assigning as relationships to satisfy the gao-rexford

UNIVERSITÀ DEGLI STUDI ROMA TRE

Dipartimento di Informatica e Automazione

Luca Cittadini (Roma Tre University)

Giuseppe Di Battista (Roma Tre University)

Thomas Erlebach (University of Leicester)

Maurizio Patrignani (Roma Tre University)

Massimo Rimondini (Roma Tre University)

Assigning AS Relationships to Satisfy the Gao-Rexford

Conditions

2010 Oct. 5 - 8IEEE

AS Relationships

2ICNP 2010 - Oct 7th

AS Relationships


Provider

Customer

$

AS Relationships


$Provider

Customer

$

AS Relationships


$ 10010

10010

01010

01010

Provider

Customer

AS Relationships


Provider

Customer

PeerPeer

AS Relationships


Provider

Customer

Cust.

Cust. Cust.

Cust.

Cust.

PeerPeer

ICNP 2010 - Oct 7th 10[1] L. Gao, J. Rexford. Stable Internet Routing without Global Coordination. SIGMETRICS, 2000

The Gao-Rexford[1] Conditions

Cust.

Prov.

PeerPeer



Cust.

Prov.

PeerPeer

1



Cust.

Prov.

PeerPeer

1

Acyclic



Cust.

Prov.

PeerPeer

1 2

Acyclic



Cust.

Prov.

PeerPeer

1 2

Acyclic Valley-free



Cust.

Prov.

PeerPeer

1 2 3

Acyclic Valley-free



Cust.

Prov.

PeerPeer

1 2 3

Acyclic Valley-freePrefer

customer

Literature(and a bit of motivation)

Safety [2] is important...

ICNP 2010 - Oct 7th 28

[2] T. Griffin, F. Shepherd, G. Wilfong. The Stable Paths Problem and Interdomain Routing. ToN, 2002.


Safety [2] is important......but hard to check [3], [4]



[3] A. Fabrikant, C. Papadimitriou. The Complexity of Game Dynamics: BGP Oscillations, Sink Equilibria, and beyond. SODA 2008.

[4] T. Griffin, G. Wilfong. An Analysis of BGP Convergence Properties. SIGCOMM 1999.


Safety [2] is important......but hard to check [3], [4]Achieved by different approaches




[4] T. Griffin, G. Wilfong. An Analysis of BGP Convergence Properties. SIGCOMM 1999.



let oscillations occur, but dynamically resolve them [5], [6]




[4] T. Griffin, G. Wilfong. An Analysis of BGP Convergence Properties. SIGCOMM 1999.[5] T. Griffin, G. Wilfong. A Safe Path Vector Protocol. INFOCOM 2000.[6] C. Ee, V. Ramachandran, B.-G. Chun, K. Lakshminarayanan, S. Shenker. Resolving Inter-

domain Policy Disputes. SIGCOMM 2007.



let oscillations occur, but dynamically resolve them [5], [6]limit policy expressiveness [7], [2]





domain Policy Disputes. SIGCOMM 2007.[7] N. Feamster, R. Johari, H. Balakrishnan. Implications of Autonomy for the Expressiveness of

Policy Routing. ToN, 2007.



let oscillations occur, but dynamically resolve them [5], [6]limit policy expressiveness [7], [2]GR







Motivation(and a bit of literature)

A GR-compliant network......preserves autonomy of each AS in configuring local policies




...is safe and robust [8]


[8] L. Gao, T. Griffin, J. Rexford. Inherently Safe Backup Routing with BGP. INFOCOM 2001




...has a convergence time that is roughly bounded by a constant [9]


[8] L. Gao, T. Griffin, J. Rexford. Inherently Safe Backup Routing with BGP. INFOCOM 2001[9] R. Sami, M. Schapira, A. Zohar. Searching for Stability in Interdomain Routing. INFOCOM 2009




...has a convergence time that is roughly bounded by a constant [9]

Remark:GR compliance is regarded as a possible explanation for Internet stability [2]


[8] L. Gao, T. Griffin, J. Rexford. Inherently Safe Backup Routing with BGP. INFOCOM 2001[9] R. Sami, M. Schapira, A. Zohar. Searching for Stability in Interdomain Routing. INFOCOM 2009

Other “Grail Seekers”

[10]: relationship inference heuristic


[10] L. Gao. On Inferring Autonomous System Relationships in the Internet. ToN, 2001.



