N-point Set Research Articles

Counting Plane Graphs: Cross-Graph Charging Schemes

We study cross-graph charging schemes for graphs drawn in the plane. These are charging schemes where charge is moved across vertices of different graphs. Such methods have recently been used to obtain various properties of triangulations that are embedded in a fixed set of points in the plane. We generalize this method to obtain results for various other types of graphs that are embedded in the plane. Specifically, we obtain a new bound ofO*(187.53N) (where theO*(⋅) notation hides polynomial factors) for the maximum number of crossing-free straight-edge graphs that can be embedded in any specific set ofNpoints in the plane (improving upon the previous best upper bound 207.85Nin Hoffmann, Schulz, Sharir, Sheffer, Tóth and Welzl [14]). We also derive upper bounds for numbers of several other types of plane graphs (such as connected and bi-connected plane graphs), and obtain various bounds on the expected vertex-degrees in graphs that are uniformly chosen from the set of all crossing-free straight-edge graphs that can be embedded in a specific point set.We then apply the cross-graph charging-scheme method to graphs that allow certain types of crossings. Specifically, we consider graphs with no set ofkpairwise crossing edges (more commonly known ask-quasi-planar graphs). Fork=3 andk=4, we prove that, for any setSofNpoints in the plane, the number of graphs that have a straight-edgek-quasi-planar embedding overSis only exponential inN.

Approximating Tverberg Points in Linear Time for Any Fixed Dimension

Let P be a d-dimensional n-point set. A Tverberg-partition of P is a partition of P into r sets P_1, ..., P_r such that the convex hulls conv(P_1), ..., conv(P_r) have non-empty intersection. A point in the intersection of the conv(P_i)'s is called a Tverberg point of depth r for P. A classic result by Tverberg implies that there always exists a Tverberg partition of size n/(d+1), but it is not known how to find such a partition in polynomial time. Therefore, approximate solutions are of interest. We describe a deterministic algorithm that finds a Tverberg partition of size n/4(d+1)^3 in time d^{O(log d)} n. This means that for every fixed dimension we can compute an approximate Tverberg point (and hence also an approximate centerpoint) in linear time. Our algorithm is obtained by combining a novel lifting approach with a recent result by Miller and Sheehy (2010).

Approximating majority depth

We consider the problem of approximating the majority depth (Liu and Singh, 1993) of a point q with respect to an n-point set, S, by random sampling. At the heart of this problem is a data structures question: How can we preprocess a set of n lines so that we can quickly test whether a randomly selected vertex in the arrangement of these lines is above or below the median level. We describe a Monte Carlo data structure for this problem that can be constructed in O(nlogn) time, can answer queries in O((logn)4/3) expected time, and answers correctly with high probability.

Sparse Euclidean Spanners with Tiny Diameter

In STOC’95, Arya et al. [1995] showed that for any set of n points in R d , a (1 + ε )-spanner with diameter at most 2 (respectively, 3) and O ( n log n ) edges (respectively, O ( n log log n ) edges) can be built in O ( n log n ) time. Moreover, it was shown in Arya et al. [1995] and Narasimhan and Smid [2007] that for any k ≥ 4, one can build in O ( n (log n )2 k α k ( n )) time a (1 + ε )-spanner with diameter at most 2 k and O ( n 2 k α k ( n )) edges. The function α k is the inverse of a certain function at the k /2 th level of the primitive recursive hierarchy, where α 0 ( n ) = n /2 , α 1 ( n ) = √ n , α 2 ( n ) = log n , α 3 ( n ) = log log n , α 4 ( n ) = log* n , α 5 ( n ) = 12 log* n , ..., etc. It is also known [Narasimhan and Smid 2007] that if one allows quadratic time, then these bounds can be improved. Specifically, for any k ≥ 4, a (1 + ε )-spanner with diameter at most k and O ( nkα k ( n )) edges can be constructed in O ( n 2 ) time [Narasimhan and Smid 2007]. A major open question in this area is whether one can construct within time O ( n log n + nkα k ( n )) a (1 + ε )-spanner with diameter at most k and O ( nkα k ( n )) edges. In this article, we answer this question in the affirmative. Moreover, in fact, we provide a stronger result. Specifically, we show that for any k ≥ 4, a (1 + ε )-spanner with diameter at most k and O ( nα k ( n )) edges can be built in optimal time O ( n log n ).

