N-point Set Research Articles

The maximum chromatic number of the disjointness graph of segments on n-point sets in the plane with $$n\le 16$$

The maximum chromatic number of the disjointness graph of segments on n-point sets in the plane with $$n\le 16$$

On extension to Fourier transforms

For n-point sets K in an LCA group Γ we obtain an estimate for the norm of “the best” extension operator from the space l∞(K) of bounded functions on K to the space A(Γ) of Fourier transforms. As a simple consequence our estimate implies that if K is an infinite closed subset of Γ then there does not exist a bounded linear extension operator from the space C0(K) of continuous functions on K vanishing at infinity to A(Γ). The latter result generalizes a result by Graham who considered the case of compact subsets K.

Lower Bounds for Semialgebraic Range Searching and Stabbing Problems

In the semialgebraic range searching problem, we are given a set of n points in ℝ d , and we want to preprocess the points such that for any query range belonging to a family of constant complexity semialgebraic sets (Tarski cells), all the points intersecting the range can be reported or counted efficiently. When the ranges are composed of simplices, the problem is well-understood: It can be solved using S(n) space and with Q(n) query time with \(S(n)Q(n)^d = \tilde{O}(n^d),\) where the \(\tilde{O}(\cdot)\) notation hides polylogarithmic factors and this trade-off is tight (up to n o (1) factors). In particular, there exist “low space” structures that use O(n) space with O ( n 1-1/ d } ) query time [ 8 , 25 ] and “fast query” structures that use O ( n d ) space with O (log n ) query time [ 9 ]. However, for general semialgebraic ranges, only “low space” solutions are known, but the best solutions [ 7 ] match the same trade-off curve as simplex queries, with O ( n ) space and \(\tilde{O}(n^{1-1/d})\) query time. It has been conjectured that the same could be done for the “fast query” case, but this open problem has stayed unresolved. Here, we disprove this conjecture. We give the first nontrivial lower bounds for semialgebraic range searching and other related problems. More precisely, we show that any data structure for reporting the points between two concentric circles, a problem that we call 2D annulus reporting, with Q ( n ) query time must use \(S(n)=\overset{\scriptscriptstyle o}{\Omega }(n^3/Q(n)^5)\) space, where the \(\overset{\scriptscriptstyle o}{\Omega }(\cdot)\) notation hides \(n^{o(1)}\) factors, meaning, for \(Q(n)=\log ^{O(1)}n\) , \(\overset{\scriptscriptstyle o}{\Omega }(n^3)\) space must be used. In addition, we study the problem of reporting the subset of input points in a polynomial slab defined by \(\lbrace (x,y)\in \mathbb {R}^2:P(x)\le y\le P(x)+w\rbrace\) , where \(P(x)=\sum _{i=0}^\Delta a_i x^i\) is a univariate polynomial of degree Δ and \(a_0, \ldots , a_\Delta , w\) are given at the query time, a problem that we call polynomial slab reporting. For this, we show a space lower bound of \(\overset{\scriptscriptstyle o}{\Omega }(n^{\Delta +1}/Q(n)^{(\Delta +3)\Delta /2})\) , which implies that for \(Q(n)=\log ^{O(1)}n\) , we must use \(\overset{\scriptscriptstyle o}{\Omega }(n^{\Delta +1})\) space. We also consider the dual semialgebraic stabbing problems of semialgebraic range searching and present lower bounds for them. In particular, we show that in linear space, any data structure that solves 2D annulus stabbing problems must use \(\Omega (n^{2/3})\) query time. Note that this almost matches the upper bound obtained by lifting 2D annuli to 3D. Like semialgebraic range searching, we also present lower bounds for general polynomial slab stabbing problems. Again, our lower bounds are almost tight for linear size data structures in this case.

