Extremal problems on triangle areas in two and three dimensions

Any set of n points in convex position in the plane induces at most 2n congruent copies of a fixed isosceles triangle. Furthermore, at most 2n–4 congruent isosceles right triangles can be induced by a set of n points in convex position, and in strictly convex position at most n congruent isosceles right triangles can be induced.

On the stretch factor of Delaunay triangulations of points in convex position

In this paper we give a lower bound for the Erd\H os–Szekeres number in higher dimensions. Namely, in two different ways we construct, for every $n>d\ge 2$, a configuration of $n$ points in general position in $\R^d$ containing at most $c_d(\log n)^{d-1}$ points in convex position. (Points in $\R^d$ are in convex position if none of them lies in the convex hull of the others.)

We consider preprocessing a set S of n points in the plane that are in convex position into a data structure supporting queries of the following form: given a point q and a directed line l in the plane, report the point of S that is farthest from (or, alternatively, nearest to) the point q subject to being to the left of line l. We present two data structures for this problem. The first data structure uses O(n1+e) space and preprocessing time, and answers queries in O(21/e log n) time. The second data structure uses O(n log3n) 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(n2) space. In the process of developing the second data structure, we develop a new representation of nearest-point and farthest-point Voronoi diagrams of points in convex position. This representation supports insertion of new points in counterclockwise order using only O(log n) amortized pointer changes, subject to supporting O(log n)-time point-location queries, even though every such update may make Θ(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) pointer changes per operation while keeping O(log n)-time point-location queries.

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.

Long alternating paths in bicolored point sets

Given n red and n blue points in convex position in the plane, we show that there exists a noncrossing alternating path of length \(n+c\sqrt{n\over \log n}\). We disprove a conjecture of Erdős by constructing an example without any such path of length greater than \({4\over 3}n+c'\sqrt{n}\).KeywordsPlanar GraphOuter SegmentHamiltonian CycleAlternate PathBlue PointThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

We study several problems concerning convex polygons whose vertices lie in a Cartesian product of two sets of n real numbers (for short, grid). First, we prove that every such grid contains a convex polygon with Omega(log n) vertices and that this bound is tight up to a constant factor. We generalize this result to d dimensions (for a fixed d in N), and obtain a tight lower bound of Omega(log^{d-1}n) for the maximum number of points in convex position in a d-dimensional grid. Second, we present polynomial-time algorithms for computing the longest convex polygonal chain in a grid that contains no two points with the same x- or y-coordinate. We show that the maximum size of such a convex polygon can be efficiently approximated up to a factor of 2. Finally, we present exponential bounds on the maximum number of convex polygons in these grids, and for some restricted variants. These bounds are tight up to polynomial factors.

Lower bounds on the maximum number of non-crossing acyclic graphs

Computing farthest neighbors on a convex polytope

This paper starts the investigation of a constrained version of the point-set embed-dability problem. Let G = (V,E) be a planar graph with n vertices, G′ = (V′,E′) a subgraph of G, and S a set of n distinct points in the plane. We study the problem of computing a point-set embedding of G on S subject to the constraint that G′ is drawn with straight-line edges. Different drawing algorithms are presented that guarantee small curve complexity of the resulting drawing, i.e. a small number of bends per edge. It is proved that: • If G′ is an outerplanar graph and S is any set of points in convex position, a point-set embedding of G on S can be computed such that the edges of E\E′ have at most 4 bends each. • If S is any set of points in general position and G′ is a face of G or if it is a simple path, the curve complexity of the edges of E\E′ is at most 8. • If S is in general position and G′ is a set of k disjoint paths, the curve complexity of the edges of E \ E′ is O(2k).

Suppose there are N not necessarily distinct points on a plane in such a position that any triangle (degenerate or non-degenerate) determined by these points has perimeter length at most 1. Denote by m the number of triangles with maximal perimeter length, (briefly, the number of maximal triangles), and put f(N) = max m where the maximum is taken over all permissible configurations. At the Colloquim on Graph Theory in Calgary, 1969, P. Erdös proposed the problem of determining f(N). He conjectured that the following construction gives the maximal number: place approximately half of the points at a point A and the others at B where AB = 1/2. The aim of this note is to prove this conjecture.

On the maximum area pentagon in a planar point set

Sylvester [7] proposed the following question in 1893. If a finite set of points in a plane is such that on the line determined by any two points of the set there is always a third point of the set, is the set collinear? Equivalently, given a finite planar set of non-collinear points, does there exist a line containing exactly two of the points?

Answering range queries is a problem of fundamental importance in spatial information retrieval and computational geometry. The objective is to store a set of n points P in R d, each associated with a weight, so that it is possible to count, or more generally to compute some function of the weights of the points lying inside a given query range. Range searching is among the most heavily studied problems, and many search structures have been proposed and analyzed [1, 7]. There is a spectrum of space-time tradeoffs. The most relevant work to ours involves halfspace range counting queries, which Matouˇsek [6] has shown can be answered in n/m 1/d time from a data structure of space O(m). Nearly matching lower bounds were given by by Brönnimann, Chazelle and Pach [4] (or BCP). Given the relatively high complexity of range searching, it is natural to consider the problem in the context of approximation. We are given an approximation parameter ε> 0 and assume that ranges are bounded. Let η denote a range, and let diam(η) denote its diameter. All the points that lie in the range must be counted, and any of the points that lie within distance ε · diam(η) of the range’s boundary may be counted as well. Arya and Mount [3] showed that in any fixed dimension d with O(n log n) preprocessing time and O(n) space, ε-approximate range queries for any bounded convex range can be answered in time O(log n+1/ε d−1) [3]. Later, Chazelle, Liu, and Magen [5] considered approximate halfspace range and Euclidean ball searching in the high dimensional setting. Ignoring polylogarithmic factors, they showed that is possible to answer queries in O(d/ε 2) time with O(dn O(1/ε2) ) space.