Download presentation

Presentation is loading. Please wait.

Published byHeather King Modified over 2 years ago

1
Random Geometric Graph Diameter in the Unit Disk Robert B. Ellis, Texas A&M University coauthors Jeremy L. Martin, University of Minnesota Catherine Yan, Texas A&M University

2
Definition of G p (λ,n) Fix 1 ≤ p ≤ ∞. p=1 λ p=2 λ p=∞ λ p=1p=2 p=∞ Randomly place vertices V n :={ v 1,v 2,…,v n } in unit disk D (independent identical uniform distributions) {u,v} is an edge iff ||u-v|| p ≤ λ. B1(u,λ)B1(u,λ) u B2(u,λ)B2(u,λ)B∞(u,λ)B∞(u,λ)

3
Motivation Simulate wireless multi-hop networks, Mobile ad hoc networks Provide an alternative to the Erdős-Rényi model for testing heuristics: Traveling salesman, minimal matching, minimal spanning tree, partitioning, clustering, etc. Model systems with intrinsic spatial relationships

4
Sample of History Clark, Colbourn, Johnson (1990): independent set (NPC), maximum clique (P), case p=2 and non-random Appel, Russo (1997): distribution of max/min vertex degree Penrose (1999): k-connectivity min degree k. Diaz, Penrose, Petit, Serna (2000-01): asymptotic optimal cost in minimum bisection, minimum vertex separation, other layout problems An authority: Random Geometric Graphs, Penrose (2003)

5
Notation. “Almost Always (a.a.), G p (λ,n) has property P” means: If then G p (λ,n) is superconnected Connectivity Regime From now on, we take λ of the form where c is constant. If then G p (λ,n) is subconnected/disconnected

6
Threshold for Connectivity Thm (Penrose, `99). Connectivity threshold = min degree 1 threshold. Specifically, X u := event that u is an isolated vertex. Ignoring boundary effects, Second moment method:

7
Major Question: Diameter of G p (λ,n) Assume G p (λ,n) is connected. Determine Lower bound. Define diam p (D) := ℓ p -diameter of unit disk D 2Ddiam Assume G p (λ,n) is connected. Then almost always,

8
Sharpened Lower Bound Prop. Let c>a p -1/2, and choose h(n) such that h(n)/n -2/3 ∞. Then a.a., h(n) << λ Picture for 1≤p≤2 Line ℓ 2 -distance = 2-2h(n) ℓ p -distance = (2-2h(n))2 1/p-1/2 Proof: examine probability that both caps have a vertex

9
Diameter Upper Bound, c> a p -1/2 “Lozenge” Lemma (extended from Penrose). Let c>a p -1/2. There exists a k>0 such that a.a., for all u,v in G p (λ,n), u and v are connected inside the convex hull of B 2 (u,kλ) U B 2 (v,kλ). u v kλkλ ||u-v|| p (k+2 -1/2 )λ B p (·,λ/2) Corollary. Let c>a p -1/2. There exists a K>0 (independent of p) such that almost always, for all u,v in G p (λ,n),

10
Diameter Upper Bound: A Spoke Construction B p (·,λ/2) ℓ 2 -distance=r A p * (r, λ/2):=min area of intersection of two ℓ p -balls of radius λ/2 with centers at Euclidean distance r Vertices in consecutive gray regions are joined by an edge. # ℓ p -balls in spoke: 2/r

11
Diameter Upper Bound: A Spoke Construction (con’t) u v u’ v’ Building a path from u to v: Instantiate Θ(log n) spokes. Suppose every gray region has a vertex. Use “lozenge lemma” to get from u to u’, and v to v’ on nearby spokes. Use spokes to meet at center.

12
A Diameter Upper Bound Theorem. Let 1≤p≤∞ and r = min{λ2 -1/2-1/p, λ/2}. Suppose that Then almost always, diam(G p (λ,n)) ≤ (2·diam p (D)+o(1)) ∕ λ. Proof Sketch. M := #gray regions in all spokes = Θ((2/r)·log n). Pr[a single gray region has no vertex] ≤ (1-A p * (r, λ/2)/π) n.

13
Two Improvements 1.Increase average distance of two gray regions in spoke, letting r min{λ2 1/2-1/p, λ}. 2.Allow o(1/λ) gray regions to have no vertex and use “lozenge lemma” to take K-step detours around empty regions. Theorem. Let 1≤p≤∞, h(n)/n -2/3 ∞, and c > a p -1/2. Then almost always, diam p (D)(1-h(n))/λ ≤ diam(G p (λ,n)) ≤ diam p (D)(1+o(1))/λ. rellis@math.tamu.eduhttp://www.math.tamu.edu/~rellis/ martin@math.umn.eduhttp://www.math.umn.edu/~martin/ cyan@math.tamu.eduhttp://www.math.tamu.edu/~cyan/

Similar presentations

OK

What is the next line of the proof? a). Let G be a graph with k vertices. b). Assume the theorem holds for all graphs with k+1 vertices. c). Let G be a.

What is the next line of the proof? a). Let G be a graph with k vertices. b). Assume the theorem holds for all graphs with k+1 vertices. c). Let G be a.

© 2017 SlidePlayer.com Inc.

All rights reserved.

Ads by Google

Ppt on next generation 2-stroke engines Slideshare net download ppt on pollution Ppt on different aspects of environment and society Ppt on mission to mars Best ppt on french revolution Ppt on carbon and its compounds Ppt on cross-sectional study in epidemiology Ppt on fibonacci numbers Download ppt on sets for class 11 Free ppt on green revolution