Long Alternating Paths in Bicolored Point Sets

Long alternating paths in bicolored point sets

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

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.

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).

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.

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

We consider some problems on red and blue points in the plane lattice. An L-line segment in the plane lattice consists of a vertical line segment and a horizontal line segment having a common endpoint. There are some results on geometric graphs on a set of red and blue points in the plane. We show that some similar results also hold for a set of red and blue points in the plane lattice using L-line segments instead of line segments. For example, we show that if n red points and n blue points are given in the plane lattice in general position, then there exists a noncrossing geometric perfect matching covering them, each of whose edges is an L-line segment and connects a red point and a blue point.

Given an n-vertex outer-planar graph G and a set P of n points in the plane, we present an O(n log3n) time and O(n) space algorithm to compute a straight-line embedding of G in P, improving upon the algorithm in [GMPP91, CU96] that requires O(n2) time. Our algorithm is near-optimal as there is an Ω(n log n) lower bound for the problem [BMS95]. We present a simpler O(nd) time and O(n) space algorithm to compute a straight-line embedding of G in P where log n ≤ d ≤ 2n is the length of the longest vertex disjoint path in the dual of G. Therefore, the time complexity of the simpler algorithm varies between O(n log n) and O(n2) depending on the value of d. More efficient algorithms are presented for certain restricted cases. If the dual of G is a path, then an optimal Θ(n log n) time algorithm is presented. If the given point set is in convex position then we show that O(n) time suffices.

This paper describes the theory and algorithms of fuzzy distance transform (FDT). Fuzzy distance is defined as the length of the shortest path between two points. The length of a path in a fuzzy subset is defined as the integration of fuzzy membership values of the points along the path. The shortest path between two points is the one with the minimum length among all (infinitely many) paths between the two points. It is demonstrated that, unlike in the binary case, the shortest path in a fuzzy subset is not necessarily a straight-line segment. The support of a fuzzy subset is the set of points with nonzero membership values. It is shown that, for any fuzzy subset, fuzzy distance is a metric for the interior of its support. FDT is defined as the process on a fuzzy subset that assigns at each point the smallest fuzzy distance from the boundary of the support. The theoretical framework of FDT in continuous space is extended to digital spaces and a dynamic programming-based algorithm is presented for its computation. Several potential medical imaging applications are presented including the quantification of blood vessels and trabecular bone thickness in the regime of limited special resolution.

We present a new game, Dots & Polygons, played on a planar point set. We prove that its NP-hard and discuss strategies for the case when the point set is in convex position.