Convex Hulls of Random Order Types

We establish the following two main results on order types of points in general position in the plane (realizable simple planar order types, realizable uniform acyclic oriented matroids of rank 3): (a) The number of extreme points in an n -point order type, chosen uniformly at random from all such order types, is on average 4+ o (1). For labeled order types, this number has average \(4- \mbox{$\frac{8}{n^2 - n +2}$}\) and variance at most 3. (b) The (labeled) order types read off a set of n points sampled independently from the uniform measure on a convex planar domain, smooth or polygonal, or from a Gaussian distribution are concentrated, i.e., such sampling typically encounters only a vanishingly small fraction of all order types of the given size. Result (a) generalizes to arbitrary dimension d for labeled order types with the average number of extreme points 2 d + o (1) and constant variance. We also discuss to what extent our methods generalize to the abstract setting of uniform acyclic oriented matroids. Moreover, our methods show the following relative of the Erdős-Szekeres theorem: for any fixed k , as n → ∞, a proportion 1 - O (1/ n ) of the n -point simple order types contain a triangle enclosing a convex k -chain over an edge. For the unlabeled case in (a), we prove that for any antipodal, finite subset of the two-dimensional sphere, the group of orientation preserving bijections is cyclic, dihedral, or one of A 4 , S 4 , or A 5 (and each case is possible). These are the finite subgroups of SO (3) and our proof follows the lines of their characterization by Felix Klein.

Testing Polynomials for Vanishing on Cartesian Products of Planar Point Sets: Collinearity Testing and Related Problems

We present subquadratic algorithms, in the algebraic decision-tree model of computation, for detecting whether there exists a triple of points, belonging to three respective sets A, B, and C of points in the plane, that satisfy a certain polynomial equation or two equations. The best known instance of such a problem is testing for the existence of a collinear triple of points in \(A\times B\times C\), a classical 3SUM-hard problem that has so far defied any attempt to obtain a subquadratic solution, whether in the (uniform) real RAM model, or in the algebraic decision-tree model. While we are still unable to solve this problem, in full generality, in subquadratic time, we obtain such a solution, in the algebraic decision-tree model, that uses roughly \(O(n^{28/15})\) constant-degree polynomial sign tests, for the special case where two of the sets lie on two respective one-dimensional curves and the third is placed arbitrarily in the plane. Our technique is fairly general, and applies to many other problems where we seek a triple that satisfies a single polynomial equation, e.g., determining whether \(A\times B\times C\) contains a triple spanning a unit-area triangle. This result extends recent work by Barba et al. (2017) and by Chan (2018), where all three sets A, B, and C are assumed to be one-dimensional. As a second application of our technique, we again have three n-point sets A, B, and C in the plane, and we want to determine whether there exists a triple \((a,b,c) \in A\times B\times C\) that simultaneously satisfies two independent real polynomial equations. For example, this is the setup when testing for collinearity in the complex plane, when each of the sets A, B, C lies on some constant-degree algebraic curve. We show that problems of this kind can be solved with roughly \(O(n^{24/13})\) constant-degree polynomial sign tests.

The density of sets containing large similar copies of finite sets

We prove that if E ⊆ ℝd (d ≥ 2) is a Lebesgue-measurable set with density larger than (n − 2)/(n − 1), then E contains similar copies of every n-point set P at all sufficiently large scales. Moreover, ‘sufficiently large’ can be taken to be uniform over all P with prescribed size, minimum separation and diameter. On the other hand, we construct an example to show that the density required to guarantee all large similar copies of n-point sets tends to 1 at a rate 1 − O(n−1/5 log n).

