N-point Set Research Articles

Enumerating Order Types for Small Point Sets with Applications

Order types are a means to characterize the combinatorial properties of a finite point configuration. In particular, the crossing properties of all straight-line segments spanned by a planar n-point set are reflected by its order type. We establish a complete and reliable data base for all possible order types of size n=10 or less. The data base includes a realizing point set for each order type in small integer grid representation. To our knowledge, no such project has been carried out before. We substantiate the usefulness of our data base by applying it to several problems in computational and combinatorial geometry. Problems concerning triangulations, simple polygonalizations, complete geometric graphs, and k-sets are addressed. This list of applications is not meant to be exhaustive. We believe our data base to be of value to many researchers who wish to examine their conjectures on small point configurations.

Geometric Minimum Spanning Trees via Well-Separated Pair Decompositions

Let S be a set of n points in ℜ d . We present an algorithm that uses the well-separated pair decomposition and computes the minimum spanning tree of S under any L p or polyhedral metric. A theoretical analysis shows that it has an expected running time of O(n log n ) for uniform point distributions; this is verified experimentally. Extensive experimental results show that this approach is practical. Under a variety of input distributions, the resulting implementation is robust and performs well for points in higher dimensional space.

Sequences of spanning trees and a fixed tree theorem

Let T S be the set of all crossing-free spanning trees of a planar n-point set S. We prove that T S contains, for each of its members T, a length-decreasing sequence of trees T 0,…, T k such that T 0= T, T k =MST( S), T i does not cross T i−1 for i=1,…, k, and k=O(log n). Here MST( S) denotes the Euclidean minimum spanning tree of the point set S. As an implication, the number of length-improving and planar edge moves needed to transform a tree T∈ T S into MST( S) is only O( nlog n). Moreover, it is possible to transform any two trees in T S into each other by means of a local and constant-size edge slide operation. Applications of these results to morphing of simple polygons are possible by using a crossing-free spanning tree as a skeleton description of a polygon.

Rim-finite, arc-free subsets of the plane

We investigate properties of rim-finite subsets of the plane (those which have topological bases whose elements have finite boundaries), which are also arc-free. Recently (see [K. Bouhjar, J.J. Dijkstra, Preprint], [K. Bouhjar, J.J. Dijkstra, J. van Mill, Topology Appl., to appear], [M.N. Charatonik, W.J. Charatonik, Comment. Math. Univ. Carolin., to appear], [D.L. Fearnley, J.W. Lamoreaux, Proc. Amer. Math. Soc., to appear] and [L.D. Loveland, S.M. Loveland, Houston J. Math. 23 (1997) 485–497]) there has been considerable research regarding n-point sets (sets which intersect each line in exactly n-points). These spaces are rim-finite (since the interior of a triangle has its boundary contained in a union of three lines, each of which has n points of the space), and our investigation provides a direction to generalize them. One of our main theorems seems to generalize all known results regarding the dimension of n-point sets (see, for example, [K. Bouhjar, J.J. Dijkstra, J. van Mill, Topology Appl., to appear], [D.L. Fearnley, J.W. Lamoreaux, Proc. Amer. Math. Soc., to appear] and [J. Kulesza, Proc. Amer. Math. Soc. 116 (1992) 551–553]), and beyond that has, as corollaries, the solutions to problems of Bouhjar and Dijkstra [Preprint], and L.D. Loveland and S.M. Loveland [Houston J. Math. 23 (1997) 485–497]. In Bouhjar and Dijkstra [Preprint] it is asked if all n-point sets which are arc-free must be zero-dimensional, and our result gives a positive answer. In [L.D. Loveland, S.M. Loveland, Houston J. Math. 23 (1997) 485–497] it is asked whether a connected 2-GM set must contain an arc, and again we give a positive answer.Another main theorem states that if X is a subset of R2 such that there is a nonnegative integer n so that every straight interval of length 1 has a local basis of open sets with boundaries which intersect X in a set of cardinality less than or equal to n, then either X is zero-dimensional or X contains an arc. We produce an example which demonstrates that, essentially, our theorem cannot be improved. The “straight interval of length 1” cannot be replaced by “point”, because our example has a base of open sets whose boundaries have cardinality less than or equal to 72 and contains no arcs, yet has dimension 1. This example seems to be the first of a positive dimensional, rim-finite and arc-free separable metric space.

ON ENUMERATING AND SELECTING DISTANCES

Given an n-point set, the problems of enumerating the k closest pairs and selecting the k-th smallest distance are revisited. For the enumeration problem, we give simpler randomized and deterministic algorithms with O(n log n+k) running time in any fixed-dimensional Euclidean space. For the selection problem, we give a randomized algorithm with running time O(n log n+n2/3k1/3 log 5/3n) in the Euclidean plane. We also describe output-sensitive results for halfspace range counting that are of use in more general distance selection problems. None of our algorithms requires parametric search.