[11]: a valley-free assignment can be achieved efficiently


[10] L. Gao. On Inferring Autonomous System Relationships in the Internet. ToN, 2001.[11] G. Di Battista, T. Erlebach, A. Hall, M. Patrignani, M. Pizzonia, T. Schank. Computing the

Types of the Relationships between Autonomous Systems. ToN, 2007.




[12]: a valley-free+acyclic assignment can be achieved efficiently



Types of the Relationships between Autonomous Systems. ToN, 2007.[12] S. Kosub, M. G. Maaß, H. Täubig. Acyclic Type-of-Relationship Problems on the Internet.

CAAN 2006.




[12]: a valley-free+acyclic assignment can be achieved efficiently

[13]: distributed detection of the GR conditions (with known relationships)



Types of the Relationships between Autonomous Systems. ToN, 2007.[12] S. Kosub, M. G. Maaß, H. Täubig. Acyclic Type-of-Relationship Problems on the Internet.

CAAN 2006.[13] S. Epstein, K. Mattar, I. Matta. Principles of Safe Policy Routing Dynamics. ICNP 2009.

44

ProblemGAO-REXFORD-CHECK

ICNP 2010 - Oct 7th

Instance: (model of) a BGP configuration

45


ICNP 2010 - Oct 7th


router bgp 100

!

neighbor 140.222.1.1 route-map FIX-WEIGHT in

neighbor 140.222.1.1 remote-as 1

!

ip as-path access-list 200 permit ^690$

ip as-path access-list 200 permit ^1800

!

route-map FIX-WEIGHT permit 10

match as-path 200

set local-preference 250

set weight200

46


ICNP 2010 - Oct 7th


47


ICNP 2010 - Oct 7th


48


ICNP 2010 - Oct 7th


49


ICNP 2010 - Oct 7th


50


ICNP 2010 - Oct 7th


Question: Can the network be partially oriented to a customer-provider graph that is GR-compliant?

Results


Results

1. Polynomial algorithm forGAO-REXFORD-CHECK


Results


GAO-REXFORD-STRICT-CHECK:

same as GAO-REXFORD-CHECK, but peers are preferred to providers


Results





Results




2. NP-hardness ofGAO-REXFORD-STRICT-CHECK


Models (briefly)


1 2

Models (briefly)


3 4

21

0

1 2

Models (briefly)


3 4

21

0

420430

21020

13010

30

1 2

Models (briefly)


3 4

21

0

420430

21020

13010

30

1 2

Stable Paths Problem(SPP) [2]

[2] T. G. Griffin, F. B. Shepherd, G. Wilfong. The Stable Paths Problem and Interdomain Routing. ToN, 2002.

Models (briefly)


3 4

21

0

420430

21020

13010

30

1 2



1 0

2

3

4

210

10134...

342...310

421...

Models (briefly)


3 4

21

0

420430

21020

13010

30

1 2



Succinct SPP(SSPP)

1 0

2

3

4

210

10134...

342...310

421...

Models (briefly)


3 4

21

0

420430

21020

13010

30

1 2


Succinct SPP(SSPP)

1 0

2

3

4

210

10134...

342...310

421...

