Point-Set Embedding in Three Dimensions

Point-Set Embedding in Three Dimensions

Given a graph G with n vertices and m edges, and a set P of n points on a three-dimensional integer grid, the 3D Point-Set Embeddability problem is to determine a (three-dimensional) crossing-free drawing of G with vertices located at P and with edges drawn as polylines with bend-points at integer grid points. We solve a variant of the problem in which the points of P lie on a plane. The result...

Efficient Approximation Algorithms for Point-set Diameter in Higher Dimensions

We study the problem of computing the diameter of a  set of $n$ points in $d$-dimensional Euclidean space for a fixed dimension $d$, and propose a new $(1+varepsilon)$-approximation algorithm with $O(n+ 1/varepsilon^{d-1})$ time and $O(n)$ space, where $0 < varepsilonleqslant 1$. We also show that the proposed algorithm can be modified to a $(1+O(varepsilon))$-approximation algorithm with $O(n+...

Orthogeodesic Point-Set Embedding of Trees

Let S be a set of N grid points in the plane, and let G a graph with n vertices (n ≤ N). An orthogeodesic point-set embedding of G on S is a drawing of G such that each vertex is drawn as a point of S and each edge is a chain of horizontal and vertical segments with bends on grid points whose length is equal to the Manhattan distance of its end vertices. We study the following problem. Given a ...

On Embedding an Outer-Planar Graph in a Point Set

An efficient approximation for point set diameter in higher dimensions

In this paper, we study the problem of computing the diameter of a set of n points in d-dimensional Euclidean space, for a fixed dimension d, and propose a new (1 + ε)-approximation algorithm with O(n + 1/ε) time and O(n) space, where 0 < ε 6 1. We also show that the running time of the proposed algorithm can be improved to O(n + 1/εd−2). Our algorithm is very simple in implementation, and does...