treewidth and treelength of internetmausoto/talks/200811_talk_poitiers.pdf · 2013-01-24 ·...

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

Outline

1 IntroductionTree Decomposition

2 Data Source

3 Treelength

4 TreewidthLower BoundUpper Bound

5 Conclusions

Outline

2 Data Source

3 Treelength

5 Conclusions

Outline

2 Data Source

3 Treelength

5 Conclusions

Outline

2 Data Source

3 Treelength

5 Conclusions

Outline

2 Data Source

3 Treelength

5 Conclusions

Introduction Data Source Treelength Treewidth Conclusions

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.

Compute bounds on treewidth and treelength on Internetgraph.

Motivation

Compute bounds on treewidth and treelength on Internetgraph.

Motivation

Introduction Data Source Treelength Treewidth Conclusions Tree Decomposition

Outline

2 Data Source

3 Treelength

5 Conclusions

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 )

is the minimum over all treedecompositions.

a b dd f g

Tree Decomposition

- bags {Xi ⊆ V , i ∈ I}

- tree T = (I ,F )

a b dd f g

Tree Decomposition

a b dd f g

Tree Decomposition

a b dd f g

Tree Decomposition

a b dd f g

Tree Decomposition

a b dd f g

Tree Decomposition

a b dd f g

Tree Decomposition

a b dd f g

Tree Decomposition

Treewidth

tw(G ) is the minimum width over alltree decompositions.

a b dd f g

Tree Decomposition

Treelength

tl(G ) is the minimum length over alltree decompositions.

a b dd f g

Algorithmic Motivation

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.

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.

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.

Outline

2 Data Source

3 Treelength

5 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

Internet Routing

AS4AS5

Router network.

Autonomous Systems.

*> 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

Internet Routing

AS4AS5

Router network.

Autonomous Systems.

*> 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

Internet Routing

AS4AS5

Router network.

Autonomous Systems.

*> 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

Internet Routing

AS4AS5

Router network.

Autonomous Systems.

*> 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

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.

Traceroute

Traceroute:

Set of monitors.Destinations list.Sends probe packet obtainingconsecutive router links.Router graph.

Traceroute

Traceroute:Set of monitors.

Destinations list.Sends probe packet obtainingconsecutive router links.Router graph.

Traceroute

Traceroute:Set of monitors.Destinations list.

Sends probe packet obtainingconsecutive router links.Router graph.

Traceroute

Traceroute:Set of monitors.Destinations list.Sends probe packet obtainingconsecutive router links.

Router graph.

Traceroute

Router graph.

Traceroute

Router graph.

Traceroute

Router graph.

Traceroute

Traceroute:Set of monitors.Destinations list.Sends probe packet obtainingconsecutive router links.Router graph.

Traceroute

Map each router to its AS.

AS graph.

Traceroute

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

100000

1 10 100 1000 10000

Node Degree

Degree Distribution

ASitdk

Graph size

Router AS

Monitors 23 13List size 865K 7.4M|V | 192244 27289|E | 609066 55771

100000

1 10 100 1000 10000

Node Degree

Degree Distribution

ASitdk

Graph size

Router AS

100000

1 10 100 1000 10000

Node Degree

Degree Distribution

ASitdk

Graph size

Router AS

100000

1 10 100 1000 10000

Node Degree

Degree Distribution

ASitdk

Graph size

Router AS

100000

1 10 100 1000 10000

Node Degree

Degree Distribution

ASitdk

Graph decomposition

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%

Graph decomposition

Router AS

# Connect. comp. 308 1

# Biconnect. comp. 53104 10505Biconnect.comp. size 2 95.3% 99.8%

68.8% 61.4%|E | 541081 45221

88.8% 81.0%

Graph decomposition

Router AS

# Connect. comp. 308 1# Biconnect. comp. 53104 10505

Biconnect.comp. size 2 95.3% 99.8%

68.8% 61.4%|E | 541081 45221

88.8% 81.0%

Graph decomposition

Router AS

68.8% 61.4%|E | 541081 45221

88.8% 81.0%

Graph decomposition

Router AS

68.8% 61.4%|E | 541081 45221

88.8% 81.0%

Outline

2 Data Source

3 Treelength

5 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

Treelength

Treelength Bounds

Router AS

Treelength

tl(Ck) = dk/3e.

G is k−chordal ⇒ tl(G ) ≤ k/2 + 3.