On directional convexity

Motivated by problems from calculus of variations and partial differential equations, we investigate geometric properties of D-convexity. A function f: R d → R is called D-convex, where D is a set of vectors in R d, if its restriction to each line parallel to a nonzero v ∈ D is convex. The D-convex hull of a compact set A ⊂ R d, denoted by coD(A), is the intersection of the zero sets of all nonnegative D-convex functions that are zero on A. It also equals the zero set of the D-convex envelope of the distance function of A. We give an example of an n-point set A ⊂ R 2 where the D-convex envelope of the distance function is exponentially close to zero at points lying relatively far from coD(A), showing that the definition of the D-convex hull can be very nonrobust. For separate convexity in R 3 (where D is the orthonormal basis of R 3), we construct arbitrarily large finite sets A with coD(A) ≠ A whose proper subsets are all equal to their D-convex hull. This implies the existence of analogous sets for rank-one convexity and for quasiconvexity on 3 × 3 (or larger) matrices.

Three-point sets

We prove that a planar set which meets each line in exactly three points cannot contain a continuum and cannot be an F σ -set. We also present some results on extending and splitting n-point sets. Our results imply that there is a four-point set which contains an arc.

A Four-Point Set That Cannot Be Split

If n is an integer greater than one then an n-point set is a subset of the plane that meets every line in precisely n points. A planar set is called a partial n-point set if it meets every line in at most n points. Using a well-ordering of the real line Mazurkiewicz [5] proved that 2-point exist. This result was generalized to n-point and beyond by Bagemihl [1] and Sierpin'ski [6]. Additional information, references, and some unsolved problems concerning these peculiar can be found in [3] and [4]. If X is an n-point set that is contained in an (n + 1)-point set Y then Y X must be a set that meets each line in exactly one point. Since such one-point sets obviously do not exist it is impossible for an (n + 1)-point set to contain an n-point set. In contrast, Bouhjar et al. [2] prove that if 2 < n < k 2 then each n-point set is contained in some

No 𝑛-point set is 𝜎-compact

Let n n be an integer greater than 1. We prove that there exist no F σ F_\sigma -subsets of the plane that intersect every line in precisely n n points.

More planar two-center algorithms

This paper considers the planar Euclidean two-center problem: given a planar n-point set S, find two congruent circular disks of the smallest radius covering S. The main result is a deterministic algorithm with running time O( nlog 2 nlog 2log n), improving the previous O( nlog 9 n) bound of Sharir and almost matching the randomized O( nlog 2 n) bound of Eppstein. If a point in the intersection of the two disks is given, then we can solve the problem in O( nlog n) time with high probability.

On the Discrepancy for Boxes and Polytopes

We consider so-called Tusnady’s problem in dimension d: Given an n-point set P in R d , color the points of P red or blue in such a way that for any d-dimensional interval B, the number of red points in \(\) differs from the number of blue points in \(\) by at most Δ, where \(\) should be as small as possible. We slightly improve previous results of Beck, Bohus, and Srinivasan by showing that \(\), with a simple proof. The same asymptotic bound is shown for an analogous problem where B is allowed to be any translated and scaled copy of a fixed convex polytope A in R d . Here the constant of proportionality depends on A and we give an explicit estimate. The same asymptotic bounds also follow for the Lebesgue-measure discrepancy, which improves and simplifies results of Beck and of Karolyi.

An optimal algorithm for approximate nearest neighbor searching fixed dimensions

Consider a set of S of n data points in real d -dimensional space, R d , where distances are measured using any Minkowski metric. In nearest neighbor searching, we preprocess S into a data structure, so that given any query point q ∈ R d , is the closest point of S to q can be reported quickly. Given any positive real ϵ, data point p is a (1 +ϵ)- approximate nearest neighbor of q if its distance from q is within a factor of (1 + ϵ) of the distance to the true nearest neighbor. We show that it is possible to preprocess a set of n points in R d in O(dn log n ) time and O(dn) space, so that given a query point q ∈ R d , and ϵ &gt; 0, a (1 + ϵ)-approximate nearest neighbor of q can be computed in O ( c d , ϵ log n ) time, where c d,ϵ ≤ d ⌈1 + 6d/ϵ⌉ d is a factor depending only on dimension and ϵ. In general, we show that given an integer k ≥ 1, (1 + ϵ)-approximations to the k nearest neighbors of q can be computed in additional O(kd log n ) time.

