Points In Convex Position Research Articles

Realizations of Multiassociahedra via Rigidity

AbstractLet $$\Delta _{k}(n)$$ Δ k ( n ) denote the simplicial complex of $$(k+1)$$ ( k + 1 ) -crossing-free subsets of edges in $${\left( {\begin{array}{c}[n]\\ 2\end{array}}\right) }$$ [ n ] 2 . Here $$k,n\in \mathbb {N}$$ k , n ∈ N and $$n\ge 2k+1$$ n ≥ 2 k + 1 . Jonsson (2003) proved that [neglecting the short edges that cannot be part of any $$(k+1)$$ ( k + 1 ) -crossing], $$\Delta _{k}(n)$$ Δ k ( n ) is a shellable sphere of dimension $$k(n-2k-1)-1$$ k ( n - 2 k - 1 ) - 1 , and conjectured it to be polytopal. The same result and question arose in the work of Knutson and Miller (Adv Math 184(1):161-176, 2004) on subword complexes. Despite considerable effort, the only values of (k, n) for which the conjecture is known to hold are $$n\le 2k+3$$ n ≤ 2 k + 3 (Pilaud and Santos, Eur J Comb. 33(4):632–662, 2012. https://doi.org/10.1016/j.ejc.2011.12.003) and (2, 8) (Bokowski and Pilaud, On symmetric realizations of the simplicial complex of 3-crossing-free sets of diagonals of the octagon. In: Proceedings of the 21st annual Canadian conference on computational geometry, 2009). Using ideas from rigidity theory and choosing points along the moment curve we realize $$\Delta _{k}(n)$$ Δ k ( n ) as a polytope for $$(k,n)\in \{(2,9), (2,10) , (3,10)\}$$ ( k , n ) ∈ { ( 2 , 9 ) , ( 2 , 10 ) , ( 3 , 10 ) } . We also realize it as a simplicial fan for all $$n\le 13$$ n ≤ 13 and arbitrary k, except the pairs (3, 12) and (3, 13). Finally, we also show that for $$k\ge 3$$ k ≥ 3 and $$n\ge 2k+6$$ n ≥ 2 k + 6 no choice of points can realize $$\Delta _{k}(n)$$ Δ k ( n ) via bar-and-joint rigidity with points along the moment curve or, more generally, via cofactor rigidity with arbitrary points in convex position.

Angle-monotonicity of theta-graphs for points in convex position

Angle-monotonicity of theta-graphs for points in convex position

Improved stretch factor of Delaunay triangulations of points in convex position

Improved stretch factor of Delaunay triangulations of points in convex position

On path-greedy geometric spanners

A t-spanner is a subgraph of a graph G in which the length of the shortest path between two vertices never exceeds t times the length of the shortest path between them in G. A geometric graph is one whose vertices are points and whose edges are line segments between the corresponding points. Geometric t-spanners are t-spanners of the complete geometric graph on a given point set. Besides approximating the distance between points, we may ask a geometric t-spanner to be planar, have low degree, or low total edge length.One famous algorithm used to generate spanners is path-greedy, which scans pairs of vertices in non-decreasing order of edge length and adds the edge between them unless the current set of added edges already connects them with a path that t-approximates the edge length. Graphs from this algorithm are called path-greedy spanners. This work analyzes properties of path-greedy geometric spanners under different conditions. Specifically, we answer an open problem regarding the planarity and degree of path-greedy t-spanners for points in convex position in 2D. Further, we show a simple and efficient way to reduce the degree of a geometric spanner by adding extra points.

Enumeration of bipartite non-crossing geometric graphs

By using the Riordan array method together with combinatorial species theory, we study enumeration problems for geometric graphs which are connected bipartite non-crossing graphs with n + 1 points in convex position. It is performed from two different points of view, one is by the distance between two vertices 1 and n + 1 , and the other one is by the degree of the vertex n + 1 . As a result, we obtain a production matrix for such geometric graphs, and a formula for the number of connected bipartite graphs, which gives an answer to an open question about the connected bipartite non-crossing graphs.

Structural properties of bichromatic non-crossing matchings

Given a set of n red and n blue points in the plane, we are interested in matching red points with blue points by straight line segments so that the segments do not cross. We develop a range of tools for dealing with the non-crossing matchings of points in convex position. It turns out that the points naturally partition into groups that we refer to as orbits, with a number of properties that prove useful for studying and efficiently processing the non-crossing matchings.Bottleneck matching is a matching that minimizes the length of the longest segment. Illustrating the use of the developed tools, we solve the problem of finding bottleneck matchings of points in convex position in O(n2) time. Subsequently, combining our tools with a geometric analysis we design an O(n)-time algorithm for the case where the given points lie on a circle. The best previously known running times were O(n3) for points in convex position, and O(nlogn) for points on a circle.

Spanning Properties of Theta–Theta-6

We show that, unlike the Yao–Yao graph $$YY_6$$, the Theta–Theta graph $${\varTheta }{\varTheta }_6$$ defined by six cones is a spanner for sets of points in convex position. We also show that, for sets of points in non-convex position, the spanning ratio of $${\varTheta }{\varTheta }_6$$ is unbounded.