Size: polynomial in |V|Close to real configurations

Size: exponential in |V|Highly expressive

Models (briefly)


1 2


Succinct SPP(SSPP)



Models (briefly)


1 2


Succinct SPP(SSPP)


unique mapping

by chainingpath fragments


Models (briefly)


1 2


Succinct SPP(SSPP)


Our results hold in both models

unique mapping

by chainingpath fragments


A Polynomial Time Algorithmfor GAO-REXFORD-CHECK

Approach

Input: instance of (S)SPP


Approach


Consider relationiff prefers some path starting

with to some path starting with

take the transitive closure


Approach






uvxuyuwz

v

u

x

y

w z

Approach





interpretation: reads


Approach





interpretation: reads

Can the input graph be partially oriented to an acyclic customer-provider graph such that paths are valley-free and constraints are honored?


Approach

Inspired by [12]


[12] S. Kosub, M. G. Maaß, H. Täubig. Acyclic Type-of-Relationship Problems on the Internet. CAAN 2006

Approach

Inspired by [12]Find a that



v

Approach

Inspired by [12]Find a that• never appears as an internal node in any paths



v w x y

Approach


• does not have incoming edges



v w x y

Approach


• does not have incoming edges– one must exist in any GR-compliant orientation



v w x y

Approach



Orient edges away from



v w x y

Approach






v w x y

Approach




Recursive call



v w x y

Approach




Recursive call



w x y

Approach




Recursive call

Not that easy due to constraints...