Exact primitives for smallest enclosing ellipses

The problem of finding the unique closed ellipsoid of smallest volume enclosing an n-point set P in d-space (known as the Löwner-John ellipsoid of P (John, 1948)) is an instance of convex programming and can be solved by general methods in time O( n) if the dimension is fixed (Welzl, 1991; Matoušek et al., 1992; Dyer, 1992; Adler and Shamir, 1993). The problem-specific parts of these methods are encapsulated in primitive operations that deal with subproblems of constant size. We derive explicit formulae for the primitive operations of Welzl's randomized method (Welzl, 1991) in dimension d = 2. Compared to previous ones (Silverman and Titterington, 1980; Post, 1982; Schönherr, 1994), these formulae are simpler and faster to evaluate, and they only contain rational expressions, allowing for an exact solution. rights reserved.

On discrepancy bounds via dual shatter function

Recently, it has been shown that tight or almost tight upper bounds for the discrepancy of many geometrically denned set systems can be derived from simple combinatorial parameters of these set systems. Namely, if the primal shatter function of a set system ℛ on an n-point set X is bounded by const. md, then the discrepancy disc (ℛ) = O(n(d−1)/2d) (which is known to be tight), and if the dual shatter function is bounded by const. md, then disc We prove that for d = 2, 3, the latter bound also cannot be improved in general. We also show that bounds on the shatter functions alone do not imply the average (L1)-discrepancy to be much smaller than the maximum discrepancy: this contrasts results of Beck and Chen for certain geometric cases. In the proof we give a construction of a certain asymptotically extremal bipartite graph, which may be of independent interest.

Efficient construction of a bounded-degree spanner with low weight

LetS be a set ofn points in ℝ d and lett>1 be a real number. At-spanner forS is a graph having the points ofS as its vertices such that for any pairp, q of points there is a path between them of length at mostt times the Euclidean distance betweenp andq. An efficient implementation of a greedy algorithm is given that constructs at-spanner having bounded degree such that the total length of all its edges is bounded byO (logn) times the length of a minimum spanning tree forS. The algorithm has running timeO (n log d n). Applying recent results of Das, Narasimhan, and Salowe to thist-spanner gives anO(n log d n)-time algorithm for constructing at-spanner having bounded degree and whose total edge length is proportional to the length of a minimum spanning tree forS. Previously, noo(n 2)-time algorithms were known for constructing at-spanner of bounded degree. In the final part of the paper, an application to the problem of distance enumeration is given.

A New Subgraph of Minimum Weight Triangulations.

In this paper, two sufficient conditions for identifying a subgraph ofminimum weight triangulation of a planar point set are presented. Theseconditions are based on local geometric properties of an edge to beidentified. Unlike the previous known sufficient conditions for identifyingsubgraphs, such as Keil‘sβ-skeleton and Yang and Xu‘s double circles, The localgeometric requirement in our conditions is not necessary symmetric withrespect to the edge to be identified. The identified subgraph is differentfrom all the known subgraphs including the newly discovered subgraph:so-called the intersection of local-optimal triangulations by Dickerson etal. An O(n3) time algorithm for finding this subgraph from a set ofn points is presented.

On Linear-Time Deterministic Algorithms for Optimization Problems in Fixed Dimension

We show that with recently developed derandomization techniques, one can convert Clarkson's randomized algorithm for linear programming in fixed dimension into a linear-time deterministic algorithm. The constant of proportionality isdO(d), which is better than those for previously known algorithms. We show that the algorithm works in a fairly general abstract setting, which allows us to solve various other problems, e.g., computing the minimum-volume ellipsoid enclosing a set ofnpoints and finding the maximum volume ellipsoid in the intersection ofnhalfspaces.

Hamiltonian triangulations for fast rendering

High-performance rendering engines are often pipelined; their speed is bounded by the rate at which triangulation data can be sent into the machine. An ordering such that consecutive triangles share a face, which reduces the data rate, exists if and only if the dual graph of the triangulation contains a Hamiltonian path. We (1) show thatany set ofn points in the plane has a Hamiltonian triangulation; (2) prove that certain nondegenerate point sets do not admit asequential triangulation; (3) test whether a polygonP has a Hamiltonian triangulation in time linear in the size of its visibility graph; and (4) show how to add Steiner points to a triangulation to create Hamiltonian triangulations that avoid narrow angles.

Ramsey-remainder

We investigate the following Ramsey-type problem. Given a natural number k,determine the smallest integer rr(k)such that, if nis sufficiently large with respect to k,and Sis any set of npoints in general position in the plane, then all but at most rr(k)points of Scan be partitioned into convex sets of sizes ⩾ k.We provide estimates on rr(k)which are best possible if a classic conjecture of Erdos and Szekeres on the Ramsey number for convex sets is valid. We also prove that in several types of combinatorial structures, the corresponding ‘Ramsey-remainder’ rr(k)is equal to the off-diagonal Ramsey number r(k, k-1) minus 1.

A deterministic algorithm for the three-dimensional diameter problem

We give a deterministic algorithm for computing the diameter of an n-point set in three dimensions with O( n log c n) running time, where c is a constant.