Treelength Bounds

Router AS

Treelength

Treelength Bounds

Router AS

Treelength

Treelength Bounds

Router AS

Treelength

Treelength Bounds

Router AS

Layer Tree [Chepoi, Dragan. Europ. J. Combin. 2000]

L0 ∪ L11

L11 ∪ L2

L21 ∪ L3

L11 ∪ L2

L22 ∪ L3

1 L22 ∪ L3

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

L0 ∪ L11

L11 ∪ L2

L21 ∪ L3

L11 ∪ L2

L22 ∪ L3

1 L22 ∪ L3

BFS-Layering

O(|E |+ |V |)

1 Construct a BFS tree.2 From bottom, put two

vertices in the same bag if

tl(G ) ≤ length(T ) ≤ 3 tl(G ) + 1

L0 ∪ L11

L11 ∪ L2

L21 ∪ L3

L11 ∪ L2

L22 ∪ L3

1 L22 ∪ L3

BFS-Layering

O(|E |+ |V |)

are in the same layer and

exists a path of verticesfurther from root.

tl(G ) ≤ length(T ) ≤ 3 tl(G ) + 1

L0 ∪ L11

L11 ∪ L2

L21 ∪ L3

L11 ∪ L2

L22 ∪ L3

1 L22 ∪ L3

BFS-Layering

O(|E |+ |V |)

tl(G ) ≤ length(T ) ≤ 3 tl(G ) + 1

L0 ∪ L11

L11 ∪ L2

L21 ∪ L3

L11 ∪ L2

L22 ∪ L3

1 L22 ∪ L3

BFS-Layering

O(|E |+ |V |)

tl(G ) ≤ length(T ) ≤ 3 tl(G ) + 1

L0 ∪ L11

L11 ∪ L2

L21 ∪ L3

L11 ∪ L2

L22 ∪ L3

1 L22 ∪ L3

BFS-Layering

O(|E |+ |V |)

tl(G ) ≤ length(T ) ≤ 3 tl(G ) + 1

L0 ∪ L11

L11 ∪ L2

L21 ∪ L3

L11 ∪ L2

L22 ∪ L3

1 L22 ∪ L3

BFS-Layering

O(|E |+ |V |)

tl(G ) ≤ length(T ) ≤ 3 tl(G ) + 1

L0 ∪ L11

L11 ∪ L2

L21 ∪ L3

L11 ∪ L2

L22 ∪ L3

1 L22 ∪ L3

BFS-Layering

O(|E |+ |V |)

tl(G ) ≤ length(T ) ≤ 3 tl(G ) + 1

L0 ∪ L11

L11 ∪ L2

L21 ∪ L3

L11 ∪ L2

L22 ∪ L3

1 L22 ∪ L3

BFS-Layering

O(|E |+ |V |)

tl(G ) ≤ length(T ) ≤ 3 tl(G ) + 1

L0 ∪ L11

L11 ∪ L2

L21 ∪ L3

L11 ∪ L2

L22 ∪ L3

1 L22 ∪ L3

BFS-Layering

O(|E |+ |V |)

tl(G ) ≤ length(T ) ≤ 3 tl(G ) + 1

L0 ∪ L11

L11 ∪ L2

L21 ∪ L3

L11 ∪ L2

L22 ∪ L3

1 L22 ∪ L3

BFS-Layering

O(|E |+ |V |)

tl(G ) ≤ length(T ) ≤ 3 tl(G ) + 1

L31 L3

L21 L2

L0 ∪ L11

L11 ∪ L2

L21 ∪ L3

L11 ∪ L2

L22 ∪ L3

1 L22 ∪ L3

BFS-Layering

O(|E |+ |V |)

tl(G ) ≤ length(T ) ≤ 3 tl(G ) + 1

L31 L3

L21 L2

L0 ∪ L11

L11 ∪ L2

L21 ∪ L3

L11 ∪ L2

L22 ∪ L3

1 L22 ∪ L3

BFS-Layering

O(|E |+ |V |)

tl(G ) ≤ length(T ) ≤ 3 tl(G ) + 1

L31 L3

L21 L2

L0 ∪ L11

L11 ∪ L2

L21 ∪ L3

L11 ∪ L2

L22 ∪ L3

1 L22 ∪ L3

BFS-Layering

O(|E |+ |V |)

tl(G ) ≤ length(T ) ≤ 3 tl(G ) + 1

L31 L3

L21 L2

L0 ∪ L11

L11 ∪ L2

L21 ∪ L3

L11 ∪ L2

L22 ∪ L3

1 L22 ∪ L3

BFS-Layering O(|E |+ |V |)1 Construct a BFS tree.2 From bottom, put two

tl(G ) ≤ length(T ) ≤ 3 tl(G ) + 1

L31 L3

L21 L2

L0 ∪ L11

L11 ∪ L2

L21 ∪ L3

L11 ∪ L2

L22 ∪ L3

1 L22 ∪ L3

BFS-Layering O(|E |+ |V |)1 Construct a BFS tree.2 From bottom, put two

tl(G ) ≤ length(T ) ≤ 3 tl(G ) + 1

Treelength: Bounds

Router graph AS

BFS-Layer 10 6

MCS - 56δ − 2 58 28

Diameter 19 8

Treelength: Bounds

Router graph AS

BFS-Layer 10 6MCS - 5

6δ − 2 58 28Diameter 19 8

Treelength: Bounds

Router graph AS

6δ − 2 58 28

Diameter 19 8

Treelength: Bounds

Router graph AS

6δ − 2 58 28Diameter 19 8

Introduction Data Source Treelength Treewidth Conclusions Lower Bound Upper Bound