The Algorithm


v

The Algorithm


v

u

The Algorithm


c d

b

a

e

v

fu

The Algorithm

ICNP 2010 - Oct 7th 100

c d

b

a

e

v

fu

The Algorithm

ICNP 2010 - Oct 7th 101

c d

b

a

e

v

fu

The Algorithm

ICNP 2010 - Oct 7th 102

c d

b

a

e

v

fu

The Algorithm

ICNP 2010 - Oct 7th 103

c d

b

a

e

v

fu

The Algorithm

ICNP 2010 - Oct 7th 104

c d

b

a

e

v

fu

The Algorithm

ICNP 2010 - Oct 7th 105

c d

b

a

e

v

fu

The Algorithm

ICNP 2010 - Oct 7th 106

c d

b

a

e

v

fu

The Algorithm

ICNP 2010 - Oct 7th 107

c d

b

a

e

v

fu

The Algorithm

ICNP 2010 - Oct 7th 108

c d

b

a

e

v

fu

The Algorithm

ICNP 2010 - Oct 7th 109

c d

b

a

e

v

fu

The Algorithm

ICNP 2010 - Oct 7th 110

c d

b

a

e

v

fu

The Algorithm

ICNP 2010 - Oct 7th 111

c d

b

a

e

v

fu

The Algorithm

ICNP 2010 - Oct 7th 112

c d

b

a

e

v

fu

The Algorithm

ICNP 2010 - Oct 7th 113

c d

b

a

e

v

valleyfree

fu

Forced Orientations

All edges in

and if

and if

and if

ICNP 2010 - Oct 7th 114

Forced Orientations

All edges in

and if

and if

and if

ICNP 2010 - Oct 7th 115

by

Forced Orientations

All edges in

and if

and if

and if

ICNP 2010 - Oct 7th 116

by valleyfreeness

Forced Orientations

All edges in

and if

and if

and if

ICNP 2010 - Oct 7th 117

Forced Orientations

ICNP 2010 - Oct 7th 118

cd

b

a

v

eu

Forced Orientations

ICNP 2010 - Oct 7th 119

cd

b

a

v

eu

Forced Orientations

ICNP 2010 - Oct 7th 120

cd

b

a

v

eu

Forced Orientations

ICNP 2010 - Oct 7th 121

cd

b

a

v

eu

Forced Orientations

ICNP 2010 - Oct 7th 122

cd

b

a

v

eu

Forced Orientations

All edges in

and if

and if

and if

ICNP 2010 - Oct 7th 123

Forced Orientations

ICNP 2010 - Oct 7th 124

cd

b

a

v

eu

Forced Orientations

ICNP 2010 - Oct 7th 125

b

a

v

cd

u

Forced Orientations

ICNP 2010 - Oct 7th 126

b

a

v

cd

u

Forced Orientations

ICNP 2010 - Oct 7th 127

b

a

v

cd

u

Forced Orientations

ICNP 2010 - Oct 7th 128

b

a

v

cd

u

Unconstrained Orientations

ICNP 2010 - Oct 7th 129

c

b

a

v

d

e

fu


ICNP 2010 - Oct 7th 130

c

b

a

v

d

e

fu


ICNP 2010 - Oct 7th 131

c

b

a

v

d

e

fu


ICNP 2010 - Oct 7th 132

c

b

a

v

d

e

fu


ICNP 2010 - Oct 7th 133

c

b

a

v

d

e

fu


ICNP 2010 - Oct 7th 134

c

b

a

v

d

e

fu


ICNP 2010 - Oct 7th 135

c

b

a

v

d

e

fu


ICNP 2010 - Oct 7th 136

c

b

a

v

d

e

fu

Recursivecall

After Recursion

ICNP 2010 - Oct 7th 137

Recursivecall

After Recursion

No valid orientation?

Return “no valid orientation”

Otherwise...

ICNP 2010 - Oct 7th 138

Recursivecall

After Recursion

ICNP 2010 - Oct 7th 139

Recursivecall

c

b

a

v

d

e

fu

After Recursion

ICNP 2010 - Oct 7th 140

Recursivecall

c

b

a

v

d

e

fu

After Recursion

ICNP 2010 - Oct 7th 141

Recursivecall

c

b

a

v

d

e

fu

After Recursion

ICNP 2010 - Oct 7th 142

Recursivecall

c

b

a

v

d

e

fu

After Recursion

ICNP 2010 - Oct 7th 143

Recursivecall

c

b

a

v

d

e

fu

After Recursion

ICNP 2010 - Oct 7th 144

Recursivecall

c

b

a

v

d

e

fu

peer-to-peer

About the Algorithm

Solves GAO-REXFORD-CHECK with constraints

ICNP 2010 - Oct 7th 145

About the Algorithm


edges are oriented only if...

