Point Sets In Convex Position Research Articles

Algebraic k-Sets and Generally Neighborly Embeddings

Given a set S of n points in $${{\mathbb {R}}}^d$$ , a k-set is a subset of k points of S that can be strictly separated by a hyperplane from the remaining $$n-k$$ points. Similarly, one may consider k-facets, which are hyperplanes that pass through d points of S and have k points on one side. A notorious open problem is to determine the asymptotics of the maximum number of k-sets. In this paper we study a variation on the k-set/k-facet problem with hyperplanes replaced by algebraic surfaces. In stark contrast to the original k-set/k-facet problem, there are some natural families of algebraic curves for which the number of k-facets can be counted exactly. For example, we show that the number of halving conic sections for any set of $$2n+5$$ points in general position in the plane is $$2\left( {\begin{array}{c}n+2\\ 2\end{array}}\right) ^2$$ . To understand the limits of our argument we study a class of maps we call generally neighborly embeddings, which map generic point sets into neighborly position. Additionally, we give a simple argument which improves the best known bound on the number of k-sets/k-facets for point sets in convex position.

Competitive Online Routing on Delaunay Triangulations

Let [Formula: see text] be a graph, [Formula: see text] be a source node and [Formula: see text] be a target node. The sequence of adjacent nodes (graph walk) visited by a routing algorithm is a [Formula: see text]-competitive route if its length in [Formula: see text] is at most [Formula: see text] times the length of the shortest path from [Formula: see text] to [Formula: see text] in [Formula: see text]. We present a [Formula: see text]-competitive online routing algorithm on the Delaunay triangulation of an arbitrary given set of points in the plane. This improves the competitive ratio on Delaunay triangulations from the previous best of [Formula: see text]. We also present a [Formula: see text]-competitive online routing algorithm for Delaunay triangulations of point sets in convex position.

Characteristic polynomials of production matrices for geometric graphs

An n×n production matrix for a class of geometric graphs has the property that the numbers of these geometric graphs on up to n vertices can be read off from the powers of the matrix. Recently, we obtained such production matrices for non-crossing geometric graphs on point sets in convex position [Huemer, C., A. Pilz, C. Seara, and R.I. Silveira, Production matrices for geometric graphs, Electronic Notes in Discrete Mathematics 54 (2016) 301–306]. In this note, we determine the characteristic polynomials of these matrices. Then, the Cayley-Hamilton theorem implies relations among the numbers of geometric graphs with different numbers of vertices. Further, relations between characteristic polynomials of production matrices for geometric graphs and Fibonacci numbers are revealed.

Degree four plane spanners: Simpler and better

Let $\mathcal{P}$ be a set of $n$ points embedded in the plane, and let $\mathcal{C}$ be the complete Euclidean graph whose point-set is $\mathcal{P}$. Each edge in $\mathcal{C}$ between two points $p$, $q$ is realized as the line segment $[pq]$ and is assigned a weight equal to the Euclidean distance $|pq|$. In this paper, we show how to construct in $O(n \lg n)$ time a plane spanner of $\mathcal{C}$ of maximum degree at most $4$ and of stretch factor at most $20$. This improves a long sequence of results on the construction of bounded degree plane spanners of $\mathcal{C}$. Our result matches the smallest known upper bound of $4$ by Bonichon et al. on the maximum degree while significantly improving their stretch factor upper bound from $156.82$ to $20$. The construction of our spanner is based on Delaunay triangulations defined with respect to the equilateral-triangle distance, and uses a different approach than that used by Bonichon et al. Our approach leads to a simple and intuitive construction of a well-structured spanner and reveals useful structural properties of Delaunay triangulations defined with respect to the equilateral-triangle distance. The structure of the constructed spanner implies that when $\mathcal{P}$ is in convex position, the maximum degree of the spanner is at most $3$. Combining the above degree upper bound with the fact that $3$ is a lower bound on the maximum degree of any plane spanner of $\mathcal{C}$ when the point-set $\mathcal{P}$ is in convex position, the results in this paper give a tight bound of $3$ on the maximum degree of plane spanners of $\mathcal{C}$ for point-sets in convex position.

Production matrices for geometric graphs

We present production matrices for non-crossing geometric graphs on point sets in convex position, which allow us to derive formulas for the numbers of such graphs. Several known identities for Catalan numbers, Ballot numbers, and Fibonacci numbers arise in a natural way, and also new formulas are obtained, such as a formula for the number of non-crossing geometric graphs with root vertex of given degree. The characteristic polynomials of some of these production matrices are also presented. The proofs make use of generating trees and Riordan arrays.

Drawing Hamiltonian Cycles with no Large Angles

Let $n \geq 4$ be even. It is shown that every set $S$ of $n$ points in the plane can be connected by a (possibly self-intersecting) spanning tour (Hamiltonian cycle) consisting of $n$ straight-line edges such that the angle between any two consecutive edges is at most $2\pi/3$. For $n=4$ and $6$, this statement is tight. It is also shown that every even-element point set $S$ can be partitioned into at most two subsets, $S_1$ and $S_2$, each admitting a spanning tour with no angle larger than $\pi/2$. Fekete and Woeginger conjectured that for sufficiently large even $n$, every $n$-element set admits such a spanning tour. We confirm this conjecture for point sets in convex position. A much stronger result holds for large point sets randomly and uniformly selected from an open region bounded by finitely many rectifiable curves: for any $\epsilon>0$, these sets almost surely admit a spanning tour with no angle larger than $\epsilon$.

