N-element Point Set Research Articles

The curse of dimensionality for the Lp-discrepancy with finite p

The Lp-discrepancy is a quantitative measure for the irregularity of distribution of an N-element point set in the d-dimensional unit-cube, which is closely related to the worst-case error of quasi-Monte Carlo algorithms for numerical integration. It's inverse for dimension d and error threshold ε∈(0,1) is the minimal number of points in [0,1)d such that the minimal normalized Lp-discrepancy is less or equal ε. It is well known, that the inverse of L2-discrepancy grows exponentially fast with the dimension d, i.e., we have the curse of dimensionality, whereas the inverse of L∞-discrepancy depends exactly linearly on d. The behavior of inverse of Lp-discrepancy for general p∉{2,∞} has been an open problem for many years. In this paper we show that the Lp-discrepancy suffers from the curse of dimensionality for all p in (1,2] which are of the form p=2ℓ/(2ℓ−1) with ℓ∈N.This result follows from a more general result that we show for the worst-case error of numerical integration in an anchored Sobolev space with anchor 0 of once differentiable functions in each variable whose first derivative has finite Lq-norm, where q is an even positive integer satisfying 1/p+1/q=1.

Peeling Sequences

Given an n-element point set in the plane, in how many ways can it be peeled off until no point remains? Only one extreme point can be removed at a time. The answer obviously depends on the point set. If the points are in convex position, there are exactly n! ways, which is the maximum number of ways for n points. But what is the minimum number?

Secure pseudorandom bit generators and point sets with low star-discrepancy

The star-discrepancy is a quantitative measure for the irregularity of distribution of a point set in the unit cube that is intimately linked to the integration error of quasi-Monte Carlo algorithms. These popular integration rules are nowadays also applied to very high-dimensional integration problems. Hence multi-dimensional point sets of reasonable size with low discrepancy are badly needed. A seminal result from Heinrich, Novak, Wasilkowski and Woźniakowski shows the existence of a positive number C such that for every dimension d there exists an N-element point set in [0,1)d with star-discrepancy of at most Cd∕N. This is a pure existence result and explicit constructions of such point sets would be very desirable. The proofs are based on random samples of N-element point sets which are difficult to realize for practical applications.In this paper we propose to use secure pseudorandom bit generators for the generation of point sets with star-discrepancy of order O(d∕N). This proposal is supported theoretically and by means of numerical experiments.

Improved dispersion bounds for modified Fibonacci lattices

We study the dispersion of point sets in the unit square; i.e. the size of the largest axes-parallel box amidst such point sets. It is known that lim infN→∞Ndisp(N,2)∈54,2, where disp(N,2) is the minimal possible dispersion for an N-element point set in the unit square. The upper bound 2 is obtained by an explicit point construction—the well-known Fibonacci lattice. In this paper we find a modification of this point set such that its dispersion is significantly lower than the dispersion of the Fibonacci lattice. Our main result will imply that lim infN→∞Ndisp(N,2)≤φ3∕5=1.894427...

Distinct distances in planar point sets with forbidden 4-point patterns

A point set in a metric space is said to have property Φ(4,5) if every 4 elements determine at least 5 distinct distances. According to an old conjecture of Erdős (1986 or earlier), a set of n points in the Euclidean plane satisfying this restriction determines Ω(n2) distinct distances. This property (restriction) is shown to be equivalent to forbidding eight 4-element patterns, πi, i=1,…,8 (described in Section 2, Lemma 1). The existence of n-element point sets without the three patterns π1,π2,π3, that determine only o(n2) distinct distances was previously known. Here we exhibit n-element point sets without the seven patterns π1,π3,π4,π5,π6,π7,π8, that determine only o(n2) distinct distances. The existence of point sets missing all eight forbidden patterns and determining only o(n2) distinct distances remains open.

Extremal distributions of discrepancy functions

The irregularities of a distribution of N points in the unit interval are often measured with various notions of discrepancy. The discrepancy function can be defined with respect to intervals of the form [0,t)⊂[0,1) or arbitrary subintervals of the unit interval. In the former case, it is a well known fact in discrepancy theory that the N-element point set in the unit interval with the lowest L2 or L∞ norm of the discrepancy function is the centered regular grid ΓN≔2n+12N:n=0,1,…,N−1. We show a stronger result on the distribution of discrepancy functions of point sets in [0,1], which basically says that the distribution of the discrepancy function of ΓN is in some sense minimal among all N-element point sets. As a consequence, we can extend the above result to rearrangement-invariant norms, including Lp, Orlicz and Lorentz norms.We study the same problem for the discrepancy notions with respect to arbitrary subintervals. In this case, we will observe that we have to deal with integrals of convolutions of functions. To this end, we prove a general upper bound on such expressions, which might be of independent interest as well.

Semi-algebraic Ramsey numbers

Given a finite point set P⊂Rd, a k-ary semi-algebraic relation E on P is a set of k-tuples of points in P determined by a finite number of polynomial equations and inequalities in kd real variables. The description complexity of such a relation is at most t if the number of polynomials and their degrees are all bounded by t. The Ramsey number Rkd,t(s,n) is the minimum N such that any N-element point set P in Rd equipped with a k-ary semi-algebraic relation E of complexity at most t contains s members such that every k-tuple induced by them is in E or n members such that every k-tuple induced by them is not in E.We give a new upper bound for Rkd,t(s,n) for k≥3 and s fixed. In particular, we show that for fixed integers d,t,sR3d,t(s,n)≤2no(1), establishing a subexponential upper bound on R3d,t(s,n). This improves the previous bound of 2nC1 due to Conlon, Fox, Pach, Sudakov, and Suk where C1 depends on d and t, and improves upon the trivial bound of 2nC2 which can be obtained by applying classical Ramsey numbers where C2 depends on s. As an application, we give new estimates for a recently studied Ramsey-type problem on hyperplane arrangements in Rd. We also study multi-color Ramsey numbers for triangles in our semi-algebraic setting, achieving some partial results.