ICNP 2010 - Oct 7th 146

About the Algorithm



• ...constrained

ICNP 2010 - Oct 7th 147

About the Algorithm



• ...constrained

• ...this does not introduce conflicts

ICNP 2010 - Oct 7th 148

About the Algorithm



• ...constrained


Solves GAO-REXFORD-CHECK

ICNP 2010 - Oct 7th 149

About the Algorithm



• ...constrained



Polynomial

ICNP 2010 - Oct 7th 150

About the Algorithm



• ...constrained



Polynomial

ICNP 2010 - Oct 7th 151

steps before recursion

steps after recursion

About the Algorithm



• ...constrained



Polynomial

ICNP 2010 - Oct 7th 152



one vertex removed at each call

About the Algorithm



• ...constrained



Polynomial

Works pretty much the same in the succinct model

ICNP 2010 - Oct 7th 153



one vertex removed at each call

An NP-hardness proof forGAO-REXFORD-STRICT-CHECK

Proof Outline

3SAT GAO-REXFORD-STRICT-CHECK

ICNP 2010 - Oct 7th 155

Proof Outline


satisfying assignment Gao-Rexford-Strict-compliant orientation

ICNP 2010 - Oct 7th 156

Proof Outline


satisfying assignment Gao-Rexford-Strict-compliant orientation

Gao-Rexford-Strict-compliant orientation: obtained by introducing a forced cycle

ICNP 2010 - Oct 7th 157

== =

u1 u2 u3

v3v2v1

== =

u1 u2 u3

v3v2v1

forcingconfiguration

== =

u1 u2 u3

v3v2v1

de-couplingconfiguration

== =

x1

x2

x3

u1 u2 u3

v3v2v1

variablegadget

== =

x1

x2

x3

u1 u2 u3

v3v2v1

tapconfiguration

== =

x1

x2

x3

u1 u2 u3

v3v2v1

clausegadget

== =

x1

x2

x3

u1 u2 u3

v3v2v1

=

v3v2v1

= =

x1

x2

x3

u1 u2 u3

The Forcing Configuration

ICNP 2010 - Oct 7th 175

u

w x

v


ICNP 2010 - Oct 7th 176

u

w x

v

0 0


ICNP 2010 - Oct 7th 177

u

w x

v

0 0


ICNP 2010 - Oct 7th 178

u

w x

v

+

+

+

0

+

+

+

0


ICNP 2010 - Oct 7th 179

u

w x

v

+

+

+

0

+

+

+

0


ICNP 2010 - Oct 7th 180

u

w x

v

+

+

+

0

+

+

+

0


ICNP 2010 - Oct 7th 181

u

w x

v

+

+

+

0

+

+

+

0


ICNP 2010 - Oct 7th 182

u

w x

v

+

+

+

0

+

+

+

0


ICNP 2010 - Oct 7th 183

u

w x

v

+

+

+

0

+

+

+

0

peer-to-peer


ICNP 2010 - Oct 7th 184

u

w x

v

+

+

+

0

+

+

+

0

peer-to-peer


ICNP 2010 - Oct 7th 185

u

w x

v

+

+

+

0

+

+

+

0


ICNP 2010 - Oct 7th 186

u

w x

v

+

+

+

0

+

+

+

0

== =

x1

x2

x3

v3v2v1

u1 u2 u3

The Symmetric Configuration

ICNP 2010 - Oct 7th 189

u v w


ICNP 2010 - Oct 7th 190

u v

00 0

0

w


ICNP 2010 - Oct 7th 191

u v

+

0

+

0

+

0+

0

w


ICNP 2010 - Oct 7th 192

u v

+

0

+

0

+

0+

0

w


ICNP 2010 - Oct 7th 193

u v

+

0

P1

+

0

+

0+

0

P3

P2 P4w


ICNP 2010 - Oct 7th 194

u v

+

0

P1

+

0

+

0+

0

P3

overall rank:P1

P3

P4

P2

P2 P4w


ICNP 2010 - Oct 7th 195

u v

+

0

P1

+

0

+

0+

0

P3

overall rank:P1

P3

P4

P2

P2 P4w


ICNP 2010 - Oct 7th 196

u v

+

0

P1

+

0

+

0+

0

P3

overall rank:P1

P3

P4

P2

P2 P4w


ICNP 2010 - Oct 7th 197

u v

+

0

P1

+

0

+

0+

0

P3

overall rank:P1

P3

P4

P2

P2 P4w


ICNP 2010 - Oct 7th 198

u v

+

0

P1

+

0

+

0+

0

P3

overall rank:P1

P3

P4

P2

P2 P4w


ICNP 2010 - Oct 7th 199

u v

+

0

P1

+

0

+

0+

0

P3

P2 P4w

v w

The De-Coupling Configuration

ICNP 2010 - Oct 7th 200

u

+

0

P1

+

0

+

0+

0

P3

P2 P4v w

v w


ICNP 2010 - Oct 7th 201

u

+

0

P1

+

0

+

0+

0

P3

P2 P4xv w

v w


ICNP 2010 - Oct 7th 202

u

+