The field of definition of point sets in [formula omitted

Let K be a perfect field of characteristic not equal to two, K¯ an algebraic closure of K and let GK be the Galois group of the extension K¯/K. Let T be an n-point set in P1(K¯). The field of moduli of T is contained in each field of definition but it is not necessarily a field of definition. In this paper we show that point sets of odd cardinality n⩾5 in P1(K¯) with field of moduli K are defined over their field of moduli. We, also, show that, except for the special case of the 4-point sets, this does not hold in general for point sets of even cardinality n⩾6. Finally we prove that the following local-to-global principle holds for point sets of cardinality n⩾5: if an n-point set T in P1(Q¯) is defined over Qp for each prime p (including the prime at ∞), then it is defined over Q. From this result we derive an analogue local-to-global principle for hyperelliptic curves.

The number of distinct distances from a vertex of a convex polygon

Erd\H{o}s conjectured in 1946 that every n-point set P in convex position in the plane contains a point that determines at least floor(n/2) distinct distances to the other points of P. The best known lower bound due to Dumitrescu (2006) is 13n/36 - O(1). In the present note, we slightly improve on this result to (13/36 + eps)n - O(1) for eps = 1/23000. Our main ingredient is an improved bound on the maximum number of isosceles triangles determined by P.

Exact and Approximation Algorithms for Geometric and Capacitated Set Cover Problems

First, we study geometric variants of the standard set cover motivated by assignment of directional antenna and shipping with deadlines, providing the first known polynomial-time exact solutions. Next, we consider the following general (non-necessarily geometric) capacitated set cover problem. There is given a set of elements with real weights and a family of sets of the elements. One can use a set if it is a subset of one of the sets in the family and the sum of the weights of its elements is at most one. The goal is to cover all the elements with the allowed sets. We show that any polynomial-time algorithm that approximates the uncapacitated version of the set cover problem with ratio r can be converted to an approximation algorithm for the capacitated version with ratio r+1.357. In particular, the composition of these two results yields a polynomial-time approximation algorithm for the problem of covering a set of customers represented by a weighted n-point set with a minimum number of antennas of variable angular range and fixed capacity with ratio 2.357. This substantially improves on the best known approximation ratio for the latter antenna problem equal to 3. Furthermore, we provide a PTAS for the dual problem where the number of sets (e.g., antennas) to use is fixed and the task is to minimize the maximum set load, in case the sets correspond to line intervals or arcs. Finally, we discuss the approximability of the generalization of the antenna problem to include several base stations for antennas, and in particular show its APX-hardness already in the uncapacitated case.

Robust surface registration using N-points approximate congruent sets

Scans acquired by 3D sensors are typically represented in a local coordinate system. When multiple scans, taken from different locations, represent the same scene these must be registered to a common reference frame. We propose a fast and robust registration approach to automatically align two scans by finding two sets of N-points, that are approximately congruent under rigid transformation and leading to a good estimate of the transformation between their corresponding point clouds. Given two scans, our algorithm randomly searches for the best sets of congruent groups of points using a RANSAC-based approach. To successfully and reliably align two scans when there is only a small overlap, we improve the basic RANSAC random selection step by employing a weight function that approximates the probability of each pair of points in one scan to match one pair in the other. The search time to find pairs of congruent sets of N-points is greatly reduced by employing a fast search codebook based on both binary and multi-dimensional lookup tables. Moreover, we introduce a novel indicator of the overlapping region quality which is used to verify the estimated rigid transformation and to improve the alignment robustness. Our framework is general enough to incorporate and efficiently combine different point descriptors derived from geometric and texture-based feature points or scene geometrical characteristics. We also present a method to improve the matching effectiveness of texture feature descriptors by extracting them from an atlas of rectified images recovered from the scan reflectance image. Our algorithm is robust with respect to different sampling densities and also resilient to noise and outliers. We demonstrate its robustness and efficiency on several challenging scan datasets with varying degree of noise, outliers, extent of overlap, acquired from indoor and outdoor scenarios.

Point sets that minimize [formula omitted]-edges, 3-decomposable drawings, and the rectilinear crossing number of [formula omitted

There are two properties shared by all known crossing-minimizing geometric drawings of Kn, for n a multiple of 3. First, the underlying n-point set of these drawings minimizes the number of (≤k)-edges, that means, has exactly 3(k+22)(≤k)-edges, for all 0≤k<n/3. Second, all such drawings have the n points divided into three groups of equal size; this last property is captured under the concept of 3-decomposability. In this paper we show that these properties are closely related: every n-point set with exactly 3(k+22)(≤k)-edges for all 0≤k<n/3, is 3-decomposable. The converse, however, is easy to see that it is false. As an application, we prove that the rectilinear crossing number of K30 is 9726.