Covering Paths for Planar Point Sets

Given n points in the plane, a covering path is a polygonal path that visits all the points. If no three points are collinear, every covering path requires at least n/2 segments, and nź1 straight line segments obviously suffice even if the covering path is required to be noncrossing. We show that every set of n points in the plane admits a (possibly self-crossing) covering path consisting of n/2+O(n/logn) straight line segments. If the path is required to be noncrossing, we prove that (1źź)n straight line segments suffice for a small constant ź>0, and we exhibit n-element point sets that require at least 5n/9źO(1) segments in every such path. Further, the analogous question for noncrossing covering trees is considered and similar bounds are obtained. Finally, it is shown that computing a noncrossing covering path for n points in the plane requires Ω(nlogn) time in the worst case.

Disjoint empty disks supported by a point set

For a planar point-set P, let D(P) be the minimum number of pairwise-disjoint empty disks such that each point in P lies on the boundary of some disk. Further define D(n) as the maximum of D(P) over all n-element point sets. Hosono and Urabe recently conjectured that $${D(n) = \lceil n/2 \rceil}$$ . Here we show that $${D(n) \geq n/2 + n/236 - O(\sqrt{n})}$$ and thereby disprove this conjecture.

On the sum of the squared multiplicities of the distances in a point set over finite spaces

We study a finite analog of a conjecture of Erd?s on the sum of the squared multiplicities of the distances determined by an n-element point set. Our result is based on an estimate of the number of hinges in spectral graphs.

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 Erdős 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 ( n 44 / 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 ( n 7 / 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 ( n log n ) triangles of unit area. (ii) The number of triangles of minimum (nonzero) area determined by n points is at most 2 3 ( n 2 − n ) ; there exist n-element point sets (for arbitrarily large n) that span ( 6 / π 2 − o ( 1 ) ) n 2 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 ( n log n ) minimum-area triangles (in contrast to (ii), where collinearities are allowed and a quadratic lower bound holds). In 3-space we prove an O ( n 17 / 7 β ( 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 ( n 8 / 3 ) , is an old result of Erdős and Purdy from 1971. We further show, for point sets in 3-space: (i) The number of minimum nonzero area triangles is at most n 2 + O ( n ) , and this is worst-case optimal, up to a constant factor. (ii) There are n-element point sets that span Ω ( n 4 / 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 ( n 4 / 3 + ε ) , for any ε > 0 . (iii) Every set of n points, not all on a line, determines at least Ω ( n 2 / 3 / β ( n ) ) triangles of distinct areas, which share a common side.

The Effect of Corners on the Complexity of Approximate Range Searching

Given an n-element point set in ℝd , the range searching problem involves preprocessing these points so that the total weight, or for our purposes the semigroup sum, of the points lying within a given query range η can be determined quickly. In e-approximate range searching we assume that η is bounded, and the sum is required to include all the points that lie within η and may additionally include any of the points lying within distance e⋅diam(η) of η’s boundary. In this paper we contrast the complexity of approximate range searching based on properties of the semigroup and range space. A semigroup (S,+) is idempotent if x+x=x for all x∈S, and it is integral if for all k≥2, the k-fold sum x+⋅⋅⋅+x is not equal to x. Recent research has shown that the computational complexity of approximate spherical range searching is significantly lower for idempotent semigroups than it is for integral semigroups in terms of the dependencies on e. In this paper we consider whether these results can be generalized to other sorts of ranges. We show that, as with integrality, allowing sharp corners on ranges has an adverse effect on the complexity of the problem. In particular, we establish lower bounds on the worst-case complexity of approximate range searching in the semigroup arithmetic model for ranges consisting of d-dimensional unit hypercubes under rigid motions. We show that for arbitrary (including idempotent) semigroups and linear space, the query time is at least $\varOmega(1/{\varepsilon }^{d-2\sqrt{d}})$. In the case of integral semigroups we prove a tighter lower bound of Ω(1/e d−2). These lower bounds nearly match existing upper bounds for arbitrary semigroups. In contrast, we show that the improvements offered by idempotence do apply to smooth convex ranges. We say that a range is smooth if at every boundary point there is an incident Euclidean sphere that lies entirely within the range whose radius is proportional to the range’s diameter. We show that for smooth ranges and idempotent semigroups, e-approximate range queries can be answered in O(log n+(1/e)(d−1)/2log (1/e)) time using O(n/e) space. We show that this is nearly tight by presenting a lower bound of Ω(log n+(1/e)(d−1)/2). This bound is in the decision-tree model and holds irrespective of space.

On distinct distances among points in general position and other related problems

A set of points in the plane is said to be in general position if no three of them are collinear and no four of them are cocircular. If a point set determines only distinct vectors, it is called parallelogram free. We show that there exist n-element point sets in the plane in general position, and parallelogram free, that determine only O(n 2/√log n) distinct distances. This answers a question of Erdős, Hickerson and Pach. We then revisit an old problem of Erdős: given any n points in the plane (or in d dimensions), how many of them can one select so that the distances which are determined are all distinct? — and provide (make explicit) some new bounds in one and two dimensions. Other related distance problems are also discussed.