Page 1
Outline
Treewidth and Treelength of Internet
Fabien de Montgolfier Mauricio Soto Laurent Viennot
LIAFAUniversite Paris Diderot - Paris 7 CNRS
GANGINRIA Paris Rocquencourt
Reunion ALADDIN , Poitiers / Novembre 2008
Fabien de Montgolfier, Mauricio Soto, Laurent Viennot Treewidth and Treelength of Internet 1 / 27
Page 2
Outline
Outline
1 IntroductionTree Decomposition
2 Data Source
3 Treelength
4 TreewidthLower BoundUpper Bound
5 Conclusions
Fabien de Montgolfier, Mauricio Soto, Laurent Viennot Treewidth and Treelength of Internet 2 / 27
Page 3
Outline
Outline
1 IntroductionTree Decomposition
2 Data Source
3 Treelength
4 TreewidthLower BoundUpper Bound
5 Conclusions
Fabien de Montgolfier, Mauricio Soto, Laurent Viennot Treewidth and Treelength of Internet 2 / 27
Page 4
Outline
Outline
1 IntroductionTree Decomposition
2 Data Source
3 Treelength
4 TreewidthLower BoundUpper Bound
5 Conclusions
Fabien de Montgolfier, Mauricio Soto, Laurent Viennot Treewidth and Treelength of Internet 2 / 27
Page 5
Outline
Outline
1 IntroductionTree Decomposition
2 Data Source
3 Treelength
4 TreewidthLower BoundUpper Bound
5 Conclusions
Fabien de Montgolfier, Mauricio Soto, Laurent Viennot Treewidth and Treelength of Internet 2 / 27
Page 6
Outline
Outline
1 IntroductionTree Decomposition
2 Data Source
3 Treelength
4 TreewidthLower BoundUpper Bound
5 Conclusions
Fabien de Montgolfier, Mauricio Soto, Laurent Viennot Treewidth and Treelength of Internet 2 / 27
Page 7
Introduction Data Source Treelength Treewidth Conclusions
Goal
Goal
Compute
bounds on
treewidth and treelength on Internetgraph.
Understand Internet graph structure.
Motivation
Small treewidth graphs have polynomial (linear) algorithms forNP-hard problems.
Small treelength graphs have compact routing protocols.
Fabien de Montgolfier, Mauricio Soto, Laurent Viennot Treewidth and Treelength of Internet 3 / 27
Page 8
Introduction Data Source Treelength Treewidth Conclusions
Goal
Goal
Compute bounds on treewidth and treelength on Internetgraph.
Understand Internet graph structure.
Motivation
Small treewidth graphs have polynomial (linear) algorithms forNP-hard problems.
Small treelength graphs have compact routing protocols.
Fabien de Montgolfier, Mauricio Soto, Laurent Viennot Treewidth and Treelength of Internet 3 / 27
Page 9
Introduction Data Source Treelength Treewidth Conclusions
Goal
Goal
Compute bounds on treewidth and treelength on Internetgraph.
Understand Internet graph structure.
Motivation
Small treewidth graphs have polynomial (linear) algorithms forNP-hard problems.
Small treelength graphs have compact routing protocols.
Fabien de Montgolfier, Mauricio Soto, Laurent Viennot Treewidth and Treelength of Internet 3 / 27
Page 10
Introduction Data Source Treelength Treewidth Conclusions Tree Decomposition
Outline
1 IntroductionTree Decomposition
2 Data Source
3 Treelength
4 TreewidthLower BoundUpper Bound
5 Conclusions
Fabien de Montgolfier, Mauricio Soto, Laurent Viennot Treewidth and Treelength of Internet 4 / 27
Page 11
Introduction Data Source Treelength Treewidth Conclusions Tree Decomposition
Tree Decomposition
Tree Decomposition
- bags {Xi ⊆ V , i ∈ I}- tree T = (I ,F )
1 every node is in a bag.
2 every edge has its ends in a bag.
3 bags containing a vertex formconnected subtree.
- Width: maxi |Xi | − 1
- Length: maxi diamG (Xi )
Tree
is the minimum over all treedecompositions.
a
b
c
d
e
f
g
a c d
a b dd f g
b e
Fabien de Montgolfier, Mauricio Soto, Laurent Viennot Treewidth and Treelength of Internet 5 / 27
Page 12
Introduction Data Source Treelength Treewidth Conclusions Tree Decomposition
Tree Decomposition
Tree Decomposition
- bags {Xi ⊆ V , i ∈ I}
- tree T = (I ,F )
1 every node is in a bag.
2 every edge has its ends in a bag.
3 bags containing a vertex formconnected subtree.
- Width: maxi |Xi | − 1
- Length: maxi diamG (Xi )
Tree
is the minimum over all treedecompositions.
a
b
c
d
e
f
g
a c d
a b dd f g
b e
Fabien de Montgolfier, Mauricio Soto, Laurent Viennot Treewidth and Treelength of Internet 5 / 27
Page 13
Introduction Data Source Treelength Treewidth Conclusions Tree Decomposition
Tree Decomposition
Tree Decomposition
- bags {Xi ⊆ V , i ∈ I}- tree T = (I ,F )
1 every node is in a bag.
2 every edge has its ends in a bag.
3 bags containing a vertex formconnected subtree.
- Width: maxi |Xi | − 1
- Length: maxi diamG (Xi )
Tree
is the minimum over all treedecompositions.
a
b
c
d
e
f
g
a c d
a b dd f g
b e
Fabien de Montgolfier, Mauricio Soto, Laurent Viennot Treewidth and Treelength of Internet 5 / 27
Page 14
Introduction Data Source Treelength Treewidth Conclusions Tree Decomposition
Tree Decomposition
Tree Decomposition
- bags {Xi ⊆ V , i ∈ I}- tree T = (I ,F )
1 every node is in a bag.
2 every edge has its ends in a bag.
3 bags containing a vertex formconnected subtree.
- Width: maxi |Xi | − 1
- Length: maxi diamG (Xi )
Tree
is the minimum over all treedecompositions.
a
b
c
d
e
f
g
a c d
a b dd f g
b e
Fabien de Montgolfier, Mauricio Soto, Laurent Viennot Treewidth and Treelength of Internet 5 / 27
Page 15
Introduction Data Source Treelength Treewidth Conclusions Tree Decomposition
Tree Decomposition
Tree Decomposition
- bags {Xi ⊆ V , i ∈ I}- tree T = (I ,F )
1 every node is in a bag.
2 every edge has its ends in a bag.
3 bags containing a vertex formconnected subtree.
- Width: maxi |Xi | − 1
- Length: maxi diamG (Xi )
Tree
is the minimum over all treedecompositions.
a
b
c
d
e
f
g
a c d
a b dd f g
b e
a b d
Fabien de Montgolfier, Mauricio Soto, Laurent Viennot Treewidth and Treelength of Internet 5 / 27
Page 16
Introduction Data Source Treelength Treewidth Conclusions Tree Decomposition
Tree Decomposition
Tree Decomposition
- bags {Xi ⊆ V , i ∈ I}- tree T = (I ,F )
1 every node is in a bag.
2 every edge has its ends in a bag.
3 bags containing a vertex formconnected subtree.
- Width: maxi |Xi | − 1
- Length: maxi diamG (Xi )
Tree
is the minimum over all treedecompositions.
a
b
c
d
e
f
g
d
a c d
a b dd f g
b e
a c d
a b dd f g
Fabien de Montgolfier, Mauricio Soto, Laurent Viennot Treewidth and Treelength of Internet 5 / 27
Page 17
Introduction Data Source Treelength Treewidth Conclusions Tree Decomposition
Tree Decomposition
Tree Decomposition
- bags {Xi ⊆ V , i ∈ I}- tree T = (I ,F )
1 every node is in a bag.
2 every edge has its ends in a bag.
3 bags containing a vertex formconnected subtree.
- Width: maxi |Xi | − 1
- Length: maxi diamG (Xi )
Tree
is the minimum over all treedecompositions.
a
b
c
d
e
f
g
a c d
a b dd f g
b e
Fabien de Montgolfier, Mauricio Soto, Laurent Viennot Treewidth and Treelength of Internet 5 / 27
Page 18
Introduction Data Source Treelength Treewidth Conclusions Tree Decomposition
Tree Decomposition
Tree Decomposition
- bags {Xi ⊆ V , i ∈ I}- tree T = (I ,F )
1 every node is in a bag.
2 every edge has its ends in a bag.
3 bags containing a vertex formconnected subtree.
- Width: maxi |Xi | − 1
- Length: maxi diamG (Xi )
Tree
is the minimum over all treedecompositions.
a
b
c
d
e
f
g
da
a c d
a b dd f g
b e
a c d
Fabien de Montgolfier, Mauricio Soto, Laurent Viennot Treewidth and Treelength of Internet 5 / 27
Page 19
Introduction Data Source Treelength Treewidth Conclusions Tree Decomposition
Tree Decomposition
Tree Decomposition
- bags {Xi ⊆ V , i ∈ I}- tree T = (I ,F )
1 every node is in a bag.
2 every edge has its ends in a bag.
3 bags containing a vertex formconnected subtree.