0

P1

+

0

+

0+

0

P3

P2 P4x

+

0

+

0

+

0+

0

v w

v w


ICNP 2010 - Oct 7th 203

u

+

0

P1

+

0

+

0+

0

P3

P2 P4x

+

0

+

0

+

0+

0

v w

v w


ICNP 2010 - Oct 7th 204

u

+

0

P1

+

0

+

0+

0

P3

P2 P4x

+

0

+

0

+

0+

0

v w

v w


ICNP 2010 - Oct 7th 205

u

+

0

P1

+

0

+

0+

0

P3

P2 P4x

+

0

+

0

+

0+

0

v w

prevents directed cycles

v w


ICNP 2010 - Oct 7th 206

u

+

0

P1

+

0

+

0+

0

P3

P2 P4x

+

0

+

0

+

0+

0

u x

v w

v w


ICNP 2010 - Oct 7th 207

u

+

0

P1

+

0

+

0+

0

P3

P2 P4x

+

0

+

0

+

0+

0

u x=v w

== =

x1

x2

x3

v3v2v1

u1 u2 u3

The Variable Gadget

ICNP 2010 - Oct 7th 210

u x=v w

The Variable Gadget

ICNP 2010 - Oct 7th 211

0

u x=v w

The Variable Gadget

ICNP 2010 - Oct 7th 212

0

u x=v w

The Variable Gadget

ICNP 2010 - Oct 7th 213

0

u x=v w

+

The Variable Gadget

ICNP 2010 - Oct 7th 214

0

u x=v w

+

“true/false” path

The Variable Gadget

ICNP 2010 - Oct 7th 215

0

u x=v w

+


The Variable Gadget

ICNP 2010 - Oct 7th 216

0

u x=v w

+


The Variable Gadget

ICNP 2010 - Oct 7th 217

0

u x=v w

+


The Variable Gadget

ICNP 2010 - Oct 7th 218

0

u x=v w

+

xi=false


The Variable Gadget

ICNP 2010 - Oct 7th 219

0

u x=v w

+


The Variable Gadget

ICNP 2010 - Oct 7th 220

0

u x=v w

+


The Variable Gadget

ICNP 2010 - Oct 7th 221

0

u x=v w

+

xi=true


The Variable Gadget

ICNP 2010 - Oct 7th 222

0

u x=v w

+

peer-to-peer prevented by valley-freeness


== =

x1

x2

x3

v3v2v1

u1 u2 u3

The Tap Configuration

ICNP 2010 - Oct 7th 225

=

v

u


ICNP 2010 - Oct 7th 226

=

v

u

+

0


ICNP 2010 - Oct 7th 227

=

v

u

+

0

+

0


ICNP 2010 - Oct 7th 228

=

v

u

+

0

+

0


ICNP 2010 - Oct 7th 229

=

v

u

+

0

+

0


ICNP 2010 - Oct 7th 230

=

v

u

+

0

+

0


ICNP 2010 - Oct 7th 231

=

v

u

+

0

+

0


ICNP 2010 - Oct 7th 232

=

v

u

+

0

+

0


ICNP 2010 - Oct 7th 233

=

v

u


ICNP 2010 - Oct 7th 234

=

v

u0

+

+


ICNP 2010 - Oct 7th 235

=

v

u0

+

+

+

0


ICNP 2010 - Oct 7th 236

=

v

u0

+

+

+

0


ICNP 2010 - Oct 7th 237

=

v

u0

+

+

+

0


ICNP 2010 - Oct 7th 238

=

v

u0

+

+

+

0


ICNP 2010 - Oct 7th 239

=

v

u0

+

+

+

0

== =

x1

x2

x3

u1 u2 u3

v3v2v1

0

+

+

0

+

0

+

0

+

+

0

+

+

0

NP-hardness Proof

Polynomial construction

Also valid in the succinct model (with minor tweaks)

248

NP-hardness Proof

Polynomial construction

Also valid in the succinct model (with minor tweaks)

249

Would not work with the original Gao-Rexford conditions

NP-hardness Proof

250

u v

+

0

P1

+

0

+

0+

0

P3

P1

P3

P4

P2

P2 P4w x


NP-hardness Proof

251

u v

+

0

P1

+

0

+

0+

0

P3

P1

P3

P4

P2

P2 P4w x


NP-hardness Proof

252

u v

+

0

P1

+

0

+

0+

0

P3

P1

P3

P4

P2

P2 P4w x


NP-hardness Proof

253

u v

+

0

P1

+

0

+

0+

0

P3

P1

P3

P4

P2

P2 P4w x


NP-hardness Proof

254

u v

+

0

P1

+

0

+

0+

0

P3

P1

P3

P4

P2

P2 P4w x

xi=false


NP-hardness Proof

255

u v

+

0

P1

+

0

+

0+

0

P3

P1

P3

P4

P2

P2 P4w x

xi=false


NP-hardness Proof

256

u v

+

0

P1

+

0

+

0+

0

P3

P1

P3

P4

P2

P2 P4w x

xi=false


NP-hardness Proof

257

u v

+

0

P1

+

0

+

0+

0

P3

P1

P3

P4

P2

P2 P4w x

xi=true


Concluding Remarks

Concluding Remarks

Our contribution:

Applicability:

Open Problems:

259

Concluding Remarks

Our contribution:

Applicability:

Open Problems:

260

(in 4 words):

Concluding Remarks

Our contribution:feasibility of checking GR• relevant for routing stability

Applicability:

Open Problems:

261

(in 4 words):

Concluding Remarks


Applicability:

Open Problems:

262

(in 4 words):

(hints):

Concluding Remarks


Applicability:network simulators

iBGP, confederations

Open Problems:

263

(in 4 words):

(hints):

Concluding Remarks


Applicability:network simulators

iBGP, confederations

Open Problems:backup routing policies?

complexity of other conditions (no DW, etc.)?

other models (e.g., [13])

264

(in 4 words):

[13] T. Griffin, J. Sobrinho. Metarouting. SIGCOMM 2005.

(hints):

どもありがとうございます。

(should read:“thank you very much”)

ICNP 2010 - Oct 7th 266

Running Example

ICNP 2010 - Oct 7th 267

10

4105205320

3203520

210202410

0

1 2 3

4 5

Running Example

ICNP 2010 - Oct 7th 268

10

4105205320

3203520

210202410

0

1 2 3

4 5

Running Example

ICNP 2010 - Oct 7th 269

10

4105205320

3203520

210202410

0

1 2 3

4 5

Running Example

ICNP 2010 - Oct 7th 270

0

1 2 3

4 5

Running Example

ICNP 2010 - Oct 7th 271

0

1 2 3

4 5

Running Example

ICNP 2010 - Oct 7th 272

0

1 2 3

4 5

not traversed by any pathsno incoming edge

Running Example

ICNP 2010 - Oct 7th 273

0

1 2 3

4 5

H10=

L10=

F10={(1,2),(1,4)}

Running Example

ICNP 2010 - Oct 7th 274

0

1 2 3

4 5

H20={(2,1)}

L20={(2,4)}

F20={(2,5),(2,3)}

Running Example

ICNP 2010 - Oct 7th 275

0

1 2 3

4 5

H20={(2,1)}

L20={(2,4)}

F20={(2,5),(2,3)}

0

Running Example

ICNP 2010 - Oct 7th 276

1 2 3

4 5

Recursion

0

Running Example

ICNP 2010 - Oct 7th 277

1 2 3

4 5

Recursion

0

Running Example

ICNP 2010 - Oct 7th 278

1 2 3

4 5

H41=

L41=

F41={(4,2)}

Recursion

0

Running Example

ICNP 2010 - Oct 7th 279

1 2 3

4 5

H21=

L21= {(2,4)}

F21={(2,5),(2,3)}

Recursion

1

0

Running Example

ICNP 2010 - Oct 7th 280

3

4 5

2

Recursion

1

0

Running Example

ICNP 2010 - Oct 7th 281

3

4 5

2

Recursion

1

0

Running Example

ICNP 2010 - Oct 7th 282

3

4 5

H42=

L42=

F42=

2

Recursion

1

0

Running Example

ICNP 2010 - Oct 7th 283

3

4 5

H52=

L52= {(5,3)}

F52={(5,3)}

2

Recursion

1

0

Running Example

ICNP 2010 - Oct 7th 284

3

4 5

H52=

L52= {(5,3)}

F52={(5,3)}

2

Recursion

1

0

Running Example

ICNP 2010 - Oct 7th 285

3

4 5

H52=

L52= {(5,3)}

F52={(5,3)}

2

Recursion

1

0

Running Example

ICNP 2010 - Oct 7th 286

3

4 5

H32=

L32= {(3,5)}

F32={(3,5)}

2

Recursion

1

0

Running Example

ICNP 2010 - Oct 7th 287

3

4 5