Outline

2 Data Source

3 Treelength

5 Conclusions

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

Treewidth

tw(Kn) = n − 1

Treewidth Bounds

Router AS

Treewidth

tw(Kn) = n − 1

Treewidth Bounds

Router AS

Treewidth

tw(Kn) = n − 1

Treewidth Bounds

Router AS

Treewidth

tw(Kn) = n − 1

Treewidth Bounds

Router AS

Treewidth

tw(Kn) = n − 1

Treewidth Bounds

Router AS

Treewidth

tw(Kn) = n − 1

Treewidth Bounds

Router AS

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.

Theorem

Maximum Minimum Degree(MMD): Delete vertex ofminimum degree.

Maximum Minimum Degree(MMD+): Contract vertex ofminimum degree.

Theorem

Router AS

MMD 3 2MCS 34 26

MMD+Min-d 218 63MMD+Least-c 372 82

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

Fill-in graph G+π

end forG+π =

⋃n−1i=0 G i

π : a b c e d f g

Fill-in graph G+π

v = π(i)

Make NG i−1(v) a cliqueG i = G i−1 − v

end forG+π =

⋃n−1i=0 G i

π : a b c e d f g

Fill-in graph G+π

v = π(i)Make NG i−1(v) a clique

G i = G i−1 − vend forG+π =

⋃n−1i=0 G i

π : a b c e d f g

Fill-in graph G+π

end forG+π =

⋃n−1i=0 G i

π : a b c e d f g

Fill-in graph G+π

end forG+π =

⋃n−1i=0 G i

π : a b c e d f g

Fill-in graph G+π

end forG+π =

⋃n−1i=0 G i

π : a b c e d f g

Fill-in graph G+π

end forG+π =

⋃n−1i=0 G i

π : a b c e d f g

Fill-in graph G+π

end forG+π =

⋃n−1i=0 G i

π : a b c e d f g

Fill-in graph G+π

end forG+π =

⋃n−1i=0 G i

π : a b c e d f g

Fill-in graph G+π

end forG+π =

⋃n−1i=0 G i

π : a b c e d f g

Fill-in graph G+π

end forG+π =

⋃n−1i=0 G i

π : a b c e d f g

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.

Triangulated Graph

Theorem [70’]

G is chordal.

Corollary

tw(G ) is the minimum over all

minimal

triangulations of G of themaximum clique minus one.

Triangulated Graph

Theorem [70’]

G is chordal.

Corollary

tw(G ) is the minimum over all minimal triangulations of G of themaximum clique minus one.

Triangulated Graph

Theorem [70’]

G is chordal.

Corollary

tl(G ) is the minimum over all minimal triangulations of G of thediameter (on G ) of a bag.

Treewidth: Upper Bound

Upper Bound Heuristic

1 Generate a permutation π.

2 Construct G+π .

3 Compute tw(G+π ).

LexBFS

Minimum Degree : Takevertex with minimumdegree.

Minimum Fill-In : Takevertex that generate lessnew edges.

π : c a e b d f g

a b dd f g

2 Construct G+π .

LexBFS

π : c a e b d f g

a b dd f g

2 Construct G+π .

LexBFS

π : c a e b d f g

a b dd f g

2 Construct G+π .

LexBFS

π : c a e b d f g

a b dd f g

2 Construct G+π .

LexBFS

π : c a e b d f g

a b dd f g

Router AS

LexBFS - 912MCS-M - 912MCS - 473

Minimum Degree -Minimum Fill-Inn -

Outline

2 Data Source

3 Treelength

5 Conclusions

Future Work

Compute other parameters.

Better Bounds by

Graph decomposition.Graph Prepossessing.

Future Work

Better Bounds by

Future Work

Better Bounds by

Graph decomposition.

Graph Prepossessing.

Future Work

Better Bounds by

Summary

Router AS

|V | 192244 27289|E | 609066 55771

Diameter 19 8

TreelengthUB 10 5LB 3 2

TreewidthLB 372 82UB - 472

treewidth and treelength of internetmausoto/talks/200811_talk_poitiers.pdf · 2013-01-24 ·...

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