Almost all Delaunay triangulations have stretch factor greater than [formula omitted

Consider the Delaunay triangulation T of a set P of points in the plane as a Euclidean graph, in which the weight of every edge is its length. It has long been conjectured that the stretch factor in T of any pair p , p ′ ∈ P , which is the ratio of the length of the shortest path from p to p ′ in T over the Euclidean distance ‖ p p ′ ‖ , can be at most π / 2 ≈ 1.5708 . In this paper, we show how to construct point sets in convex position with stretch factor >1.5810 and in general position with stretch factor >1.5846. Furthermore, we show that a sufficiently large set of points drawn independently from any distribution will in the limit approach the worst-case stretch factor for that distribution.

Upward straight-line embeddings of directed graphs into point sets

In this paper we study the problem of computing an upward straight-line embedding of a planar DAG (directed acyclic graph) G into a point set S, i.e. a planar drawing of G such that each vertex is mapped to a point of S, each edge is drawn as a straight-line segment, and all the edges are oriented according to a common direction. In particular, we show that no biconnected DAG admits an upward straight-line embedding into every point set in convex position. We provide a characterization of the family of DAGs that admit an upward straight-line embedding into every convex point set such that the points with the largest and the smallest y-coordinate are consecutive in the convex hull of the point set. We characterize the family of DAGs that contain a Hamiltonian directed path and that admit an upward straight-line embedding into every point set in general position. We also prove that a DAG whose underlying graph is a tree does not always have an upward straight-line embedding into a point set in convex position and we describe how to construct such an embedding for a DAG whose underlying graph is a path. Finally, we give results about the embeddability of some sub-classes of DAGs whose underlying graphs are trees on point set in convex and in general position.

Weak ε-nets and interval chains

We construct weak ε-nets of almost linear size for certain types of point sets. Specifically, for planar point sets in convex position we construct weak 1/r-nets of size O(rα(r)), where α(r) denotes the inverse Ackermann function. For point sets along the moment curve in ℝ d we construct weak 1/r-nets of size r · 2 poly(α(r)) , where the degree of the polynomial in the exponent depends (quadratically) on d. Our constructions result from a reduction to a new problem, which we call stabbing interval chains with j-tuples. Given the range of integers N = [1, n], an interval chain of length k is a sequence of k consecutive, disjoint, nonempty intervals contained in N. A j-tuple $\bar{P}$ = (p1,…,pj) is said to stab an interval chain C = I 1 …I k if each p i falls on a different interval of C. The problem is to construct a small-size family Z of j-tuples that stabs all k-interval chains in N. Let z (j) k (n) denote the minimum size of such a family Z. We derive almost-tight upper and lower bounds for z (j) k (n) for every fixed j; our bounds involve functions α m (n) of the inverse Ackermann hierarchy. Specifically, we show that for j = 3 we have z (3) k (n) = Θ(nα $\lfloor$k/2$\rfloor$ (n)) for all k ≥ 6. For each j≥4, we construct a pair of functions Pʹ j (m), Qʹ j (m), almost equal asymptotically, such that z (j) Pʹ j(m)(n) = O(nα m (n)) and z (j) Qʹ j(m)(n) = Ω(nα m (n)).

Multitriangulations as Complexes of Star Polygons

Maximal $(k+1)$-crossing-free graphs on a planar point set in convex position, that is, $k$-triangulations, have received attention in recent literature, with motivation coming from several interpretations of them. We introduce a new way of looking at $k$-triangulations, namely as complexes of star polygons. With this tool we give new, direct, proofs of the fundamental properties of $k$-triangulations, as well as some new results. This interpretation also opens-up new avenues of research, that we briefly explore in the last section.

A note on a theorem of Perles concerning non-crossing paths in convex geometric graphs

During the last decade a Turán-type result of Perles about the length of the longest non-crossing paths in convex geometric graphs has been receiving some attention in the community studying geometric graphs. In this note we prove that it implies a theorem of Merino, Salazar and Urrutia about the length of the longest alternating paths for a multicoloured point set in convex position. We also give an alternative proof of Perles's theorem based on some ideas from the Merino et al. paper.

On the length of longest alternating paths for multicoloured point sets in convex position

Let P be a set of points in R 2 in general position such that each point is coloured with one of k colours. An alternating path of P is a simple polygonal whose edges are straight line segments joining pairs of elements of P with different colours. In this paper we prove the following: suppose that each colour class has cardinality s and P is the set of vertices of a convex polygon. Then P always has an alternating path with at least ( k - 1 ) s elements. Our bound is asymptotically sharp for odd values of k .

Convexity minimizes pseudo-triangulations

The number of minimum pseudo-triangulations is minimized for point sets in convex position.