arXiv subjects

Otfried Cheong

Publications and source records attributed to Otfried Cheong.

34 records · Page 2Linked to original sources

arXiv2013

On the Number of Edges of Fan-Crossing Free Graphs

A graph drawn in the plane with n vertices is k-fan-crossing free for k > 1 if there are no k+1 edges $g,e_1,...e_k$, such that $e_1,e_2,...e_k$ have a common endpoint and $g$ crosses all $e_i$. We prove a tight bound of 4n-8 on the maximum number of edges of a 2-fan-crossing free graph, and a tight 4n-9 bound for a straight-edge drawing. For k > 2, we prove an upper bound of 3(k-1)(n-2) edges. We also discuss generalizations to monotone graph properties.

cs.CG↗

arXiv2012

The Cost of Bounded Curvature

We study the motion-planning problem for a car-like robot whose turning radius is bounded from below by one and which is allowed to move in the forward direction only (Dubins car). For two robot configurations $σ, σ'$, let $\ell(σ, σ')$ be the shortest bounded-curvature path from $σ$ to $σ'$. For $d \geq 0$, let $\ell(d)$ be the supremum of $\ell(σ, σ')$, over all pairs $(σ, σ')$ that are at Euclidean distance $d$. We study the function $\dub(d) = \ell(d) - d$, which expresses the difference between the bounded-curvature path length and the Euclidean distance of its endpoints. We show that $\dub(d)$ decreases monotonically from $\dub(0) = 7π/3$ to $\dub(\ds) = 2π$, and is constant for $d \geq \ds$. Here $\ds \approx 1.5874$. We describe pairs of configurations that exhibit the worst-case of $\dub(d)$ for every distance $d$.

cs.CG↗

arXiv2012

A Generalization of the Convex Kakeya Problem

Given a set of line segments in the plane, not necessarily finite, what is a convex region of smallest area that contains a translate of each input segment? This question can be seen as a generalization of Kakeya's problem of finding a convex region of smallest area such that a needle can be rotated through 360 degrees within this region. We show that there is always an optimal region that is a triangle, and we give an optimal Θ(n log n)-time algorithm to compute such a triangle for a given set of n segments. We also show that, if the goal is to minimize the perimeter of the region instead of its area, then placing the segments with their midpoint at the origin and taking their convex hull results in an optimal solution. Finally, we show that for any compact convex figure G, the smallest enclosing disk of G is a smallest-perimeter region containing a translate of every rotated copy of G.

cs.CG↗

arXiv2012

Single-Source Dilation-Bounded Minimum Spanning Trees

Given a set $S$ of points in the plane, a geometric network for $S$ is a graph $G$ with vertex set $S$ and straight edges. We consider a broadcasting situation, where one point $r \in S$ is a designated source. Given a dilation factor $δ$, we ask for a geometric network $G$ such that for every point $v \in S$ there is a path from $r$ to $v$ in $G$ of length at most $δ|rv|$, and such that the total edge length is minimized. We show that finding such a network of minimum total edge length is NP-hard, and give an approximation algorithm.

cs.CG↗

arXiv2010

Farthest-Polygon Voronoi Diagrams

Given a family of k disjoint connected polygonal sites in general position and of total complexity n, we consider the farthest-site Voronoi diagram of these sites, where the distance to a site is the distance to a closest point on it. We show that the complexity of this diagram is O(n), and give an O(n log^3 n) time algorithm to compute it. We also prove a number of structural properties of this diagram. In particular, a Voronoi region may consist of k-1 connected components, but if one component is bounded, then it is equal to the entire region.

cs.CG↗

arXiv2010

Reverse nearest neighbor queries in fixed dimension

Reverse nearest neighbor queries are defined as follows: Given an input point-set P, and a query point q, find all the points p in P whose nearest point in P U {q} \ {p} is q. We give a data structure to answer reverse nearest neighbor queries in fixed-dimensional Euclidean space. Our data structure uses O(n) space, its preprocessing time is O(n log n), and its query time is O(log n).

cs.CG↗

arXiv2010

Reachability by Paths of Bounded Curvature in a Convex Polygon

Let $B$ be a point robot moving in the plane, whose path is constrained to forward motions with curvature at most one, and let $P$ be a convex polygon with $n$ vertices. Given a starting configuration (a location and a direction of travel) for $B$ inside $P$, we characterize the region of all points of $P$ that can be reached by $B$, and show that it has complexity $O(n)$. We give an $O(n^2)$ time algorithm to compute this region. We show that a point is reachable only if it can be reached by a path of type CCSCS, where C denotes a unit circle arc and S denotes a line segment.

cs.CG↗

arXiv2010

Lines pinning lines

A line g is a transversal to a family F of convex polytopes in 3-dimensional space if it intersects every member of F. If, in addition, g is an isolated point of the space of line transversals to F, we say that F is a pinning of g. We show that any minimal pinning of a line by convex polytopes such that no face of a polytope is coplanar with the line has size at most eight. If, in addition, the polytopes are disjoint, then it has size at most six. We completely characterize configurations of disjoint polytopes that form minimal pinnings of a line.