No-Dimensional Tverberg Theorems and Algorithms

Tverberg’s theorem states that for any kge 2 and any set Psubset {mathbb {R}}^d of at least (d+1)(k-1)+1 points in d dimensions, we can partition P into k subsets whose convex hulls have a non-empty intersection. The associated search problem of finding the partition lies in the complexity class text {CLS} = text {PPAD} cap text {PLS}, but no hardness results are known. In the colorful Tverberg theorem, the points in P have colors, and under certain conditions, P can be partitioned into colorful sets, in which each color appears exactly once and whose convex hulls intersect. To date, the complexity of the associated search problem is unresolved. Recently, Adiprasito, Bárány, and Mustafa (SODA 2019) gave a no-dimensional Tverberg theorem, in which the convex hulls may intersect in an approximate fashion. This relaxes the requirement on the cardinality of P. The argument is constructive, but does not result in a polynomial-time algorithm. We present a deterministic algorithm that finds for any n-point set Psubset {mathbb {R}}^d and any kin {2,dots ,n} in O(ndlceil {log k}rceil ) time a k-partition of P such that there is a ball of radius O((k/sqrt{n}){text {diam}}(P)) that intersects the convex hull of each set. Given that this problem is not known to be solvable exactly in polynomial time, our result provides a remarkably efficient and simple new notion of approximation. Our main contribution is to generalize Sarkaria’s method (Israel Journal Math., 1992) to reduce the Tverberg problem to the colorful Carathéodory problem (in the simplified tensor product interpretation of Bárány and Onn) and to apply it algorithmically. It turns out that this not only leads to an alternative algorithmic proof of a no-dimensional Tverberg theorem, but it also generalizes to other settings such as the colorful variant of the problem.

Area-Optimal Simple Polygonalizations: The CG Challenge 2019

We give an overview of theoretical and practical aspects of finding a simple polygon of minimum (Min-Area) or maximum (Max-Area) possible area for a given set ofnpoints in the plane. Both problems are known to beNP-hard and were the subject of the 2019 Computational Geometry Challenge, which presented the quest of finding good solutions to more than 200 instances, ranging fromn= 10 all the way ton= 1, 000, 000.

Closest Pair Algorithms in 2D space,a commentary on complexity and reductions

One of the most challenging problems in computational geometry is closest pair of points given n points. Brute force algorithms[1] and Divide and conquer[1] have been verified and the lowest complexity of attributed to latter class of algorithms, with worst case being for the former being . We propose a method of partitioning the set of n-points based on the least area rectangle that can circumscribe these points

Upper bounds for stabbing simplices by a line

It is known that for every dimension d≥2 and every k<d there exists a constant cd,k>0 such that for every n-point set X⊂Rd there exists a k-flat that intersects at least cd,knd+1−k−o(nd+1−k) of the (d−k)-dimensional simplices spanned by X. However, the optimal values of the constants cd,k are mostly unknown. The case k=0 (stabbing by a point) has received a great deal of attention.In this paper we focus on the case k=1 (stabbing by a line). Specifically, we try to determine the upper bounds yielded by two point sets, known as the stretched grid and the stretched diagonal. Even though the calculations are independent of n, they are still very complicated, so we resort to analytical and numerical software methods. We provide strong evidence that, surprisingly, for d=4,5,6 the stretched grid yields better bounds than the stretched diagonal (unlike for all cases k=0 and for the case (d,k)=(3,1), in which both point sets yield the same bound). Our experiments indicate that the stretched grid yields c4,1≤0.00457936, c5,1≤0.000405335, and c6,1≤0.0000291323.

On β-Plurality Points in Spatial Voting Games

Let V be a set of n points in mathcal R d , called voters . A point p ∈ mathcal R d is a plurality point for V when the following holds: For every q ∈ mathcal R d , the number of voters closer to p than to q is at least the number of voters closer to q than to p . Thus, in a vote where each v ∈ V votes for the nearest proposal (and voters for which the proposals are at equal distance abstain), proposal p will not lose against any alternative proposal q . For most voter sets, a plurality point does not exist. We therefore introduce the concept of β-plurality points , which are defined similarly to regular plurality points, except that the distance of each voter to p (but not to q ) is scaled by a factor β , for some constant 0&lt; β ⩽ 1. We investigate the existence and computation of β -plurality points and obtain the following results. • Define β * d := {β : any finite multiset V in mathcal R d admits a β-plurality point. We prove that β * d = √3/2, and that 1/√ d ⩽ β * d ⩽ √ 3/2 for all d ⩾ 3. • Define β ( p, V ) := sup {β : p is a β -plurality point for V }. Given a voter set V in mathcal R 2 , we provide an algorithm that runs in O ( n log n ) time and computes a point p such that β ( p , V ) ⩾ β * b . Moreover, for d ⩾ 2, we can compute a point p with β ( p , V ) ⩾ 1/√ d in O ( n ) time. • Define β ( V ) := sup { β : V admits a β -plurality point}. We present an algorithm that, given a voter set V in mathcal R d , computes an ((1-ɛ)ċ β ( V ))-plurality point in time O n 2 ɛ 3d-2 ċ log n ɛ d-1 ċ log 2 1ɛ).

