A Randomly Embedded Random Graph is Not a Spanner
نویسنده
چکیده
Select n points uniformly at random from a unit square, and then form a random graph on these points by adding an edge joining each pair independently with probability p. We show that for every fixed > 0, if p < 1 − , then with probability approaching 1 as n becomes large, the resulting embedded graph has unbounded stretch factor.
منابع مشابه
On Geometric Spanners of Euclidean Graphs and their Applications in Wireless Networks
We consider the problem of constructing a bounded-degree planar geometric spanner for a unit disk graph modeling a wireless network. The related problem of constructing a planar geometric spanner of a Euclidean graph has been extensively studied in the literature. It is well known that the Delaunay subgraph is a planar geometric spanner with stretch factor ; however, its degree may not be bound...
متن کاملLowest Degree k-Spanner: Approximation and Hardness
A k-spanner is a subgraph in which distances are approximately preserved, up to some given stretch factor k. We focus on the following problem: Given a graph and a value k, can we find a k-spanner that minimizes the maximum degree? While reasonably strong bounds are known for some spanner problems, they almost all involve minimizing the total number of edges. Switching the objective to the degr...
متن کاملOn Geometric Spanners of Euclidean and Unit Disk Graphs
We consider the problem of constructing bounded-degree planar geometric spanners of Euclidean and unit-disk graphs. It is well known that the Delaunay subgraph is a planar geometric spanner with stretch factor Cdel ≈ 2.42; however, its degree may not be bounded. Our first result is a very simple linear time algorithm for constructing a subgraph of the Delaunay graph with stretch factor ρ = 1 + ...
متن کاملRandomized and deterministic algorithms for geometric spanners of small diameter
Let S be a set of n points in IR and let t > 1 be a real number. A t-spanner for S is a directed graph having the points of S as its vertices, such that for any pair p and q of points there is a path from p to q of length at most t times the Euclidean distance between p and q. Such a path is called a t-spanner path. The spanner diameter of such a spanner is defined as the smallest integer D suc...
متن کاملDynamic algorithms for geometric spanners of small diameter: Randomized solutions
Let S be a set of n points in IRd and let t > 1 be a real number. A t-spanner for S is a directed graph having the points of S as its vertices, such that for any pair p and q of points there is a path from p to q of length at most t times the Euclidean distance between p and q. Such a path is called a t-spanner path. The spanner diameter of such a spanner is de ned as the smallest integer D suc...
متن کامل