On Notions of Distortion and an Almost Minimum Spanning Tree with Constant Average Distortion

66 Pages·2016·1.2 MB·English

On Notions of Distortion and an Almost Minimum
Spanning Tree with Constant Average Distortion
Yair Bartal1 Arnold Filtser2 Ofer Neiman2
1 HebrewUniversityofJerusalem
2 Ben-GurionUniversityoftheNegev
January 11, 2016
Y.Bartal,A.Filtser,O.Neiman OnNotionsofDistortion January11,2016 1/21
For details: www.cs.bgu.ac.il/~arnoldf/
Warning
The talk is about an improved version of the paper.
Y.Bartal,A.Filtser,O.Neiman OnNotionsofDistortion January11,2016 2/21
Warning
The talk is about an improved version of the paper.
For details: www.cs.bgu.ac.il/~arnoldf/
Y.Bartal,A.Filtser,O.Neiman OnNotionsofDistortion January11,2016 2/21
Distortion
f has distortion t if for every x,y X,
∈
d (x,y) d (f (x),f (y)) t d (x,y) .
X Y X
≤ ≤ ·
Average distortion
1 (cid:88) dY(f(v),f(u))
(cid:0)X (cid:1) · d (v,u)
|2| v,u X X
∈
Embedding
Embedding
(X,d ),(Y,d ) metric spaces.
X Y
f : (X,d ) (Y,d ) is called an embedding.
X Y
→
Y.Bartal,A.Filtser,O.Neiman OnNotionsofDistortion January11,2016 3/21
Average distortion
1 (cid:88) dY(f(v),f(u))
(cid:0)X (cid:1) · d (v,u)
|2| v,u X X
∈
Embedding
Embedding
(X,d ),(Y,d ) metric spaces.
X Y
f : (X,d ) (Y,d ) is called an embedding.
X Y
→
Distortion
f has distortion t if for every x,y X,
∈
d (x,y) d (f (x),f (y)) t d (x,y) .
X Y X
≤ ≤ ·
Y.Bartal,A.Filtser,O.Neiman OnNotionsofDistortion January11,2016 3/21
Embedding
Embedding
(X,d ),(Y,d ) metric spaces.
X Y
f : (X,d ) (Y,d ) is called an embedding.
X Y
→
Distortion
f has distortion t if for every x,y X,
∈
d (x,y) d (f (x),f (y)) t d (x,y) .
X Y X
≤ ≤ ·
Average distortion
1 (cid:88) dY(f(v),f(u))
(cid:0)X (cid:1) · d (v,u)
|2| v,u X X
∈
Y.Bartal,A.Filtser,O.Neiman OnNotionsofDistortion January11,2016 3/21
(cid:80)
w(e)
The lightness of a H is Ψ(H) = e∈EH .
w(MST)
Spanner
Graph spanner
Given a weighted graph G = (V,E,w), a subgraph H = (V,E ,w) of G
H
is a spanner of G with distortion t if
u,v V, d (u,v) t d (u,v)
H G
∀ ∈ ≤ ·
Y.Bartal,A.Filtser,O.Neiman OnNotionsofDistortion January11,2016 4/21
Spanner
Graph spanner
Given a weighted graph G = (V,E,w), a subgraph H = (V,E ,w) of G
H
is a spanner of G with distortion t if
u,v V, d (u,v) t d (u,v)
H G
∀ ∈ ≤ ·
(cid:80)
w(e)
The lightness of a H is Ψ(H) = e∈EH .
w(MST)
Y.Bartal,A.Filtser,O.Neiman OnNotionsofDistortion January11,2016 4/21
Spanner
Graph spanner
Given a weighted graph G = (V,E,w), a subgraph H = (V,E ,w) of G
H
is a spanner of G with distortion t if
u,v V, d (u,v) t d (u,v)
H G
∀ ∈ ≤ ·
(cid:80)
w(e)
The lightness of a H is Ψ(H) = e∈EH .
w(MST)
Y.Bartal,A.Filtser,O.Neiman OnNotionsofDistortion January11,2016 4/21
The MST has lightness 1!
But unbounded average distortion...
Theorem (Abraham, Bartal and Neiman 2006)
Every weighted graph contains a spanning tree
with O(1) average distortion.
But unbounded lightness...
Lightness vs Average Distortion in Trees
G = (V,E,w) is a weighted graph.
Y.Bartal,A.Filtser,O.Neiman OnNotionsofDistortion January11,2016 5/21