An Upper Bound Asymptotically Tight for the Connectivity of the Disjointness Graph of Segments in the Plane

Let P be a set of n≥3 points in general position in the plane. The edge disjointness graph D(P) of P is the graph whose vertices are the n2 closed straight line segments with endpoints in P, two of which are adjacent in D(P) if and only if they are disjoint. In this paper we show that the connectivity of D(P) is at most 7n218+Θ(n), and that this upper bound is asymptotically tight. The proof is based on the analysis of the connectivity of D(Qn), where Qn denotes an n-point set that is almost 3-symmetric.

Invariant Metric Under Deformed Markov Embeddings with Overlapped Supports

Due to Čencov’s theorem, there exists a unique family of invariant symmetric (0, 2)-tensor fields on the space of positive probability measures on a set of n-points indexed by \(n\in \mathbb {N}\) under Markov embeddings. We deform Markov embeddings keeping sufficiency, and prove existence and uniqueness of invariant families under the embeddings.

Fine-grained Complexity Analysis of Two Classic TSP Variants

We analyze two classic variants of the T RAVELING S ALESMAN P ROBLEM ( TSP ) using the toolkit of fine-grained complexity. Our first set of results is motivated by the B ITONIC TSP problem: given a set of n points in the plane, compute a shortest tour consisting of two monotone chains. It is a classic dynamic-programming exercise to solve this problem in O ( n 2 ) time. While the near-quadratic dependency of similar dynamic programs for L ONGEST C OMMON S UBSEQUENCE and D ISCRETE F réchet D istance has recently been proven to be essentially optimal under the Strong Exponential Time Hypothesis, we show that bitonic tours can be found in subquadratic time. More precisely, we present an algorithm that solves bitonic TSP in O ( n log 2 n ) time and its bottleneck version in O ( n log 3 n ) time. In the more general pyramidal TSP problem, the points to be visited are labeled 1,… , n and the sequence of labels in the solution is required to have at most one local maximum. Our algorithms for the bitonic (bottleneck) TSP problem also work for the pyramidal TSP problem in the plane. Our second set of results concerns the popular k - OPT heuristic for TSP in the graph setting. More precisely, we study the k - OPT decision problem, which asks whether a given tour can be improved by a k - OPT move that replaces k edges in the tour by k new edges. A simple algorithm solves k - OPT in O ( n k ) time for fixed k . For 2- OPT , this is easily seen to be optimal. For k =3, we prove that an algorithm with a runtime of the form Õ( n 3−ɛ ) exists if and only if A LL -P AIRS S HORTEST P ATHS in weighted digraphs has such an algorithm. For general k - OPT , it is known that a runtime of f ( k ) · n o ( k / log k ) would contradict the Exponential Time Hypothesis. The results for k =2,3 may suggest that the actual time complexity of k - OPT is Θ ( n k ). We show that this is not the case, by presenting an algorithm that finds the best k -move in O ( n ⌊ 2 k /3 ⌋+1 ) time for fixed k ≥ 3. This implies that 4- OPT can be solved in O ( n 3 ) time, matching the best-known algorithm for 3- OPT . Finally, we show how to beat the quadratic barrier for k =2 in two important settings, namely, for points in the plane and when we want to solve 2- OPT repeatedly.

