorphisms of generalized algebraic-geometry codes

We consider a class of generalized algebraic-geometry codes based on places of the same degree of a fixed algebraic function field over a finite field . We study automorphisms of such codes which are associated with automorphisms of .     

The last subconstituent of the Hemmeter graph

We prove that when q is any odd prime power, the distance-2 graph on the set of vertices at maximal distance D from any fixed vertex of the Hemmeter graph Hem_D(q) is isomorphic to the graph Quad_D-1(q) of quadratic forms on .     

A quadratic distance bound on sliding between crossing-free spanning trees

Let S be a set of n points in the plane and let be the set of all crossing-free spanning trees of S. We show that it is possible to transform any two trees in into each other by O(n²) local and constant-size edge slide operations. Previously no polynomial upper bound for this task was known, but in [O. Aichholzer, F. Aurenhammer, F. Hurtado, Sequences of spanning trees and a fixed tree theorem, Comput. Geom.: Theory Appl. 21 (1–2) (2002) 3–20] a bound of O(n²logn) operations was conjectured.     

Fixed and periodic points in the probabilistic normed and metric spaces

In this paper, a fixed point theorem is proved, i.e., if A is a C-contraction in the Menger space (S, F) and E  S be such that is compact, then A has a fixed point. In addition, under the same condition, the existence of a periodic point of A is proved. Finally, a fixed point theorem in probabilistic normed spaces is proved.     

Non-orientable 3-manifolds of complexity up to 7

We classify all closed non-orientable -irreducible 3-manifolds with complexity up to 7, fixing two mistakes in our previous complexity-up-to-6 classification. We show that there is no such manifold with complexity less than 6, five with complexity 6 (the four flat ones and the filled Gieseking manifold, which is of type Sol), and three with complexity 7 (one manifold of type Sol, and the two manifolds of type with smallest base orbifolds).     

Space-efficient geometric divide-and-conquer algorithms

We develop a number of space-efficient tools including an approach to simulate divide-and-conquer space-efficiently, stably selecting and unselecting a subset from a sorted set, and computing the kth smallest element in one dimension from a multi-dimensional set that is sorted in another dimension. We then apply these tools to solve several geometric problems that have solutions using some form of divide-and-conquer. Specifically, we present a deterministic algorithm running in time using extra memory given inputs of size n for the closest pair problem and a randomized solution running in expected time and using extra space for the bichromatic closest pair problem. For the orthogonal line segment intersection problem, we solve the problem in time using extra space where n is the number of horizontal and vertical line segments and k is the number of intersections.     

The minimum-area spanning tree problem

Motivated by optimization problems in sensor coverage, we formulate and study the Minimum-Area Spanning Tree (mast) problem: Given a set of n points in the plane, find a spanning tree of of minimum “area”, where the area of a spanning tree is the area of the union of the n−1 disks whose diameters are the edges in . We prove that the Euclidean minimum spanning tree of is a constant-factor approximation for mast. We then apply this result to obtain constant-factor approximations for the Minimum-Area Range Assignment (mara) problem, for the Minimum-Area Connected Disk Graph (macdg) problem, and for the Minimum-Area Tour (mat) problem. The first problem is a variant of the power assignment problem in radio networks, the second problem is a related natural problem, and the third problem is a variant of the traveling salesman problem.     

The minimum weight triangulation problem with few inner points

We look at the computational complexity of 2-dimensional geometric optimization problems on a finite point set with respect to the number of inner points (that is, points in the interior of the convex hull). As a case study, we consider the minimum weight triangulation problem. Finding a minimum weight triangulation for a set of n points in the plane is not known to be NP-hard nor solvable in polynomial time, but when the points are in convex position, the problem can be solved in O(n³) time by dynamic programming. We extend the dynamic programming approach to the general problem and describe an exact algorithm which runs in O(6^kn⁵logn) time where n is the total number of input points and k is the number of inner points. If k is taken as a parameter, this is a fixed-parameter algorithm. It also shows that the problem can be solved in polynomial time if k=O(logn). In fact, the algorithm works not only for convex polygons, but also for simple polygons with k inner points.     