- Width: maxi |Xi | − 1
- Length: maxi diamG (Xi )
Treewidth
tw(G ) is the minimum width over alltree decompositions.
a
b
c
d
e
f
g
a c d
a b dd f g
b e
Fabien de Montgolfier, Mauricio Soto, Laurent Viennot Treewidth and Treelength of Internet 5 / 27
Page 20
Introduction Data Source Treelength Treewidth Conclusions Tree Decomposition
Tree Decomposition
Tree Decomposition
- bags {Xi ⊆ V , i ∈ I}- tree T = (I ,F )
1 every node is in a bag.
2 every edge has its ends in a bag.
3 bags containing a vertex formconnected subtree.
- Width: maxi |Xi | − 1
- Length: maxi diamG (Xi )
Treelength
tl(G ) is the minimum length over alltree decompositions.
a
b
c
d
e
f
g
a c d
a b dd f g
b e
Fabien de Montgolfier, Mauricio Soto, Laurent Viennot Treewidth and Treelength of Internet 5 / 27
Page 21
Introduction Data Source Treelength Treewidth Conclusions Tree Decomposition
Algorithmic Motivation
Claim
Many problems NP-hard problems become linear or polynomialtime solvable on bounded treewidth graphs:Hamiltonian Circuit, Independent Set, VertexCover...
Theorem [Dourisboure. DISC, 2004]
If tw(G ) = δ then it can be constructed a routing scheme with a6δ − 2 additive stretch and address memory of size O(δ log2 n).
Theorem [Dourisboure,Dragan,Gavoille,Yan. Theor. Comput. Sci.,2007]
If tw(G ) = δ then G has an additive 2δ-spanner (4δ-spanners) withO(δn + n log n) (O(δn)) edges.
Fabien de Montgolfier, Mauricio Soto, Laurent Viennot Treewidth and Treelength of Internet 6 / 27
Page 22
Introduction Data Source Treelength Treewidth Conclusions Tree Decomposition
Algorithmic Motivation
Claim
Many problems NP-hard problems become linear or polynomialtime solvable on bounded treewidth graphs:Hamiltonian Circuit, Independent Set, VertexCover...
Theorem [Dourisboure. DISC, 2004]
If tw(G ) = δ then it can be constructed a routing scheme with a6δ − 2 additive stretch and address memory of size O(δ log2 n).
Theorem [Dourisboure,Dragan,Gavoille,Yan. Theor. Comput. Sci.,2007]
If tw(G ) = δ then G has an additive 2δ-spanner (4δ-spanners) withO(δn + n log n) (O(δn)) edges.
Fabien de Montgolfier, Mauricio Soto, Laurent Viennot Treewidth and Treelength of Internet 6 / 27
Page 23
Introduction Data Source Treelength Treewidth Conclusions Tree Decomposition
Algorithmic Motivation
Claim
Many problems NP-hard problems become linear or polynomialtime solvable on bounded treewidth graphs:Hamiltonian Circuit, Independent Set, VertexCover...
Theorem [Dourisboure. DISC, 2004]
If tw(G ) = δ then it can be constructed a routing scheme with a6δ − 2 additive stretch and address memory of size O(δ log2 n).
Theorem [Dourisboure,Dragan,Gavoille,Yan. Theor. Comput. Sci.,2007]
If tw(G ) = δ then G has an additive 2δ-spanner (4δ-spanners) withO(δn + n log n) (O(δn)) edges.
Fabien de Montgolfier, Mauricio Soto, Laurent Viennot Treewidth and Treelength of Internet 6 / 27
Page 24
Introduction Data Source Treelength Treewidth Conclusions Tree Decomposition
Structural Motivation
Proposition [Chepoi, Dragan, Estellon, Habib, Vaxes. SoCG 2008]
If tl(G ) = δ then G is δ-hyperbolic
A δ-hyperbolic graph G = (V ,E ) has a tree decomposition oflength at most 4(4 + 3δ + δ log2 n) + 1.
Shavitt, Tankel.IEEE/ACM Trans. Netw. 2008
Internet graph can be embedded into a hyperbolic space.
Fabien de Montgolfier, Mauricio Soto, Laurent Viennot Treewidth and Treelength of Internet 7 / 27
Page 25
Introduction Data Source Treelength Treewidth Conclusions Tree Decomposition
Structural Motivation
Proposition [Chepoi, Dragan, Estellon, Habib, Vaxes. SoCG 2008]
If tl(G ) = δ then G is δ-hyperbolic
A δ-hyperbolic graph G = (V ,E ) has a tree decomposition oflength at most 4(4 + 3δ + δ log2 n) + 1.
Shavitt, Tankel.IEEE/ACM Trans. Netw. 2008
Internet graph can be embedded into a hyperbolic space.
Fabien de Montgolfier, Mauricio Soto, Laurent Viennot Treewidth and Treelength of Internet 7 / 27
Page 26
Introduction Data Source Treelength Treewidth Conclusions
Outline
1 IntroductionTree Decomposition
2 Data Source
3 Treelength
4 TreewidthLower BoundUpper Bound
5 Conclusions
Fabien de Montgolfier, Mauricio Soto, Laurent Viennot Treewidth and Treelength of Internet 8 / 27
Page 27
Introduction Data Source Treelength Treewidth Conclusions
Internet Routing
Router network.
Autonomous Systems.
Border Gateway Protocol(BGP)
Network Next Hop Metric LocPrf Weight Path
*> 192.9.9.0 134.24.127.3 0 1740 90 i
* 194.68.130.254 0 5459 5413 90 i
* 158.43.133.48 0 1849 702 701 90 i
* 193.0.0.242 0 3333 286 90 i
* 144.228.240.93 0 1239 90 i
Fabien de Montgolfier, Mauricio Soto, Laurent Viennot Treewidth and Treelength of Internet 9 / 27
Page 28
Introduction Data Source Treelength Treewidth Conclusions
Internet Routing
AS1
AS2
AS3
AS4AS5
Router network.
Autonomous Systems.
Border Gateway Protocol(BGP)
Network Next Hop Metric LocPrf Weight Path
*> 192.9.9.0 134.24.127.3 0 1740 90 i
* 194.68.130.254 0 5459 5413 90 i
* 158.43.133.48 0 1849 702 701 90 i
* 193.0.0.242 0 3333 286 90 i
* 144.228.240.93 0 1239 90 i
Fabien de Montgolfier, Mauricio Soto, Laurent Viennot Treewidth and Treelength of Internet 9 / 27
Page 29
Introduction Data Source Treelength Treewidth Conclusions
Internet Routing
AS1
AS2
AS3
AS4AS5
Router network.
Autonomous Systems.
Border Gateway Protocol(BGP)
Network Next Hop Metric LocPrf Weight Path
*> 192.9.9.0 134.24.127.3 0 1740 90 i
* 194.68.130.254 0 5459 5413 90 i
* 158.43.133.48 0 1849 702 701 90 i
* 193.0.0.242 0 3333 286 90 i
* 144.228.240.93 0 1239 90 i
Fabien de Montgolfier, Mauricio Soto, Laurent Viennot Treewidth and Treelength of Internet 9 / 27
Page 30
Introduction Data Source Treelength Treewidth Conclusions
Internet Routing
AS1
AS2
AS3
AS4AS5
Router network.
Autonomous Systems.
Border Gateway Protocol(BGP)
Network Next Hop Metric LocPrf Weight Path
*> 192.9.9.0 134.24.127.3 0 1740 90 i
* 194.68.130.254 0 5459 5413 90 i
* 158.43.133.48 0 1849 702 701 90 i
* 193.0.0.242 0 3333 286 90 i
* 144.228.240.93 0 1239 90 i
Fabien de Montgolfier, Mauricio Soto, Laurent Viennot Treewidth and Treelength of Internet 9 / 27
Page 31
Introduction Data Source Treelength Treewidth Conclusions
Internet Routing
AS1
AS2
AS3
AS4AS5
Router network.
Autonomous Systems.
Border Gateway Protocol(BGP)
Network Next Hop Metric LocPrf Weight Path
*> 192.9.9.0 134.24.127.3 0 1740 90 i
* 194.68.130.254 0 5459 5413 90 i
* 158.43.133.48 0 1849 702 701 90 i
* 193.0.0.242 0 3333 286 90 i
* 144.228.240.93 0 1239 90 i
Fabien de Montgolfier, Mauricio Soto, Laurent Viennot Treewidth and Treelength of Internet 9 / 27
Page 32
Introduction Data Source Treelength Treewidth Conclusions
Traceroute
CAIDA, The CooperativeAssociation for Internet DataAnalysis.http://www.caida.org/
Traceroute:
Set of monitors.Destinations list.Sends probe packet obtainingconsecutive router links.Router graph.
BGP tables on RouteViewhttp://www.routeviews.org/
Map each router to its AS.AS graph.
Fabien de Montgolfier, Mauricio Soto, Laurent Viennot Treewidth and Treelength of Internet 10 / 27
Page 33
Introduction Data Source Treelength Treewidth Conclusions
Traceroute
CAIDA, The CooperativeAssociation for Internet DataAnalysis.http://www.caida.org/
Traceroute:
Set of monitors.Destinations list.Sends probe packet obtainingconsecutive router links.Router graph.
BGP tables on RouteViewhttp://www.routeviews.org/
Map each router to its AS.AS graph.