Lower bounds for weak epsilon-nets and stair-convexity

A set N is called a "weak epsilon-net" (with respect to convex sets) for a finite set X in R^d if N intersects every convex set that contains at least epsilon*|X| points of X. For every fixed d>=2 and every r>=1 we construct sets X in R^d for which every weak (1/r)-net has at least Omega(r log^{d-1} r) points; this is the first superlinear lower bound for weak epsilon-nets in a fixed dimension. The construction is a "stretched grid", i.e., the Cartesian product of d suitable fast-growing finite sequences, and convexity in this grid can be analyzed using "stair-convexity", a new variant of the usual notion of convexity. We also consider weak epsilon-nets for the diagonal of our stretched grid in R^d, d>=3, which is an "intrinsically 1-dimensional" point set. In this case we exhibit slightly superlinear lower bounds (involving the inverse Ackermann function), showing that upper bounds by Alon, Kaplan, Nivasch, Sharir, and Smorodinsky (2008) are not far from the truth in the worst case. Using the stretched grid we also improve the known upper bound for the so-called "second selection lemma" in the plane by a logarithmic factor: We obtain a set T of t triangles with vertices in an n-point set in the plane such that no point is contained in more than O(t^2 / (n^3 log (n^3/t))) triangles of T.

The number of unit distances is almost linear for most norms

We prove that there exists a norm in the plane under which no n-point set determines more than O ( n log n log log n ) unit distances. Actually, most norms have this property, in the sense that their complement is a meager set in the metric space of all norms (with the metric given by the Hausdorff distance of the unit balls).

Novel graph distance matrix

We have introduced novel distance matrix for graphs, which is based on interpretation of columns of the adjacency matrix of a graph as a set of points in n-dimensional space, n being the number of vertices in the graph. Numerical values for the distances are based on the Euclidean distance between n points in n-dimensional space. In this way, we have combined the traditional representation of graphs (drawn as 2D object of no fixed geometry) with their representation in n-dimensional space, defined by a set of n-points that lead to a representation of definite geometry. The novel distance matrix, referred to as natural distance matrix, shows some structural properties and offers novel graph invariants as molecular descriptors for structure-property-activity studies. One of the novel graph descriptors is the modified connectivity index in which the bond contribution for (m, n) bond-type is given by 1/ radical(m + n), where m and n are the valence of the end vertices of the bond. The novel distance matrix (ND) can be reduced to sparse distance-adjacency matrix (DA), which can be viewed as specially weighted adjacency matrix of a graph. The quotient of the leading eigenvalues of novel distance-adjacency matrix and novel distance matrix, as illustrated on a collection of graphs of chemical interest, show parallelism with a simple measure of graph density, based on the quotient of the number of edges in a graph and the maximal possible number of edges for graphs of the same size.

A dynamic data structure for 3-D convex hulls and 2-D nearest neighbor queries

We present a fully dynamic randomized data structure that can answer queries about the convex hull of a set of n points in three dimensions, where insertions take O (log 3 n ) expected amortized time, deletions take O (log 6 n ) expected amortized time, and extreme-point queries take O (log 2 n ) worst-case time. This is the first method that guarantees polylogarithmic update and query cost for arbitrary sequences of insertions and deletions, and improves the previous O ( n ϵ )-time method by Agarwal and Matoušek a decade ago. As a consequence, we obtain similar results for nearest neighbor queries in two dimensions and improved results for numerous fundamental geometric problems (such as levels in three dimensions and dynamic Euclidean minimum spanning trees in the plane).

Selections and hyperspaces of finite sets

It is demonstrated that the hyperspace of at most ( n + 1 ) -point sets has a Vietoris continuous selection if both the hyperspace of at most n-point sets and that of exactly ( n + 1 ) -point sets have Vietoris continuous selections. This result is applied to demonstrate that the hyperspace of at most ( 2 n + 2 ) -point sets has a Vietoris continuous selection provided that one of at most ( 2 n + 1 ) -point sets has such a selection. This settles some open questions.

Blocking Visibility for Points in General Position

For a finite set P in the plane, let b(P) be the smallest possible size of a set Q, Q∩P=∅, such that every segment with both endpoints in P contains at least one point of Q. We raise the problem of estimating b(n), the minimum of b(P) over all n-point sets P with no three points collinear. We review results providing bounds on b(n) and mention some additional observations.