Proximity problems on line segments spanned by points

Finding the closest or farthest line segment (line) from a point are fundamental proximity problems. Given a set S of n points in the plane and another point q, we present optimal O(nlogn) time, O(n) space algorithms for finding the closest and farthest line segments (lines) from q among those spanned by the points in S. We further show how to apply our techniques to find the minimum (maximum) area triangle with a vertex at q and the other two vertices in S{q} in optimal O(nlogn) time and O(n) space. Finally, we give an O(nlogn) time, O(n) space algorithm to find the kth closest line from q and show how to find the k closest lines from q in O(nlogn+k) time and O(n+k) space.     

Covering point sets with two disjoint disks or squares

We study the following problem: Given a set of red points and a set of blue points on the plane, find two unit disks C_R and C_B with disjoint interiors such that the number of red points covered by C_R plus the number of blue points covered by C_B is maximized. We give an algorithm to solve this problem in O(n^8/3log²n) time, where n denotes the total number of points. We also show that the analogous problem of finding two axis-aligned unit squares S_R and S_B instead of unit disks can be solved in O(nlogn) time, which is optimal. If we do not restrict ourselves to axis-aligned squares, but require that both squares have a common orientation, we give a solution using O(n³logn) time.     

Adaptive sampling for geometric problems over data streams

Geometric coordinates are an integral part of many data streams. Examples include sensor locations in environmental monitoring, vehicle locations in traffic monitoring or battlefield simulations, scientific measurements of earth or atmospheric phenomena, etc. This paper focuses on the problem of summarizing such geometric data streams using limited storage so that many natural geometric queries can be answered faithfully. Some examples of such queries are: report the smallest convex region in which a chemical leak has been sensed, or track the diameter of the dataset, or track the extent of the dataset in any given direction. One can also pose queries over multiple streams: for instance, track the minimum distance between the convex hulls of two data streams, report when datasets A and B are no longer linearly separable, or report when points of data stream A become completely surrounded by points of data stream B, etc. These queries are easily extended to more than two streams.

In this paper, we propose an adaptive sampling scheme that gives provably optimal error bounds for extremal problems of this nature. All our results follow from a single technique for computing the approximate convex hull of a point stream in a single pass. Our main result is this: given a stream of two-dimensional points and an integer r, we can maintain an adaptive sample of at most 2r+1 points such that the distance between the true convex hull and the convex hull of the sample points is O(D/r²), where D is the diameter of the sample set. The amortized time for processing each point in the stream is O(logr). Using the sample convex hull, all the queries mentioned above can be answered approximately in either O(logr) or O(r) time.     

Linear maps preserving idempotency of products or triple jordan products of operators

Let be the algebra of all bounded linear operators on a complex Banach space . We give the concrete forms of linear surjective maps on which preserve the nonzero idempotency of either products of two operators or triple Jordan products of two operators.     

Fractal sets determined by dilation-stable processes

Let be a dilation-stable process on . We determine a Hausdorff measure function (a) such that the fractal set X[0,1]={X(t):0t1} has positive finite -measure. We also investigate the packing measure of X[0,1].     

The number of translates of a closed nowhere dense set required to cover a Polish group

For a Polish group let be the minimal number of translates of a fixed closed nowhere dense subset of required to cover . For many locally compact this cardinal is known to be consistently larger than which is the smallest cardinality of a covering of the real line by meagre sets. It is shown that for several non-locally compact groups . For example the equality holds for the group of permutations of the integers, the additive group of a separable Banach space with an unconditional basis and the group of homeomorphisms of various compact spaces.     

Sign balance for finite groups of Lie type

A product formula for the parity generating function of the number of 1’s in invertible matrices over is given. The computation is based on algebraic tools such as the Bruhat decomposition. It is somewhat surprising that the number of such matrices with odd number of 1’s is greater than the number of those with even number of 1’s. The same technique can be used to obtain a parity generating function also for symplectic matrices over . We present also a generating function for the sum of entries of matrices over an arbitrary finite field calculated in . The Mahonian distribution appears in these formulas.     

Cauchy–Riemann equations and J-symplectic forms

Let (Σ,j) be a Riemann surface. The almost complex manifolds (M,J) for which the J-holomorphic curves :Σ→M are of variational type, are characterized. This problem is related to the existence of a vertically non-degenerate closed complex 3-form on Σ×M (see Theorem 4.3 below), which determines a family of J-symplectic structures on (M,J) parametrized by Σ.