Fabien de Montgolfier, Mauricio Soto, Laurent Viennot Treewidth and Treelength of Internet 10 / 27
Page 34
Introduction Data Source Treelength Treewidth Conclusions
Traceroute
CAIDA, The CooperativeAssociation for Internet DataAnalysis.http://www.caida.org/
Traceroute:Set of monitors.
Destinations list.Sends probe packet obtainingconsecutive router links.Router graph.
BGP tables on RouteViewhttp://www.routeviews.org/
Map each router to its AS.AS graph.
Fabien de Montgolfier, Mauricio Soto, Laurent Viennot Treewidth and Treelength of Internet 10 / 27
Page 35
Introduction Data Source Treelength Treewidth Conclusions
Traceroute
CAIDA, The CooperativeAssociation for Internet DataAnalysis.http://www.caida.org/
Traceroute:Set of monitors.Destinations list.
Sends probe packet obtainingconsecutive router links.Router graph.
BGP tables on RouteViewhttp://www.routeviews.org/
Map each router to its AS.AS graph.
Fabien de Montgolfier, Mauricio Soto, Laurent Viennot Treewidth and Treelength of Internet 10 / 27
Page 36
Introduction Data Source Treelength Treewidth Conclusions
Traceroute
CAIDA, The CooperativeAssociation for Internet DataAnalysis.http://www.caida.org/
Traceroute:Set of monitors.Destinations list.Sends probe packet obtainingconsecutive router links.
Router graph.
BGP tables on RouteViewhttp://www.routeviews.org/
Map each router to its AS.AS graph.
Fabien de Montgolfier, Mauricio Soto, Laurent Viennot Treewidth and Treelength of Internet 10 / 27
Page 37
Introduction Data Source Treelength Treewidth Conclusions
Traceroute
CAIDA, The CooperativeAssociation for Internet DataAnalysis.http://www.caida.org/
Traceroute:Set of monitors.Destinations list.Sends probe packet obtainingconsecutive router links.
Router graph.
BGP tables on RouteViewhttp://www.routeviews.org/
Map each router to its AS.AS graph.
Fabien de Montgolfier, Mauricio Soto, Laurent Viennot Treewidth and Treelength of Internet 10 / 27
Page 38
Introduction Data Source Treelength Treewidth Conclusions
Traceroute
CAIDA, The CooperativeAssociation for Internet DataAnalysis.http://www.caida.org/
Traceroute:Set of monitors.Destinations list.Sends probe packet obtainingconsecutive router links.
Router graph.
BGP tables on RouteViewhttp://www.routeviews.org/
Map each router to its AS.AS graph.
Fabien de Montgolfier, Mauricio Soto, Laurent Viennot Treewidth and Treelength of Internet 10 / 27
Page 39
Introduction Data Source Treelength Treewidth Conclusions
Traceroute
CAIDA, The CooperativeAssociation for Internet DataAnalysis.http://www.caida.org/
Traceroute:Set of monitors.Destinations list.Sends probe packet obtainingconsecutive router links.
Router graph.
BGP tables on RouteViewhttp://www.routeviews.org/
Map each router to its AS.AS graph.
Fabien de Montgolfier, Mauricio Soto, Laurent Viennot Treewidth and Treelength of Internet 10 / 27
Page 40
Introduction Data Source Treelength Treewidth Conclusions
Traceroute
CAIDA, The CooperativeAssociation for Internet DataAnalysis.http://www.caida.org/
Traceroute:Set of monitors.Destinations list.Sends probe packet obtainingconsecutive router links.Router graph.
BGP tables on RouteViewhttp://www.routeviews.org/
Map each router to its AS.AS graph.
Fabien de Montgolfier, Mauricio Soto, Laurent Viennot Treewidth and Treelength of Internet 10 / 27
Page 41
Introduction Data Source Treelength Treewidth Conclusions
Traceroute
CAIDA, The CooperativeAssociation for Internet DataAnalysis.http://www.caida.org/
Traceroute:Set of monitors.Destinations list.Sends probe packet obtainingconsecutive router links.Router graph.
BGP tables on RouteViewhttp://www.routeviews.org/
Map each router to its AS.AS graph.
Fabien de Montgolfier, Mauricio Soto, Laurent Viennot Treewidth and Treelength of Internet 10 / 27
Page 42
Introduction Data Source Treelength Treewidth Conclusions
Traceroute
CAIDA, The CooperativeAssociation for Internet DataAnalysis.http://www.caida.org/
Traceroute:Set of monitors.Destinations list.Sends probe packet obtainingconsecutive router links.Router graph.
BGP tables on RouteViewhttp://www.routeviews.org/
Map each router to its AS.
AS graph.
Fabien de Montgolfier, Mauricio Soto, Laurent Viennot Treewidth and Treelength of Internet 10 / 27
Page 43
Introduction Data Source Treelength Treewidth Conclusions
Traceroute
CAIDA, The CooperativeAssociation for Internet DataAnalysis.http://www.caida.org/
Traceroute:Set of monitors.Destinations list.Sends probe packet obtainingconsecutive router links.Router graph.
BGP tables on RouteViewhttp://www.routeviews.org/
Map each router to its AS.AS graph.
Fabien de Montgolfier, Mauricio Soto, Laurent Viennot Treewidth and Treelength of Internet 10 / 27
Page 44
Introduction Data Source Treelength Treewidth Conclusions
Graph size
Router AS
Monitors 23 13List size 865K 7.4M
|V | 192244 27289|E | 609066 55771
Average degree 6.34 4.09Max. degree 1071 2616
1
10
100
1000
10000
100000
1e+06
1 10 100 1000 10000
CC
DF
Node Degree
Degree Distribution
ASitdk
Fabien de Montgolfier, Mauricio Soto, Laurent Viennot Treewidth and Treelength of Internet 11 / 27
Page 45
Introduction Data Source Treelength Treewidth Conclusions
Graph size
Router AS
Monitors 23 13List size 865K 7.4M|V | 192244 27289|E | 609066 55771
Average degree 6.34 4.09Max. degree 1071 2616
1
10
100
1000
10000
100000
1e+06
1 10 100 1000 10000
CC
DF
Node Degree
Degree Distribution
ASitdk
Fabien de Montgolfier, Mauricio Soto, Laurent Viennot Treewidth and Treelength of Internet 11 / 27
Page 46
Introduction Data Source Treelength Treewidth Conclusions
Graph size
Router AS
Monitors 23 13List size 865K 7.4M|V | 192244 27289|E | 609066 55771
Average degree 6.34 4.09Max. degree 1071 2616
1
10
100
1000
10000
100000
1e+06
1 10 100 1000 10000
CC
DF
Node Degree
Degree Distribution
ASitdk
Fabien de Montgolfier, Mauricio Soto, Laurent Viennot Treewidth and Treelength of Internet 11 / 27
Page 47
Introduction Data Source Treelength Treewidth Conclusions
Graph size
Router AS
Monitors 23 13List size 865K 7.4M|V | 192244 27289|E | 609066 55771
Average degree 6.34 4.09Max. degree 1071 2616
1
10
100
1000
10000
100000
1e+06
1 10 100 1000 10000
CC
DF
Node Degree
Degree Distribution
ASitdk
Fabien de Montgolfier, Mauricio Soto, Laurent Viennot Treewidth and Treelength of Internet 11 / 27
Page 48
Introduction Data Source Treelength Treewidth Conclusions
Graph size
Router AS
Monitors 23 13List size 865K 7.4M|V | 192244 27289|E | 609066 55771
Average degree 6.34 4.09Max. degree 1071 2616
1
10
100
1000
10000
100000
1e+06
1 10 100 1000 10000
CC
DF
Node Degree
Degree Distribution
ASitdk
Fabien de Montgolfier, Mauricio Soto, Laurent Viennot Treewidth and Treelength of Internet 11 / 27
Page 49
Introduction Data Source Treelength Treewidth Conclusions
Graph decomposition
Claim
tw(G ) (tl(G )) equals the maximum of treewidth (treelength) overbiconnected components.
Router AS
# Connect. comp. 308 1# Biconnect. comp. 53104 10505Biconnect.comp. size 2 95.3% 99.8%
Biggest biconnected component|V | 132367 16762
68.8% 61.4%|E | 541081 45221
88.8% 81.0%
Fabien de Montgolfier, Mauricio Soto, Laurent Viennot Treewidth and Treelength of Internet 12 / 27
Page 50
Introduction Data Source Treelength Treewidth Conclusions
Graph decomposition
Claim
tw(G ) (tl(G )) equals the maximum of treewidth (treelength) overbiconnected components.
Router AS
# Connect. comp. 308 1
# Biconnect. comp. 53104 10505Biconnect.comp. size 2 95.3% 99.8%
Biggest biconnected component|V | 132367 16762
68.8% 61.4%|E | 541081 45221
88.8% 81.0%
Fabien de Montgolfier, Mauricio Soto, Laurent Viennot Treewidth and Treelength of Internet 12 / 27
Page 51
Introduction Data Source Treelength Treewidth Conclusions
Graph decomposition
Claim
tw(G ) (tl(G )) equals the maximum of treewidth (treelength) overbiconnected components.