Sublinear Explicit Incremental Planar Voronoi Diagrams

A data structure is presented that explicitly maintains the graph of a Voronoi diagram of $N$ point sites in the plane or the dual graph of a convex hull of points in three dimensions while allowing insertions of new sites/points. Our structure supports insertions in $\tilde O (N^{3/4})$ expected amortized time, where $\tilde O$ suppresses polylogarithmic terms. This is the first result to achieve sublinear time insertions; previously it was shown by Allen et al. that $\Theta(\sqrt{N})$ amortized combinatorial changes per insertion could occur in the Voronoi diagram but a sublinear-time algorithm was only presented for the special case of points in convex position.

Flip distance to some plane configurations

We study an old geometric optimization problem in the plane. Given a perfect matching M on a set of n points in the plane, we can transform it to a non-crossing perfect matching by a finite sequence of flip operations. The flip operation removes two crossing edges from M and adds two non-crossing edges. Let f(M) and F(M) denote the minimum and maximum lengths of a flip sequence on M, respectively. It has been proved by Bonnet and Miltzow (2016) that f(M)=O(n2) and by van Leeuwen and Schoone (1980) that F(M)=O(n3). We prove that f(M)=O(nΔ) where Δ is the spread of the point set, which is defined as the ratio between the longest and the shortest pairwise distances. This improves the previous bound for point sets with sublinear spread. For a matching M on n points in convex position we prove that f(M)=n/2−1 and F(M)=(n/22); these bounds are tight.Any bound on F(⋅) carries over to the bichromatic setting, while this is not necessarily true for f(⋅). Let M′ be a bichromatic matching. The best known upper bound for f(M′) is the same as for F(M′), which is essentially O(n3). We prove that f(M′)⩽n−2 for points in convex position, and f(M′)=O(n2) for semi-collinear points.The flip operation can also be defined on spanning trees. For a spanning tree T on a convex point set we show that f(T)=O(nlog⁡n).

More Turán-Type Theorems for Triangles in Convex Point Sets

We study the following family of problems: Given a set of $n$ points in convex position, what is the maximum number triangles one can create having these points as vertices while avoiding certain sets of forbidden configurations. As forbidden configurations we consider all 8 ways in which a pair of triangles in such a point set can interact. This leads to 256 extremal Turán-type questions. We give nearly tight (within a $\log n$ factor) bounds for 248 of these questions and show that the remaining 8 questions are all asymptotically equivalent to Stein's longstanding tripod packing problem.

Perfect k-Colored Matchings and (k+2)-Gonal Tilings

We derive a simple bijection between geometric plane perfect matchings on 2n points in convex position and triangulations on n+2 points in convex position. We then extend this bijection to monochromatic plane perfect matchings on periodically k-colored vertices and (k+2)-gonal tilings of convex point sets. These structures are related to a generalization of Temperley–Lieb algebras and our bijections provide explicit one-to-one relations between matchings and tilings. Moreover, for a given element of one class, the corresponding element of the other class can be computed in linear time.

New results on production matrices for geometric graphs

We present novel production matrices for non-crossing partitions, connected geometric graphs, and k-angulations, which provide another way of counting the number of such objects. For instance, a formula for the number of connected geometric graphs with given root degree, drawn on a set of n points in convex position in the plane, is presented. Further, we find the characteristic polynomials and we provide a characterization of the eigenvectors of the production matrices.

Reconstruction of the path graph

Let P be a set of n≥5 points in convex position in the plane. The path graph G(P) of P is an abstract graph whose vertices are non-crossing spanning paths of P, such that two paths are adjacent if one can be obtained from the other by deleting an edge and adding another edge.We prove that the automorphism group of G(P) is isomorphic to Dn, the dihedral group of order 2n. The heart of the proof is an algorithm that first identifies the vertices of G(P) that correspond to boundary paths of P, where the identification is unique up to an automorphism of K(P) as a geometric graph, and then identifies (uniquely) all edges of each path represented by a vertex of G(P). The complexity of the algorithm is O(Nlog⁡N) where N is the number of vertices of G(P).

Data Structures for Halfplane Proximity Queries and Incremental Voronoi Diagrams

We consider preprocessing a set S of n points in convex position in the plane into a data structure supporting queries of the following form: given a point q and a directed line $$\ell $$ in the plane, report the point of S that is farthest from (or, alternatively, nearest to) the point q among all points to the left of line $$\ell $$ . We present two data structures for this problem. The first data structure uses $$O(n^{1+\varepsilon })$$ space and preprocessing time, and answers queries in $$O(2^{1/\varepsilon }\log n)$$ time, for any $$0< \varepsilon < 1$$ . The second data structure uses $$O(n \log ^3 n)$$ space and polynomial preprocessing time, and answers queries in $$O(\log n)$$ time. These are the first solutions to the problem with $$O(\log n)$$ query time and $$o(n^2)$$ space. The second data structure uses a new representation of nearest- and farthest-point Voronoi diagrams of points in convex position. This representation supports the insertion of new points in clockwise order using only $$O(\log n)$$ amortized pointer changes, in addition to $$O(\log n)$$ -time point-location queries, even though every such update may make $$\Theta (n)$$ combinatorial changes to the Voronoi diagram. This data structure is the first demonstration that deterministically and incrementally constructed Voronoi diagrams can be maintained in o(n) amortized pointer changes per operation while keeping $$O(\log n)$$ -time point-location queries.

