Convex polygons in Cartesian products

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.

Extremal problems on triangle areas in two and three dimensions

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.

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

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

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.

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).

Given a sequence of ordered convex polygons in which the adjacent polygons may intersect with each other, but the nonadjacent polygons do not intersect, a start point s, and an end point t in the plane, our goal is to obtain a shortest path that starts from s, visits each given polygon in order, and ends at t finally. We converted the touring polygons problem into the problem of computing the shortest path of visiting the disjoint line segments by analyzing the geometrical features of the given convex polygons, and preprocessing the intersection points of the jointed polygons, and using a forward partition process combined with a backward search process for finding the access edge of each convex polygon. Thus, we proposed an 0(max{n, klog⁁2k}) algorithm for solving the original problem, where n is the total number of vertices of the given polygons and k is the total number of polygons.

Let P be a convex polygon in the plane with n vertices and let Q be a convex polygon with m vertices. We prove that the maximum number of combinatorially distinct placements of Q with respect to P under translations is O(n 2+m 2+min(nm 2+n 2m)), and we give an example showing that this bound is tight in the worst case. Second, we present an O((n+m) log(n+m)) algorithm for determining a translation of Q that maximizes the area of overlap of P and Q.We also prove that the placement of Q that makes the centroids of Q and P coincide realizes an overlap of at least 9/25 of the maximum possible overlap. As an upper bound, we show an example where the overlap in this placement is 4/9 of the maximum possible overlap.

This paper analyzes the number of polygons generated by map overlay. Overlay of maps yields a number of small polygons, and often causes problems in computation, handling, and storage of derived polygons. One method to deal with these problems is to estimate the number of derived polygons before executing overlay operation. To this end, a stochastic model is proposed by which the expected number of derived polygons is calculated. The results are summarized as follows: (1) the number of polygons generated by map overlay depends on the number of original polygons and their perimeters; (2) the number of derived polygons increases with the perimeters of original polygons; (3) McAlpine and Cook's (1971) earlier method underestimated the number of derived polygons; and (4) the number of polygons generated by the overlay of maps having the same lattice system is proportional to the square of the number of overlaid maps and the number of polygons on the map, and inversely proportional to the area‐perimeter ratio of the unit cell.

Minimum convex partition of a constrained point set