Router AS
# Connect. comp. 308 1# Biconnect. comp. 53104 10505
Biconnect.comp. size 2 95.3% 99.8%
Biggest biconnected component|V | 132367 16762
68.8% 61.4%|E | 541081 45221
88.8% 81.0%
Fabien de Montgolfier, Mauricio Soto, Laurent Viennot Treewidth and Treelength of Internet 12 / 27
Page 52
Introduction Data Source Treelength Treewidth Conclusions
Graph decomposition
Claim
tw(G ) (tl(G )) equals the maximum of treewidth (treelength) overbiconnected components.
Router AS
# Connect. comp. 308 1# Biconnect. comp. 53104 10505Biconnect.comp. size 2 95.3% 99.8%
Biggest biconnected component|V | 132367 16762
68.8% 61.4%|E | 541081 45221
88.8% 81.0%
Fabien de Montgolfier, Mauricio Soto, Laurent Viennot Treewidth and Treelength of Internet 12 / 27
Page 53
Introduction Data Source Treelength Treewidth Conclusions
Graph decomposition
Claim
tw(G ) (tl(G )) equals the maximum of treewidth (treelength) overbiconnected components.
Router AS
# Connect. comp. 308 1# Biconnect. comp. 53104 10505Biconnect.comp. size 2 95.3% 99.8%
Biggest biconnected component|V | 132367 16762
68.8% 61.4%|E | 541081 45221
88.8% 81.0%
Fabien de Montgolfier, Mauricio Soto, Laurent Viennot Treewidth and Treelength of Internet 12 / 27
Page 54
Introduction Data Source Treelength Treewidth Conclusions
Outline
1 IntroductionTree Decomposition
2 Data Source
3 Treelength
4 TreewidthLower BoundUpper Bound
5 Conclusions
Fabien de Montgolfier, Mauricio Soto, Laurent Viennot Treewidth and Treelength of Internet 13 / 27
Page 55
Introduction Data Source Treelength Treewidth Conclusions
Treelength
tl(G ) = 1 ⇐⇒ G is chordal.
tl(G ) = 2 ⊇ AT-free graphs, permutation graphs,distance-hereditary graphs.
tl(Ck) = dk/3e.G is k−chordal ⇒ tl(G ) ≤ k/2 + 3.
Treelength Bounds
Router AS
Upper Bound 10 5Lower Bound 3 2
Fabien de Montgolfier, Mauricio Soto, Laurent Viennot Treewidth and Treelength of Internet 14 / 27
Page 56
Introduction Data Source Treelength Treewidth Conclusions
Treelength
tl(G ) = 1 ⇐⇒ G is chordal.
tl(G ) = 2 ⊇ AT-free graphs, permutation graphs,distance-hereditary graphs.
tl(Ck) = dk/3e.G is k−chordal ⇒ tl(G ) ≤ k/2 + 3.
Treelength Bounds
Router AS
Upper Bound 10 5Lower Bound 3 2
Fabien de Montgolfier, Mauricio Soto, Laurent Viennot Treewidth and Treelength of Internet 14 / 27
Page 57
Introduction Data Source Treelength Treewidth Conclusions
Treelength
tl(G ) = 1 ⇐⇒ G is chordal.
tl(G ) = 2 ⊇ AT-free graphs, permutation graphs,distance-hereditary graphs.
tl(Ck) = dk/3e.
G is k−chordal ⇒ tl(G ) ≤ k/2 + 3.
Treelength Bounds
Router AS
Upper Bound 10 5Lower Bound 3 2
Fabien de Montgolfier, Mauricio Soto, Laurent Viennot Treewidth and Treelength of Internet 14 / 27
Page 58
Introduction Data Source Treelength Treewidth Conclusions
Treelength
tl(G ) = 1 ⇐⇒ G is chordal.
tl(G ) = 2 ⊇ AT-free graphs, permutation graphs,distance-hereditary graphs.
tl(Ck) = dk/3e.G is k−chordal ⇒ tl(G ) ≤ k/2 + 3.
Treelength Bounds
Router AS
Upper Bound 10 5Lower Bound 3 2
Fabien de Montgolfier, Mauricio Soto, Laurent Viennot Treewidth and Treelength of Internet 14 / 27
Page 59
Introduction Data Source Treelength Treewidth Conclusions
Treelength
tl(G ) = 1 ⇐⇒ G is chordal.
tl(G ) = 2 ⊇ AT-free graphs, permutation graphs,distance-hereditary graphs.
tl(Ck) = dk/3e.G is k−chordal ⇒ tl(G ) ≤ k/2 + 3.
Treelength Bounds
Router AS
Upper Bound 10 5Lower Bound 3 2
Fabien de Montgolfier, Mauricio Soto, Laurent Viennot Treewidth and Treelength of Internet 14 / 27
Page 60
Introduction Data Source Treelength Treewidth Conclusions
Treelength
tl(G ) = 1 ⇐⇒ G is chordal.
tl(G ) = 2 ⊇ AT-free graphs, permutation graphs,distance-hereditary graphs.
tl(Ck) = dk/3e.G is k−chordal ⇒ tl(G ) ≤ k/2 + 3.
Treelength Bounds
Router AS
Upper Bound 10 5Lower Bound 3 2
Fabien de Montgolfier, Mauricio Soto, Laurent Viennot Treewidth and Treelength of Internet 14 / 27
Page 61
Introduction Data Source Treelength Treewidth Conclusions
Layer Tree [Chepoi, Dragan. Europ. J. Combin. 2000]
L0
L0 ∪ L11
L11 ∪ L2
1
L21 ∪ L3
1
L11 ∪ L2
2
L22 ∪ L3
1 L22 ∪ L3
2
BFS-Layering
O(|E |+ |V |)
1 Construct a BFS tree.
2 From bottom, put twovertices in the same bag if
are in the same layer andexists a path of verticesfurther from root.
3 Constructtree-decomposition T (G ).
Theorem [Doursboure, Gavoille.EUROCOMB 2003]
tl(G ) ≤ length(T ) ≤ 3 tl(G ) + 1
Fabien de Montgolfier, Mauricio Soto, Laurent Viennot Treewidth and Treelength of Internet 15 / 27
Page 62
Introduction Data Source Treelength Treewidth Conclusions
Layer Tree [Chepoi, Dragan. Europ. J. Combin. 2000]
L0
L0 ∪ L11
L11 ∪ L2
1
L21 ∪ L3
1
L11 ∪ L2
2
L22 ∪ L3
1 L22 ∪ L3
2
BFS-Layering
O(|E |+ |V |)
1 Construct a BFS tree.2 From bottom, put two
vertices in the same bag if
are in the same layer andexists a path of verticesfurther from root.
3 Constructtree-decomposition T (G ).
Theorem [Doursboure, Gavoille.EUROCOMB 2003]
tl(G ) ≤ length(T ) ≤ 3 tl(G ) + 1
Fabien de Montgolfier, Mauricio Soto, Laurent Viennot Treewidth and Treelength of Internet 15 / 27
Page 63
Introduction Data Source Treelength Treewidth Conclusions
Layer Tree [Chepoi, Dragan. Europ. J. Combin. 2000]
L0
L0 ∪ L11
L11 ∪ L2
1
L21 ∪ L3
1
L11 ∪ L2
2
L22 ∪ L3
1 L22 ∪ L3
2
BFS-Layering
O(|E |+ |V |)
1 Construct a BFS tree.2 From bottom, put two
vertices in the same bag if
are in the same layer and
exists a path of verticesfurther from root.
3 Constructtree-decomposition T (G ).
Theorem [Doursboure, Gavoille.EUROCOMB 2003]
tl(G ) ≤ length(T ) ≤ 3 tl(G ) + 1
Fabien de Montgolfier, Mauricio Soto, Laurent Viennot Treewidth and Treelength of Internet 15 / 27
Page 64
Introduction Data Source Treelength Treewidth Conclusions
Layer Tree [Chepoi, Dragan. Europ. J. Combin. 2000]
L0
L0 ∪ L11
L11 ∪ L2
1
L21 ∪ L3
1
L11 ∪ L2
2
L22 ∪ L3
1 L22 ∪ L3
2
BFS-Layering
O(|E |+ |V |)
1 Construct a BFS tree.2 From bottom, put two
vertices in the same bag if
are in the same layer andexists a path of verticesfurther from root.
3 Constructtree-decomposition T (G ).
Theorem [Doursboure, Gavoille.EUROCOMB 2003]
tl(G ) ≤ length(T ) ≤ 3 tl(G ) + 1
Fabien de Montgolfier, Mauricio Soto, Laurent Viennot Treewidth and Treelength of Internet 15 / 27
Page 65
Introduction Data Source Treelength Treewidth Conclusions
Layer Tree [Chepoi, Dragan. Europ. J. Combin. 2000]
L0
L0 ∪ L11
L11 ∪ L2
1
L21 ∪ L3
1
L11 ∪ L2
2
L22 ∪ L3
1 L22 ∪ L3
2
BFS-Layering
O(|E |+ |V |)
1 Construct a BFS tree.2 From bottom, put two
vertices in the same bag if
are in the same layer andexists a path of verticesfurther from root.