A SAT attack on the Erdős–Szekeres conjecture

A classical conjecture of Erdős and Szekeres states that, for every integer k≥2, every set of 2k−2+1 points in the plane in general position contains k points in convex position. In 2006, Peters and Szekeres introduced the following stronger conjecture: every red-blue coloring of the edges of the ordered complete 3-uniform hypergraph on 2k−2+1 vertices contains an ordered subhypergraph with k vertices and k−2 edges, which is a union of a red monotone path and a blue monotone path that are vertex disjoint except for their two common end-vertices.Applying a state of art SAT solver, we refute the conjecture of Peters and Szekeres. We also apply techniques of Erdős, Tuza, and Valtr to refine the Erdős–Szekeres conjecture in order to tackle it with SAT solvers.

Faster bottleneck non-crossing matchings of points in convex position

Given an even number of points in a plane, we are interested in matching all the points by straight line segments so that the segments do not cross. Bottleneck matching is a matching that minimizes the length of the longest segment. For points in convex position, we present a quadratic-time algorithm for finding a bottleneck non-crossing matching, improving upon the best previously known algorithm of cubic time complexity.

A plane 1.88-spanner for points in convex position

Let $S$ be a set of $n$ points in the plane that is in convex position. For a real number $t>1$, we say that a point $p$ in $S$ is $t$-good if for every point $q$ of $S$, the shortest-path distance between $p$ and $q$ along the boundary of the convex hull of $S$ is at most $t$ times the Euclidean distance between $p$ and $q$. We prove that any point that is part of (an approximation to) the diameter of $S$ is $1.88$-good. Using this, we show how to compute a plane $1.88$-spanner of $S$ in $O(n)$ time, assuming that the points of $S$ are given in sorted order along their convex hull. Previously, the best known stretch factor for plane spanners was $1.998$ (which, in fact, holds for any point set, i.e., even if it is not in convex position).

An Improved Upper Bound for the Erdős–Szekeres Conjecture

Let ES(n) denote the minimum natural number such that every set of ES(n) points in general position in the plane contains n points in convex position. In 1935, Erdős and Szekeres proved that ES\((n) \le {2n-4 \atopwithdelims ()n-2}+1\). In 1961, they obtained the lower bound \(2^{n-2}+1 \le \mathrm{ES}(n)\), which they conjectured to be optimal. In this paper, we prove that $$\begin{aligned} \mathrm{ES}(n) \le \Big (\begin{array}{c}{2n-5} \\ {n-2}\end{array}\Big ) -\Big (\begin{array}{c}{2n-8} \\ {n-3}+2\end{array}\Big ) \approx \frac{7}{16} \Big (\begin{array}{c}{2n-4}\\ {n-2}\end{array}\Big ). \end{aligned}$$

Erdős–Szekeres Without Induction

Let ES(n) be the minimal integer such that any set of ES(n) points in the plane in general position contains n points in convex position. The problem of estimating ES(n) was first formulated by Erd?s and Szekeres (Compos Math 2: 463---470, 1935), who proved that $$ES(n) \le \left( {\begin{array}{c}2n-4\\ n-2\end{array}}\right) +1$$ES(n)≤2n-4n-2+1. The current best upper bound, $$\lim \sup _{n \rightarrow \infty } \tfrac{ES(n)}{\left( {\begin{array}{c}2n-5\\ n-2\end{array}}\right) }\le \tfrac{29}{32}$$limsupn??ES(n)2n-5n-2≤2932, is due to Vlachos (On a conjecture of Erd?s and Szekeres, 2015). We improve this to $$\begin{aligned} \lim \sup _{n \rightarrow \infty } \tfrac{ES(n)}{\left( {\begin{array}{c}2n-5\\ n-2\end{array}}\right) }\le \tfrac{7}{8}. \end{aligned}$$limsupn??ES(n)2n-5n-2≤78.

A SAT attack on the Erdős–Szekeres conjecture

A classical conjecture of Erdős and Szekeres states that every set of 2k−2+1 points in the plane in general position contains k points in convex position. In 2006, Peters and Szekeres introduced the following stronger conjecture: every red-blue coloring of the edges of the ordered complete 3-uniform hypergraph on 2k−2+1 vertices contains an ordered k-vertex hypergraph consisting of a red and a blue monotone path that are vertex disjoint except for the common end-vertices.Applying the state of art SAT solver, we refute the conjecture of Peters and Szekeres. We also apply techniques of Erdős, Tuza, and Valtr to refine the Erdős–Szekeres conjecture in order to tackle it with SAT solvers.