Stabbing Simplices by Points and Flats

The following result was proved by Barany in 1982: For every d >= 1 there exists c_d > 0 such that for every n-point set S in R^d there is a point p in R^d contained in at least c_d n^{d+1} - O(n^d) of the simplices spanned by S. We investigate the largest possible value of c_d. It was known that c_d <= 1/(2^d(d+1)!) (this estimate actually holds for every point set S). We construct sets showing that c_d <= (d+1)^{-(d+1)}, and we conjecture this estimate to be tight. The best known lower bound, due to Wagner, is c_d >= gamma_d := (d^2+1)/((d+1)!(d+1)^{d+1}); in his method, p can be chosen as any centerpoint of S. We construct n-point sets with a centerpoint that is contained in no more than gamma_d n^{d+1}+O(n^d) simplices spanned by S, thus showing that the approach using an arbitrary centerpoint cannot be further improved. We also prove that for every n-point set S in R^d there exists a (d-2)-flat that stabs at least c_{d,d-2} n^3 - O(n^2) of the triangles spanned by S, with c_{d,d-2}>=(1/24)(1- 1/(2d-1)^2). To this end, we establish an equipartition result of independent interest (generalizing planar results of Buck and Buck and of Ceder): Every mass distribution in R^d can be divided into 4d-2 equal parts by 2d-1 hyperplanes intersecting in a common (d-2)-flat.

Traversing a Set of Points with a Minimum Number of Turns

Given a finite set of points S in źd, consider visiting the points in S with a polygonal path which makes a minimum number of turns, or equivalently, has the minimum number of segments (links). We call this minimization problem the minimum link spanning path problem. This natural problem has appeared several times in the literature under different variants. The simplest one is that in which the allowed paths are axis-aligned. Let L(S) be the minimum number of links of an axis-aligned path for S, and let Gnd be an nן×n grid in źd. Kranakis et al. (Ars Comb. 38:177---192, 1994) showed that L(Gn2)=2nź1 and $\frac{4}{3}n^{2}-O(n)\le L(G^{3}_{n})\le \frac{3}{2}n^{2}+O(n)$ and conjectured that, for all dź3, $L(G^{d}_{n})=\frac{d}{d-1}n^{d-1}\pm O(n^{d-2}).$ We prove the conjecture for d=3 by showing the lower bound for L(Gn3). For d=4, we prove that $L(G^{4}_{n})=\frac{4}{3}n^{3}\pm O(n^{5/2}).$ For general d, we give new estimates on L(Gnd) that are very close to the conjectured value. The new lower bound of $(1+\frac{1}{d})n^{d-1}-O(n^{d-2})$ improves previous result by Collins and Moret (Inf. Process. Lett. 68:317---319, 1998), while the new upper bound of $(1+\frac{1}{d-1})n^{d-1}+O(n^{d-3/2})$ differs from the conjectured value only in the lower order terms. For arbitrary point sets, we include an exact bound on the minimum number of links needed in an axis-aligned path traversing any planar n-point set. We obtain similar tight estimates (within 1) in any number of dimensions d. For the general problem of traversing an arbitrary set of points in źd with an axis-aligned spanning path having a minimum number of links, we present a constant ratio (depending on the dimension d) approximation algorithm.

Algorithms for center and Tverberg points

Given a set S of n points in R 3 , a point x in R 3 is called center point of S if every closed halfspace whose bounding hyperplane passes through x contains at least ⌈ n /4⌉ points from S . We present a near-quadratic algorithm for computing the center region , that is the set of all center points, of a set of n points in R 3 . This is nearly tight in the worst case since the center region can have Ω( n 2 ) complexity. We then consider sets S of 3 n points in the plane which are the union of three disjoint sets consisting respectively of n red, n blue, and n green points. A point x in R 2 is called a colored Tverberg point of S if there is a partition of S into n triples with one point of each color, so that x lies in all triangles spanned by these triples. We present a first polynomial-time algorithm for recognizing whether a given point is a colored Tverberg point of such a 3-colored set S .

On variants of the JohnsonLindenstrauss lemma

The JohnsonLindenstrauss lemma asserts that an n-point set in any Euclidean space can be mapped to a Euclidean space of dimension k = O(e-2 log n) so that all distances are preserved up to a multip...

Component-by-component construction of low-discrepancy point sets of small size

Abstract We investigate the problem of constructing small point sets with low star discrepancy in the s-dimensional unit cube. The size of the point set shall always be polynomial in the dimension s. Our particular focus is on extending the dimension of a given low-discrepancy point set. This results in a deterministic algorithm that constructs N-point sets with small discrepancy in a component-by-component fashion. The algorithm also provides the exact star discrepancy of the output set. Its run-time considerably improves on the run-times of the currently known deterministic algorithms that generate low-discrepancy point sets of comparable quality. We also study infinite sequences of points with infinitely many components such that all initial subsegments projected down to all finite dimensions have low discrepancy. To this end, we introduce the inverse of the star discrepancy of such a sequence, and derive upper bounds for it as well as for the star discrepancy of the projections of finite subsequences with explicitly given constants. In particular, we establish the existence of sequences whose inverse of the star discrepancy depends linearly on the dimension.