3 Constructtree-decomposition T (G ).
Theorem [Doursboure, Gavoille.EUROCOMB 2003]
tl(G ) ≤ length(T ) ≤ 3 tl(G ) + 1
Fabien de Montgolfier, Mauricio Soto, Laurent Viennot Treewidth and Treelength of Internet 15 / 27
Page 66
Introduction Data Source Treelength Treewidth Conclusions
Layer Tree [Chepoi, Dragan. Europ. J. Combin. 2000]
L0
L0 ∪ L11
L11 ∪ L2
1
L21 ∪ L3
1
L11 ∪ L2
2
L22 ∪ L3
1 L22 ∪ L3
2
BFS-Layering
O(|E |+ |V |)
1 Construct a BFS tree.2 From bottom, put two
vertices in the same bag if
are in the same layer andexists a path of verticesfurther from root.
3 Constructtree-decomposition T (G ).
Theorem [Doursboure, Gavoille.EUROCOMB 2003]
tl(G ) ≤ length(T ) ≤ 3 tl(G ) + 1
Fabien de Montgolfier, Mauricio Soto, Laurent Viennot Treewidth and Treelength of Internet 15 / 27
Page 67
Introduction Data Source Treelength Treewidth Conclusions
Layer Tree [Chepoi, Dragan. Europ. J. Combin. 2000]
L0
L0 ∪ L11
L11 ∪ L2
1
L21 ∪ L3
1
L11 ∪ L2
2
L22 ∪ L3
1 L22 ∪ L3
2
BFS-Layering
O(|E |+ |V |)
1 Construct a BFS tree.2 From bottom, put two
vertices in the same bag if
are in the same layer andexists a path of verticesfurther from root.
3 Constructtree-decomposition T (G ).
Theorem [Doursboure, Gavoille.EUROCOMB 2003]
tl(G ) ≤ length(T ) ≤ 3 tl(G ) + 1
Fabien de Montgolfier, Mauricio Soto, Laurent Viennot Treewidth and Treelength of Internet 15 / 27
Page 68
Introduction Data Source Treelength Treewidth Conclusions
Layer Tree [Chepoi, Dragan. Europ. J. Combin. 2000]
L0
L0 ∪ L11
L11 ∪ L2
1
L21 ∪ L3
1
L11 ∪ L2
2
L22 ∪ L3
1 L22 ∪ L3
2
BFS-Layering
O(|E |+ |V |)
1 Construct a BFS tree.2 From bottom, put two
vertices in the same bag if
are in the same layer andexists a path of verticesfurther from root.
3 Constructtree-decomposition T (G ).
Theorem [Doursboure, Gavoille.EUROCOMB 2003]
tl(G ) ≤ length(T ) ≤ 3 tl(G ) + 1
Fabien de Montgolfier, Mauricio Soto, Laurent Viennot Treewidth and Treelength of Internet 15 / 27
Page 69
Introduction Data Source Treelength Treewidth Conclusions
Layer Tree [Chepoi, Dragan. Europ. J. Combin. 2000]
L0
L0 ∪ L11
L11 ∪ L2
1
L21 ∪ L3
1
L11 ∪ L2
2
L22 ∪ L3
1 L22 ∪ L3
2
BFS-Layering
O(|E |+ |V |)
1 Construct a BFS tree.2 From bottom, put two
vertices in the same bag if
are in the same layer andexists a path of verticesfurther from root.
3 Constructtree-decomposition T (G ).
Theorem [Doursboure, Gavoille.EUROCOMB 2003]
tl(G ) ≤ length(T ) ≤ 3 tl(G ) + 1
Fabien de Montgolfier, Mauricio Soto, Laurent Viennot Treewidth and Treelength of Internet 15 / 27
Page 70
Introduction Data Source Treelength Treewidth Conclusions
Layer Tree [Chepoi, Dragan. Europ. J. Combin. 2000]
L0
L0 ∪ L11
L11 ∪ L2
1
L21 ∪ L3
1
L11 ∪ L2
2
L22 ∪ L3
1 L22 ∪ L3
2
BFS-Layering
O(|E |+ |V |)
1 Construct a BFS tree.2 From bottom, put two
vertices in the same bag if
are in the same layer andexists a path of verticesfurther from root.
3 Constructtree-decomposition T (G ).
Theorem [Doursboure, Gavoille.EUROCOMB 2003]
tl(G ) ≤ length(T ) ≤ 3 tl(G ) + 1
Fabien de Montgolfier, Mauricio Soto, Laurent Viennot Treewidth and Treelength of Internet 15 / 27
Page 71
Introduction Data Source Treelength Treewidth Conclusions
Layer Tree [Chepoi, Dragan. Europ. J. Combin. 2000]
L0
L0 ∪ L11
L11 ∪ L2
1
L21 ∪ L3
1
L11 ∪ L2
2
L22 ∪ L3
1 L22 ∪ L3
2
BFS-Layering
O(|E |+ |V |)
1 Construct a BFS tree.2 From bottom, put two
vertices in the same bag if
are in the same layer andexists a path of verticesfurther from root.
3 Constructtree-decomposition T (G ).
Theorem [Doursboure, Gavoille.EUROCOMB 2003]
tl(G ) ≤ length(T ) ≤ 3 tl(G ) + 1
Fabien de Montgolfier, Mauricio Soto, Laurent Viennot Treewidth and Treelength of Internet 15 / 27
Page 72
Introduction Data Source Treelength Treewidth Conclusions
Layer Tree [Chepoi, Dragan. Europ. J. Combin. 2000]
L31 L3
3L32
L21 L2
2
L11
L0
L0
L0 ∪ L11
L11 ∪ L2
1
L21 ∪ L3
1
L11 ∪ L2
2
L22 ∪ L3
1 L22 ∪ L3
2
BFS-Layering
O(|E |+ |V |)
1 Construct a BFS tree.2 From bottom, put two
vertices in the same bag if
are in the same layer andexists a path of verticesfurther from root.
3 Constructtree-decomposition T (G ).
Theorem [Doursboure, Gavoille.EUROCOMB 2003]
tl(G ) ≤ length(T ) ≤ 3 tl(G ) + 1
Fabien de Montgolfier, Mauricio Soto, Laurent Viennot Treewidth and Treelength of Internet 15 / 27
Page 73
Introduction Data Source Treelength Treewidth Conclusions
Layer Tree [Chepoi, Dragan. Europ. J. Combin. 2000]
L31 L3
3L32
L21 L2
2
L11
L0
L0
L0 ∪ L11
L11 ∪ L2
1
L21 ∪ L3
1
L11 ∪ L2
2
L22 ∪ L3
1 L22 ∪ L3
2
BFS-Layering
O(|E |+ |V |)
1 Construct a BFS tree.2 From bottom, put two
vertices in the same bag if
are in the same layer andexists a path of verticesfurther from root.
3 Constructtree-decomposition T (G ).
Theorem [Doursboure, Gavoille.EUROCOMB 2003]
tl(G ) ≤ length(T ) ≤ 3 tl(G ) + 1
Fabien de Montgolfier, Mauricio Soto, Laurent Viennot Treewidth and Treelength of Internet 15 / 27
Page 74
Introduction Data Source Treelength Treewidth Conclusions
Layer Tree [Chepoi, Dragan. Europ. J. Combin. 2000]
L31 L3
3L32
L21 L2
2
L11
L0
L0
L0 ∪ L11
L11 ∪ L2
1
L21 ∪ L3
1
L11 ∪ L2
2
L22 ∪ L3
1 L22 ∪ L3
2
BFS-Layering
O(|E |+ |V |)
1 Construct a BFS tree.2 From bottom, put two
vertices in the same bag if
are in the same layer andexists a path of verticesfurther from root.
3 Constructtree-decomposition T (G ).
Theorem [Doursboure, Gavoille.EUROCOMB 2003]
tl(G ) ≤ length(T ) ≤ 3 tl(G ) + 1
Fabien de Montgolfier, Mauricio Soto, Laurent Viennot Treewidth and Treelength of Internet 15 / 27
Page 75
Introduction Data Source Treelength Treewidth Conclusions
Layer Tree [Chepoi, Dragan. Europ. J. Combin. 2000]
L31 L3
3L32
L21 L2
2
L11
L0
L0
L0 ∪ L11
L11 ∪ L2
1
L21 ∪ L3
1
L11 ∪ L2
2
L22 ∪ L3
1 L22 ∪ L3
2
BFS-Layering O(|E |+ |V |)1 Construct a BFS tree.2 From bottom, put two
vertices in the same bag if
are in the same layer andexists a path of verticesfurther from root.
3 Constructtree-decomposition T (G ).
Theorem [Doursboure, Gavoille.EUROCOMB 2003]
tl(G ) ≤ length(T ) ≤ 3 tl(G ) + 1
Fabien de Montgolfier, Mauricio Soto, Laurent Viennot Treewidth and Treelength of Internet 15 / 27
Page 76
Introduction Data Source Treelength Treewidth Conclusions
Layer Tree [Chepoi, Dragan. Europ. J. Combin. 2000]
L31 L3
3L32
L21 L2
2
L11
L0
L0
L0 ∪ L11
L11 ∪ L2
1
L21 ∪ L3
1
L11 ∪ L2
2
L22 ∪ L3
1 L22 ∪ L3
2
BFS-Layering O(|E |+ |V |)1 Construct a BFS tree.2 From bottom, put two
vertices in the same bag if
are in the same layer andexists a path of verticesfurther from root.
