Optimal Algorithms to Embed Trees in a Point Set

Optimal Algorithms to Embed Trees in a Point Set

We present optimal Θ(n log n) time algorithms to solve two tree embedding problems whose solution previously took quadratic time or more: rooted-tree embeddings and degree-constrained embeddings. In the rootedtree embedding problem we are given a rooted tree T with n nodes and a set of n points P with one designated point p and are asked to find a straight-line embedding of T into P with the ro...

Journal of Graph Algorithms and Applications Optimal Algorithms to Embed Trees in a Point Set

We present optimal (n log n) time algorithms to solve two tree embedding problems whose solution previously took quadratic time or more: rooted-tree embeddings and degree-constrained embeddings. In the rootedtree embedding problem we are given a rooted tree T with n nodes and a set of n points P with one designated point p and are asked to nd a straight-line embedding of T into P with the root ...

Optimal point-to-point broadcast algorithms via lopsided trees

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 ...