Journey to the Center of the Point Set

Let P be a set of n points in R d . For a parameter α ∈ (0,1), an α-centerpoint of P is a point p ∈ R d such that all closed halfspaces containing P also contain at least α n points of P . We revisit an algorithm of Clarkson et al. [1996] that computes (roughly) a 1/(4 d 2 )-centerpoint in Õ( d 9 ) randomized time, where Õ hides polylogarithmic terms. We present an improved algorithm that can compute centerpoints with quality arbitrarily close to 1/ d 2 and runs in randomized time Õ( d 7 ). While the improvements are (arguably) mild, it is the first refinement of the algorithm by Clarkson et al. [1996] in over 20 years. The new algorithm is simpler, and the running time bound follows by a simple random walk argument, which we believe to be of independent interest. We also present several new applications of the improved centerpoint algorithm.

Approximation algorithms for spherical k-means problem using local search scheme

In the spherical k-means problem (SKMP), which is a well-studied clustering problem in text mining, we are given an n-point set D in d-dimensional unit sphere Sd, and an integer k≤n. The goal is to find a center subset S⊂Sd with |S|≤k that minimizes the sum of cosine dissimilarity measure for each point in D to the nearest center. We prove that any γ-approximation algorithm for the k-means problem (KMP) can be adapted to the SKMP with 2γ-approximation ratio. It follows that there is a local search (18+ϵ)-approximation algorithm for the SKMP, by leveraging the classical local search (9+ϵ)-approximation algorithm for the KMP. Therefore, an interesting problem arises, that is whether there exists an approximation algorithm using local search scheme directly for the SKMP. In this paper, we present a local search approximation algorithm for the SKMP and prove its performance guarantee is (2(4+7)+ϵ). We also conduct numerical computation to show the efficiency of the local search approximation algorithm by single-swap operation in the end.

A constant FPT approximation algorithm for hard-capacitated k-means

Hard-capacitated k-means (HCKM) is one of the fundamental problems remaining open in combinatorial optimization and engineering. In HCKM, one is required to partition a given n-point set into k disjoint clusters with known capacity so as to minimize the sum of within-cluster variances. It is known to be at least APX-hard, and most of the work on it has been done from a meta heuristic or bi-criteria approximation perspective. To the best our knowledge, no constant approximation algorithm or existence proof of such an algorithm is known. As our main contribution, we propose an FPT(k) approximation algorithm with constant performance guarantee for HCKM in this paper.

An Improved Analysis of Local Search for Max-Sum Diversification

We present new techniques to analyze natural local search algorithms for several variants of the max-sum diversification problem which, in its most basic form, is as follows: given an n-point set [Formula: see text] and an integer k, select k points in X so that the sum of all of their [Formula: see text] Euclidean distances is maximized. This problem has recently received a lot of attention in the context of information retrieval and web search. We focus on distances of negative type, a class that includes Euclidean distances of unbounded dimension, as well as several other natural distances, including nonmetric ones. We prove that local search over these distances provides simple and fast polynomial-time approximation schemes (PTASs) for variants that are constrained by a matroid or even a matroid intersection, and asymptotically optimal O(1)-approximations when combining the sum-of-distances objective with a monotone submodular function.

Point sets with nontrivial automorphism group

Let be the field of complex numbers and n and k natural numbers satisfying and Let be the subset of nth symmetric product of with itself consisting of n-point sets in general position and let We describe the dimension of the points in which correspond to n-point sets whose automorphism group contains a subgroup of prime order. From this result, we prove that the points in which correspond to n-point sets with nontrivial automorphism group form a proper subset

The Steiner Subratio in Banach Spaces

For every n = 2, 3,…, the minimum of the Steiner subratio is found for n-point sets in Banach spaces, and an example of a Banach space is constructed for which this minimum is attained. An example of a Banach space for which the minimum possible Steiner subratio equals 1/2 is also constructed.