3 Constructtree-decomposition T (G ).
Theorem [Doursboure, Gavoille.EUROCOMB 2003]
tl(G ) ≤ length(T ) ≤ 3 tl(G ) + 1
Fabien de Montgolfier, Mauricio Soto, Laurent Viennot Treewidth and Treelength of Internet 15 / 27
Page 77
Introduction Data Source Treelength Treewidth Conclusions
Treelength: Bounds
Router graph AS
BFS-Layer 10 6
MCS - 56δ − 2 58 28
Diameter 19 8
Fabien de Montgolfier, Mauricio Soto, Laurent Viennot Treewidth and Treelength of Internet 16 / 27
Page 78
Introduction Data Source Treelength Treewidth Conclusions
Treelength: Bounds
Router graph AS
BFS-Layer 10 6MCS - 5
6δ − 2 58 28Diameter 19 8
Fabien de Montgolfier, Mauricio Soto, Laurent Viennot Treewidth and Treelength of Internet 16 / 27
Page 79
Introduction Data Source Treelength Treewidth Conclusions
Treelength: Bounds
Router graph AS
BFS-Layer 10 6MCS - 5
6δ − 2 58 28
Diameter 19 8
Fabien de Montgolfier, Mauricio Soto, Laurent Viennot Treewidth and Treelength of Internet 16 / 27
Page 80
Introduction Data Source Treelength Treewidth Conclusions
Treelength: Bounds
Router graph AS
BFS-Layer 10 6MCS - 5
6δ − 2 58 28Diameter 19 8
Fabien de Montgolfier, Mauricio Soto, Laurent Viennot Treewidth and Treelength of Internet 16 / 27
Page 81
Introduction Data Source Treelength Treewidth Conclusions Lower Bound Upper Bound
Outline
1 IntroductionTree Decomposition
2 Data Source
3 Treelength
4 TreewidthLower BoundUpper Bound
5 Conclusions
Fabien de Montgolfier, Mauricio Soto, Laurent Viennot Treewidth and Treelength of Internet 17 / 27
Page 82
Introduction Data Source Treelength Treewidth Conclusions Lower Bound Upper Bound
Treewidth
tw(G ) = 1 ⇐⇒ G is a tree
tw(G ) ≤ 2 ⇐⇒ G has no K 4 as minor (Series parallelgraphs, cycles, outerplanar, . . . )
tw(Kn) = n − 1
tw(grid(n ×m)) = min{n,m}
Treewidth Bounds
Router AS
Lower Bound 372 82Upper Bound - 473
Dijk, van den Heuvel, Slob, Bodlaender.www.treewidth.com
Fabien de Montgolfier, Mauricio Soto, Laurent Viennot Treewidth and Treelength of Internet 18 / 27
Page 83
Introduction Data Source Treelength Treewidth Conclusions Lower Bound Upper Bound
Treewidth
tw(G ) = 1 ⇐⇒ G is a tree
tw(G ) ≤ 2 ⇐⇒ G has no K 4 as minor (Series parallelgraphs, cycles, outerplanar, . . . )
tw(Kn) = n − 1
tw(grid(n ×m)) = min{n,m}
Treewidth Bounds
Router AS
Lower Bound 372 82Upper Bound - 473
Dijk, van den Heuvel, Slob, Bodlaender.www.treewidth.com
Fabien de Montgolfier, Mauricio Soto, Laurent Viennot Treewidth and Treelength of Internet 18 / 27
Page 84
Introduction Data Source Treelength Treewidth Conclusions Lower Bound Upper Bound
Treewidth
tw(G ) = 1 ⇐⇒ G is a tree
tw(G ) ≤ 2 ⇐⇒ G has no K 4 as minor (Series parallelgraphs, cycles, outerplanar, . . . )
tw(Kn) = n − 1
tw(grid(n ×m)) = min{n,m}
Treewidth Bounds
Router AS
Lower Bound 372 82Upper Bound - 473
Dijk, van den Heuvel, Slob, Bodlaender.www.treewidth.com
Fabien de Montgolfier, Mauricio Soto, Laurent Viennot Treewidth and Treelength of Internet 18 / 27
Page 85
Introduction Data Source Treelength Treewidth Conclusions Lower Bound Upper Bound
Treewidth
tw(G ) = 1 ⇐⇒ G is a tree
tw(G ) ≤ 2 ⇐⇒ G has no K 4 as minor (Series parallelgraphs, cycles, outerplanar, . . . )
tw(Kn) = n − 1
tw(grid(n ×m)) = min{n,m}
Treewidth Bounds
Router AS
Lower Bound 372 82Upper Bound - 473
Dijk, van den Heuvel, Slob, Bodlaender.www.treewidth.com
Fabien de Montgolfier, Mauricio Soto, Laurent Viennot Treewidth and Treelength of Internet 18 / 27
Page 86
Introduction Data Source Treelength Treewidth Conclusions Lower Bound Upper Bound
Treewidth
tw(G ) = 1 ⇐⇒ G is a tree
tw(G ) ≤ 2 ⇐⇒ G has no K 4 as minor (Series parallelgraphs, cycles, outerplanar, . . . )
tw(Kn) = n − 1
tw(grid(n ×m)) = min{n,m}
Treewidth Bounds
Router AS
Lower Bound 372 82Upper Bound - 473
Dijk, van den Heuvel, Slob, Bodlaender.www.treewidth.com
Fabien de Montgolfier, Mauricio Soto, Laurent Viennot Treewidth and Treelength of Internet 18 / 27
Page 87
Introduction Data Source Treelength Treewidth Conclusions Lower Bound Upper Bound
Treewidth
tw(G ) = 1 ⇐⇒ G is a tree
tw(G ) ≤ 2 ⇐⇒ G has no K 4 as minor (Series parallelgraphs, cycles, outerplanar, . . . )
tw(Kn) = n − 1
tw(grid(n ×m)) = min{n,m}
Treewidth Bounds
Router AS
Lower Bound 372 82Upper Bound - 473
Dijk, van den Heuvel, Slob, Bodlaender.www.treewidth.com
Fabien de Montgolfier, Mauricio Soto, Laurent Viennot Treewidth and Treelength of Internet 18 / 27
Page 88
Introduction Data Source Treelength Treewidth Conclusions Lower Bound Upper Bound
Treewidth
tw(G ) = 1 ⇐⇒ G is a tree
tw(G ) ≤ 2 ⇐⇒ G has no K 4 as minor (Series parallelgraphs, cycles, outerplanar, . . . )
tw(Kn) = n − 1
tw(grid(n ×m)) = min{n,m}
Treewidth Bounds
Router AS
Lower Bound 372 82Upper Bound - 473
Dijk, van den Heuvel, Slob, Bodlaender.www.treewidth.com
Fabien de Montgolfier, Mauricio Soto, Laurent Viennot Treewidth and Treelength of Internet 18 / 27
Page 89
Introduction Data Source Treelength Treewidth Conclusions Lower Bound Upper Bound
Treewidth: Lower Bound
Theorem
For all tree-decomposition of G exists a vertex who has all itsneighbors in the same bags.
Min Degree: tw(G ) ≥ minv∈V (G) d(v)
Theorem
If H is a minor of G then tw(H) ≤ tw(G ).
Maximum Minimum Degree(MMD): Delete vertex ofminimum degree.Maximum Minimum Degree(MMD+): Contract vertex ofminimum degree.
Theorem [Lucena. SIAM J. Disc. Math., 2003]
tw(G ) ≥ maximum over labels on a MCS.
Fabien de Montgolfier, Mauricio Soto, Laurent Viennot Treewidth and Treelength of Internet 19 / 27
Page 90
Introduction Data Source Treelength Treewidth Conclusions Lower Bound Upper Bound
Treewidth: Lower Bound
Theorem
For all tree-decomposition of G exists a vertex who has all itsneighbors in the same bags.
Min Degree: tw(G ) ≥ minv∈V (G) d(v)
Theorem
If H is a minor of G then tw(H) ≤ tw(G ).
Maximum Minimum Degree(MMD): Delete vertex ofminimum degree.Maximum Minimum Degree(MMD+): Contract vertex ofminimum degree.
Theorem [Lucena. SIAM J. Disc. Math., 2003]
tw(G ) ≥ maximum over labels on a MCS.
Fabien de Montgolfier, Mauricio Soto, Laurent Viennot Treewidth and Treelength of Internet 19 / 27
Page 91
Introduction Data Source Treelength Treewidth Conclusions Lower Bound Upper Bound
Treewidth: Lower Bound
Theorem
For all tree-decomposition of G exists a vertex who has all itsneighbors in the same bags.
Min Degree: tw(G ) ≥ minv∈V (G) d(v)
Theorem
If H is a minor of G then tw(H) ≤ tw(G ).
Maximum Minimum Degree(MMD): Delete vertex ofminimum degree.Maximum Minimum Degree(MMD+): Contract vertex ofminimum degree.
Theorem [Lucena. SIAM J. Disc. Math., 2003]
tw(G ) ≥ maximum over labels on a MCS.
Fabien de Montgolfier, Mauricio Soto, Laurent Viennot Treewidth and Treelength of Internet 19 / 27
Page 92
Introduction Data Source Treelength Treewidth Conclusions Lower Bound Upper Bound
Treewidth: Lower Bound
Theorem
For all tree-decomposition of G exists a vertex who has all itsneighbors in the same bags.
Min Degree: tw(G ) ≥ minv∈V (G) d(v)
Theorem
If H is a minor of G then tw(H) ≤ tw(G ).