math.MG↗

arXiv2009

Set Systems and Families of Permutations with Small Traces

We study the maximum size of a set system on $n$ elements whose trace on any $b$ elements has size at most $k$. We show that if for some $b \ge i \ge 0$ the shatter function $f_R$ of a set system $([n],R)$ satisfies $f_R(b) < 2^i(b-i+1)$ then $|R| = O(n^i)$; this generalizes Sauer's Lemma on the size of set systems with bounded VC-dimension. We use this bound to delineate the main growth rates for the same problem on families of permutations, where the trace corresponds to the inclusion for permutations. This is related to a question of Raz on families of permutations with bounded VC-dimension that generalizes the Stanley-Wilf conjecture on permutations with excluded patterns.

cs.DM↗

arXiv2009

Lower Bounds for Pinning Lines by Balls

A line L is a transversal to a family F of convex objects in R^d if it intersects every member of F. In this paper we show that for every integer d>2 there exists a family of 2d-1 pairwise disjoint unit balls in R^d with the property that every subfamily of size 2d-2 admits a transversal, yet any line misses at least one member of the family. This answers a question of Danzer from 1957.

cs.CG↗

arXiv2007

Sparse geometric graphs with small dilation

Given a set S of n points in R^D, and an integer k such that 0 <= k < n, we show that a geometric graph with vertex set S, at most n - 1 + k edges, maximum degree five, and dilation O(n / (k+1)) can be computed in time O(n log n). For any k, we also construct planar n-point sets for which any geometric graph with n-1+k edges has dilation Omega(n/(k+1)); a slightly weaker statement holds if the points of S are required to be in convex position.

cs.CG↗

arXiv2007

Constructing Optimal Highways

For two points $p$ and $q$ in the plane, a straight line $h$, called a highway, and a real $v>1$, we define the \emph{travel time} (also known as the \emph{city distance}) from $p$ and $q$ to be the time needed to traverse a quickest path from $p$ to $q$, where the distance is measured with speed $v$ on $h$ and with speed 1 in the underlying metric elsewhere. Given a set $S$ of $n$ points in the plane and a highway speed $v$, we consider the problem of finding a \emph{highway} that minimizes the maximum travel time over all pairs of points in $S$. If the orientation of the highway is fixed, the optimal highway can be computed in linear time, both for the $L_1$- and the Euclidean metric as the underlying metric. If arbitrary orientations are allowed, then the optimal highway can be computed in $O(n^{2} \log n)$ time. We also consider the problem of computing an optimal pair of highways, one being horizontal, one vertical.

cs.CG↗

arXiv2007

Computing a Minimum-Dilation Spanning Tree is NP-hard

In a geometric network G = (S, E), the graph distance between two vertices u, v in S is the length of the shortest path in G connecting u to v. The dilation of G is the maximum factor by which the graph distance of a pair of vertices differs from their Euclidean distance. We show that given a set S of n points with integer coordinates in the plane and a rational dilation delta > 1, it is NP-hard to determine whether a spanning tree of S with dilation at most delta exists.

cs.CG↗

arXiv2007

Aperture-Angle and Hausdorff-Approximation of Convex Figures

The aperture angle alpha(x, Q) of a point x not in Q in the plane with respect to a convex polygon Q is the angle of the smallest cone with apex x that contains Q. The aperture angle approximation error of a compact convex set C in the plane with respect to an inscribed convex polygon Q of C is the minimum aperture angle of any x in C Q with respect to Q. We show that for any compact convex set C in the plane and any k > 2, there is an inscribed convex k-gon Q of C with aperture angle approximation error (1 - 2/(k+1)) pi. This bound is optimal, and settles a conjecture by Fekete from the early 1990s. The same proof technique can be used to prove a conjecture by Brass: If a polygon P admits no approximation by a sub-k-gon (the convex hull of k vertices of P) with Hausdorff distance sigma, but all subpolygons of P (the convex hull of some vertices of P) admit such an approximation, then P is a (k+1)-gon. This implies the following result: For any k > 2 and any convex polygon P of perimeter at most 1 there is a sub-k-gon Q of P such that the Hausdorff-distance of P and Q is at most 1/(k+1) * sin(pi/(k+1)).

cs.CG↗

arXiv2007

The Hadwiger Number of Jordan Regions is Unbounded

We show that for every n > 0 there is a planar topological disk A_0 and n translates A_1, A_2, ..., A_n of A_0 such that the interiors of A_0, ... A_n are pairwise disjoint, but with each A_i touching A_0 for 1 <= i <= n.

cs.CG↗

arXiv2007

Hadwiger and Helly-type theorems for disjoint unit spheres

We prove Helly-type theorems for line transversals to disjoint unit balls in $\R^{d}$. In particular, we show that a family of $n \geq 2d$ disjoint unit balls in $\R^d$ has a line transversal if, for some ordering $\prec$ of the balls, any subfamily of 2d balls admits a line transversal consistent with $\prec$. We also prove that a family of $n \geq 4d-1$ disjoint unit balls in $\R^d$ admits a line transversal if any subfamily of size $4d-1$ admits a transversal.

cs.CG↗