Maximum Minimum Degree(MMD): Delete vertex ofminimum degree.Maximum Minimum Degree(MMD+): Contract vertex ofminimum degree.
Theorem [Lucena. SIAM J. Disc. Math., 2003]
tw(G ) ≥ maximum over labels on a MCS.
Fabien de Montgolfier, Mauricio Soto, Laurent Viennot Treewidth and Treelength of Internet 19 / 27
Page 93
Introduction Data Source Treelength Treewidth Conclusions Lower Bound Upper Bound
Treewidth: Lower Bound
Theorem
For all tree-decomposition of G exists a vertex who has all itsneighbors in the same bags.
Min Degree: tw(G ) ≥ minv∈V (G) d(v)
Theorem
If H is a minor of G then tw(H) ≤ tw(G ).
Maximum Minimum Degree(MMD): Delete vertex ofminimum degree.
Maximum Minimum Degree(MMD+): Contract vertex ofminimum degree.
Theorem [Lucena. SIAM J. Disc. Math., 2003]
tw(G ) ≥ maximum over labels on a MCS.
Fabien de Montgolfier, Mauricio Soto, Laurent Viennot Treewidth and Treelength of Internet 19 / 27
Page 94
Introduction Data Source Treelength Treewidth Conclusions Lower Bound Upper Bound
Treewidth: Lower Bound
Theorem
For all tree-decomposition of G exists a vertex who has all itsneighbors in the same bags.
Min Degree: tw(G ) ≥ minv∈V (G) d(v)
Theorem
If H is a minor of G then tw(H) ≤ tw(G ).
Maximum Minimum Degree(MMD): Delete vertex ofminimum degree.Maximum Minimum Degree(MMD+): Contract vertex ofminimum degree.
Theorem [Lucena. SIAM J. Disc. Math., 2003]
tw(G ) ≥ maximum over labels on a MCS.
Fabien de Montgolfier, Mauricio Soto, Laurent Viennot Treewidth and Treelength of Internet 19 / 27
Page 95
Introduction Data Source Treelength Treewidth Conclusions Lower Bound Upper Bound
Treewidth: Lower Bound
Theorem
For all tree-decomposition of G exists a vertex who has all itsneighbors in the same bags.
Min Degree: tw(G ) ≥ minv∈V (G) d(v)
Theorem
If H is a minor of G then tw(H) ≤ tw(G ).
Maximum Minimum Degree(MMD): Delete vertex ofminimum degree.Maximum Minimum Degree(MMD+): Contract vertex ofminimum degree.
Theorem [Lucena. SIAM J. Disc. Math., 2003]
tw(G ) ≥ maximum over labels on a MCS.
Fabien de Montgolfier, Mauricio Soto, Laurent Viennot Treewidth and Treelength of Internet 19 / 27
Page 96
Introduction Data Source Treelength Treewidth Conclusions Lower Bound Upper Bound
Treewidth: Lower Bound
Router AS
MMD 3 2MCS 34 26
MMD+Min-d 218 63MMD+Least-c 372 82
Fabien de Montgolfier, Mauricio Soto, Laurent Viennot Treewidth and Treelength of Internet 20 / 27
Page 97
Introduction Data Source Treelength Treewidth Conclusions Lower Bound Upper Bound
Elimination ordering
Given an elimination orderingπ(1), π(2), . . . , π(n) of nodes.
Fill-in graph G+π
G 0 = Gfor i = 1 to n do
v = π(i)Make NG i−1(v) a cliqueG i = G i−1 − v
end forG+π =
⋃n−1i=0 G i
An elimination order is perfect ifG+π = G .
π : a b c e d f g
a
b
c
d
e
f
g
Fabien de Montgolfier, Mauricio Soto, Laurent Viennot Treewidth and Treelength of Internet 21 / 27
Page 98
Introduction Data Source Treelength Treewidth Conclusions Lower Bound Upper Bound
Elimination ordering
Given an elimination orderingπ(1), π(2), . . . , π(n) of nodes.
Fill-in graph G+π
G 0 = Gfor i = 1 to n do
v = π(i)Make NG i−1(v) a cliqueG i = G i−1 − v
end forG+π =
⋃n−1i=0 G i
An elimination order is perfect ifG+π = G .
π : a b c e d f g
a
b
c
d
e
f
g
Fabien de Montgolfier, Mauricio Soto, Laurent Viennot Treewidth and Treelength of Internet 21 / 27
Page 99
Introduction Data Source Treelength Treewidth Conclusions Lower Bound Upper Bound
Elimination ordering
Given an elimination orderingπ(1), π(2), . . . , π(n) of nodes.
Fill-in graph G+π
G 0 = Gfor i = 1 to n do
v = π(i)
Make NG i−1(v) a cliqueG i = G i−1 − v
end forG+π =
⋃n−1i=0 G i
An elimination order is perfect ifG+π = G .
π : a b c e d f g
a
b
c
d
e
f
g
a
Fabien de Montgolfier, Mauricio Soto, Laurent Viennot Treewidth and Treelength of Internet 21 / 27
Page 100
Introduction Data Source Treelength Treewidth Conclusions Lower Bound Upper Bound
Elimination ordering
Given an elimination orderingπ(1), π(2), . . . , π(n) of nodes.
Fill-in graph G+π
G 0 = Gfor i = 1 to n do
v = π(i)Make NG i−1(v) a clique
G i = G i−1 − vend forG+π =
⋃n−1i=0 G i
An elimination order is perfect ifG+π = G .
π : a b c e d f g
a
b
c
d
e
f
g
a
Fabien de Montgolfier, Mauricio Soto, Laurent Viennot Treewidth and Treelength of Internet 21 / 27
Page 101
Introduction Data Source Treelength Treewidth Conclusions Lower Bound Upper Bound
Elimination ordering
Given an elimination orderingπ(1), π(2), . . . , π(n) of nodes.
Fill-in graph G+π
G 0 = Gfor i = 1 to n do
v = π(i)Make NG i−1(v) a cliqueG i = G i−1 − v
end forG+π =
⋃n−1i=0 G i
An elimination order is perfect ifG+π = G .
π : a b c e d f g
a
b
c
d
e
f
g
b
a
Fabien de Montgolfier, Mauricio Soto, Laurent Viennot Treewidth and Treelength of Internet 21 / 27
Page 102
Introduction Data Source Treelength Treewidth Conclusions Lower Bound Upper Bound
Elimination ordering
Given an elimination orderingπ(1), π(2), . . . , π(n) of nodes.
Fill-in graph G+π
G 0 = Gfor i = 1 to n do
v = π(i)Make NG i−1(v) a cliqueG i = G i−1 − v
end forG+π =
⋃n−1i=0 G i
An elimination order is perfect ifG+π = G .
π : a b c e d f g
a
b
c
d
e
f
gc
a
b
Fabien de Montgolfier, Mauricio Soto, Laurent Viennot Treewidth and Treelength of Internet 21 / 27
Page 103
Introduction Data Source Treelength Treewidth Conclusions Lower Bound Upper Bound
Elimination ordering
Given an elimination orderingπ(1), π(2), . . . , π(n) of nodes.
Fill-in graph G+π
G 0 = Gfor i = 1 to n do
v = π(i)Make NG i−1(v) a cliqueG i = G i−1 − v
end forG+π =
⋃n−1i=0 G i
An elimination order is perfect ifG+π = G .
π : a b c e d f g
a
b
c
d
e
f
g
e
a
b
c
Fabien de Montgolfier, Mauricio Soto, Laurent Viennot Treewidth and Treelength of Internet 21 / 27
Page 104
Introduction Data Source Treelength Treewidth Conclusions Lower Bound Upper Bound
Elimination ordering
Given an elimination orderingπ(1), π(2), . . . , π(n) of nodes.
Fill-in graph G+π
G 0 = Gfor i = 1 to n do
v = π(i)Make NG i−1(v) a cliqueG i = G i−1 − v
end forG+π =
⋃n−1i=0 G i
An elimination order is perfect ifG+π = G .
π : a b c e d f g
a
b
c
d
e
f
g
da
b
c
e
Fabien de Montgolfier, Mauricio Soto, Laurent Viennot Treewidth and Treelength of Internet 21 / 27
Page 105
Introduction Data Source Treelength Treewidth Conclusions Lower Bound Upper Bound
Elimination ordering
Given an elimination orderingπ(1), π(2), . . . , π(n) of nodes.
Fill-in graph G+π
G 0 = Gfor i = 1 to n do
v = π(i)Make NG i−1(v) a cliqueG i = G i−1 − v
end forG+π =
⋃n−1i=0 G i
An elimination order is perfect ifG+π = G .
π : a b c e d f g
a
b
c
d
e
f
g
fa
b
c
e
d
Fabien de Montgolfier, Mauricio Soto, Laurent Viennot Treewidth and Treelength of Internet 21 / 27
Page 106
Introduction Data Source Treelength Treewidth Conclusions Lower Bound Upper Bound
Elimination ordering
Given an elimination orderingπ(1), π(2), . . . , π(n) of nodes.
Fill-in graph G+π
G 0 = Gfor i = 1 to n do
v = π(i)Make NG i−1(v) a cliqueG i = G i−1 − v
end forG+π =
⋃n−1i=0 G i
An elimination order is perfect ifG+π = G .
π : a b c e d f g
a
b
c
d
e
f
gg
a
b
c
e
d f
Fabien de Montgolfier, Mauricio Soto, Laurent Viennot Treewidth and Treelength of Internet 21 / 27
Page 107
Introduction Data Source Treelength Treewidth Conclusions Lower Bound Upper Bound
Elimination ordering
Given an elimination orderingπ(1), π(2), . . . , π(n) of nodes.
Fill-in graph G+π
G 0 = Gfor i = 1 to n do
v = π(i)Make NG i−1(v) a cliqueG i = G i−1 − v
end forG+π =
⋃n−1i=0 G i
An elimination order is perfect ifG+π = G .
π : a b c e d f g
a
b
c
d
e
f
g
a
b
c
e
d f
g
a
b
c
d
e
f
g
Fabien de Montgolfier, Mauricio Soto, Laurent Viennot Treewidth and Treelength of Internet 21 / 27
Page 108
Introduction Data Source Treelength Treewidth Conclusions Lower Bound Upper Bound
Elimination ordering
Given an elimination orderingπ(1), π(2), . . . , π(n) of nodes.
Fill-in graph G+π
G 0 = Gfor i = 1 to n do
v = π(i)Make NG i−1(v) a cliqueG i = G i−1 − v
end forG+π =
⋃n−1i=0 G i
An elimination order is perfect ifG+π = G .
π : a b c e d f g
a
b
c
d
e
f
g
a
b
c
e
d f
g
a
b
c
d
e
f
g
Fabien de Montgolfier, Mauricio Soto, Laurent Viennot Treewidth and Treelength of Internet 21 / 27
Page 109
Introduction Data Source Treelength Treewidth Conclusions Lower Bound Upper Bound
Triangulated Graph
Theorem [70’]
Let G be a graph. The following are equivalent.
G is chordal.
G has a perfect elimination order.
G is the intersection graph of subtrees of a tree, i.e., G has atree decomposition such that each bag is a clique.
Corollary
tw(G ) is the minimum over all
minimal
triangulations of G of themaximum clique minus one.
Fabien de Montgolfier, Mauricio Soto, Laurent Viennot Treewidth and Treelength of Internet 22 / 27
Page 110
Introduction Data Source Treelength Treewidth Conclusions Lower Bound Upper Bound
Triangulated Graph
Theorem [70’]
Let G be a graph. The following are equivalent.
G is chordal.
G has a perfect elimination order.
G is the intersection graph of subtrees of a tree, i.e., G has atree decomposition such that each bag is a clique.
Corollary
tw(G ) is the minimum over all
minimal
triangulations of G of themaximum clique minus one.
Fabien de Montgolfier, Mauricio Soto, Laurent Viennot Treewidth and Treelength of Internet 22 / 27
Page 111
Introduction Data Source Treelength Treewidth Conclusions Lower Bound Upper Bound
Triangulated Graph
Theorem [70’]
Let G be a graph. The following are equivalent.
G is chordal.
G has a perfect elimination order.
G is the intersection graph of subtrees of a tree, i.e., G has atree decomposition such that each bag is a clique.
Corollary
tw(G ) is the minimum over all minimal triangulations of G of themaximum clique minus one.
Fabien de Montgolfier, Mauricio Soto, Laurent Viennot Treewidth and Treelength of Internet 22 / 27
Page 112
Introduction Data Source Treelength Treewidth Conclusions Lower Bound Upper Bound
Triangulated Graph
Theorem [70’]
Let G be a graph. The following are equivalent.
G is chordal.
G has a perfect elimination order.
G is the intersection graph of subtrees of a tree, i.e., G has atree decomposition such that each bag is a clique.
Corollary
tl(G ) is the minimum over all minimal triangulations of G of thediameter (on G ) of a bag.
Fabien de Montgolfier, Mauricio Soto, Laurent Viennot Treewidth and Treelength of Internet 22 / 27
Page 113
Introduction Data Source Treelength Treewidth Conclusions Lower Bound Upper Bound
Treewidth: Upper Bound
Upper Bound Heuristic
1 Generate a permutation π.
2 Construct G+π .
3 Compute tw(G+π ).
MCS
LexBFS
Minimum Degree : Takevertex with minimumdegree.
Minimum Fill-In : Takevertex that generate lessnew edges.
...
π : c a e b d f g
a
b
c
d
e
f
g
a c d
a b dd f g
b e
Fabien de Montgolfier, Mauricio Soto, Laurent Viennot Treewidth and Treelength of Internet 23 / 27
Page 114
Introduction Data Source Treelength Treewidth Conclusions Lower Bound Upper Bound
Treewidth: Upper Bound
Upper Bound Heuristic
1 Generate a permutation π.
2 Construct G+π .
3 Compute tw(G+π ).
MCS
LexBFS
Minimum Degree : Takevertex with minimumdegree.
Minimum Fill-In : Takevertex that generate lessnew edges.
...
π : c a e b d f g
a
b
c
d
e
f
g
a c d
a b dd f g
b e
Fabien de Montgolfier, Mauricio Soto, Laurent Viennot Treewidth and Treelength of Internet 23 / 27
Page 115
Introduction Data Source Treelength Treewidth Conclusions Lower Bound Upper Bound
Treewidth: Upper Bound
Upper Bound Heuristic
1 Generate a permutation π.
2 Construct G+π .
3 Compute tw(G+π ).
MCS
LexBFS
Minimum Degree : Takevertex with minimumdegree.
Minimum Fill-In : Takevertex that generate lessnew edges.
...
π : c a e b d f g
a
b
c
d
e
f
g
a c d
a b dd f g
b e
Fabien de Montgolfier, Mauricio Soto, Laurent Viennot Treewidth and Treelength of Internet 23 / 27
Page 116
Introduction Data Source Treelength Treewidth Conclusions Lower Bound Upper Bound
Treewidth: Upper Bound
Upper Bound Heuristic
1 Generate a permutation π.
2 Construct G+π .
3 Compute tw(G+π ).
MCS
LexBFS
Minimum Degree : Takevertex with minimumdegree.
Minimum Fill-In : Takevertex that generate lessnew edges.
...
π : c a e b d f g
a
b
c
d
e
f
g
a c d
a b dd f g
b e
Fabien de Montgolfier, Mauricio Soto, Laurent Viennot Treewidth and Treelength of Internet 23 / 27
Page 117
Introduction Data Source Treelength Treewidth Conclusions Lower Bound Upper Bound
Treewidth: Upper Bound
Upper Bound Heuristic
1 Generate a permutation π.
2 Construct G+π .
3 Compute tw(G+π ).
MCS
LexBFS
Minimum Degree : Takevertex with minimumdegree.
Minimum Fill-In : Takevertex that generate lessnew edges.
...
π : c a e b d f g
a
b
c
d
e
f
g
a c d
a b dd f g
b e
Fabien de Montgolfier, Mauricio Soto, Laurent Viennot Treewidth and Treelength of Internet 23 / 27
Page 118
Introduction Data Source Treelength Treewidth Conclusions Lower Bound Upper Bound
Treewidth: Upper Bound
Router AS
LexBFS - 912MCS-M - 912MCS - 473
Minimum Degree -Minimum Fill-Inn -
Fabien de Montgolfier, Mauricio Soto, Laurent Viennot Treewidth and Treelength of Internet 24 / 27
Page 119
Introduction Data Source Treelength Treewidth Conclusions
Outline
1 IntroductionTree Decomposition
2 Data Source
3 Treelength
4 TreewidthLower BoundUpper Bound
5 Conclusions
Fabien de Montgolfier, Mauricio Soto, Laurent Viennot Treewidth and Treelength of Internet 25 / 27
Page 120
Introduction Data Source Treelength Treewidth Conclusions
Future Work
Compute other parameters.
Better Bounds by
Graph decomposition.Graph Prepossessing.
Fabien de Montgolfier, Mauricio Soto, Laurent Viennot Treewidth and Treelength of Internet 26 / 27
Page 121
Introduction Data Source Treelength Treewidth Conclusions
Future Work
Compute other parameters.
Better Bounds by
Graph decomposition.Graph Prepossessing.
Fabien de Montgolfier, Mauricio Soto, Laurent Viennot Treewidth and Treelength of Internet 26 / 27
Page 122
Introduction Data Source Treelength Treewidth Conclusions
Future Work
Compute other parameters.
Better Bounds by
Graph decomposition.
Graph Prepossessing.
Fabien de Montgolfier, Mauricio Soto, Laurent Viennot Treewidth and Treelength of Internet 26 / 27
Page 123
Introduction Data Source Treelength Treewidth Conclusions
Future Work
Compute other parameters.
Better Bounds by
Graph decomposition.Graph Prepossessing.
Fabien de Montgolfier, Mauricio Soto, Laurent Viennot Treewidth and Treelength of Internet 26 / 27
Page 124
Introduction Data Source Treelength Treewidth Conclusions
Summary
Router AS
|V | 192244 27289|E | 609066 55771
Average degree 6.34 4.09Max. degree 1071 2616
Diameter 19 8
TreelengthUB 10 5LB 3 2
TreewidthLB 372 82UB - 472
Fabien de Montgolfier, Mauricio Soto, Laurent Viennot Treewidth and Treelength of Internet 27 / 27