H32=

L32= {(3,5)}

F32={(3,5)}

2

Recursion

1

0

Running Example

ICNP 2010 - Oct 7th 288

3

4 5

H32=

L32= {(3,5)}

F32={(3,5)}

2

Recursion

3

4

21

0

Running Example

ICNP 2010 - Oct 7th 289

4

3

5

Recursion

3

4

21

0

Running Example

ICNP 2010 - Oct 7th 290

3

5

Recursion

3

4

21

0

Running Example

ICNP 2010 - Oct 7th 291

3

5

Recursion

3

4

21

0

Running Example

ICNP 2010 - Oct 7th 292

3

5

H53=

L53=

F53=

Recursion

3

4

21

0

Running Example

ICNP 2010 - Oct 7th 293

5

Recursion

3

4

21

0

Running Example

ICNP 2010 - Oct 7th 294

5

Recursion

3

4

21

0

Running Example

ICNP 2010 - Oct 7th 295

5

Back from recursion

3

4

21

0

Running Example

ICNP 2010 - Oct 7th 296

3

5

Back from recursion

3

4

21

0

Running Example

ICNP 2010 - Oct 7th 297

3

5

Back from recursion

3

4

21

0

Running Example

ICNP 2010 - Oct 7th 298

3

5

H53=

L53=

F53=

Back from recursion

3

4

21

0

Running Example

ICNP 2010 - Oct 7th 299

3

5

H53=

L53=

F53=

Back from recursion

3

4

21

0

Running Example

ICNP 2010 - Oct 7th 300

3

5

H53=

L53=

F53=

Back from recursion

3

4

21

0

Running Example

ICNP 2010 - Oct 7th 301

4

3

5

Back from recursion

1

0

Running Example

ICNP 2010 - Oct 7th 302

5

3

4

2

Back from recursion

1

0

Running Example

ICNP 2010 - Oct 7th 303

5

3

4

2

Back from recursion

1

0

Running Example

ICNP 2010 - Oct 7th 304

5

3

H42=

L42=

F42=

4

2

Back from recursion

1

0

Running Example

ICNP 2010 - Oct 7th 305

5

3

H42=

L42=

F42=

4

2

Back from recursion

1

0

Running Example

ICNP 2010 - Oct 7th 306

5

3

H42=

L42=

F42=

4

2

Back from recursion

0

1

Running Example

ICNP 2010 - Oct 7th 307

5

3

4

2

Back from recursion

0

1

Running Example

ICNP 2010 - Oct 7th 308

5

3

4

2

Back from recursion

0

1

Running Example

ICNP 2010 - Oct 7th 309

5

3

4

2

Back from recursion

H41=

L41=

F41={(4,2)}

0

1

Running Example

ICNP 2010 - Oct 7th 310

5

3

4

2

Back from recursion

H41=

L41=

F41={(4,2)}

0

1

Running Example

ICNP 2010 - Oct 7th 311

5

3

4

2

Back from recursion

H41=

L41=

F41={(4,2)}

0

1

Running Example

ICNP 2010 - Oct 7th 312

5

3

4

2

Back from recursion

H21=

L21= {(2,4)}

F21={(2,5),(2,3)}

0

1

Running Example

ICNP 2010 - Oct 7th 313

5

3

4

2

Back from recursion

H21=

L21= {(2,4)}

F21={(2,5),(2,3)}

0

1

Running Example

ICNP 2010 - Oct 7th 314

5

3

4

2

Back from recursion

H21=

L21= {(2,4)}

F21={(2,5),(2,3)}

0

1

Running Example

ICNP 2010 - Oct 7th 315

5

3

4

2

Back from recursion

0

1

Running Example

ICNP 2010 - Oct 7th 316

5

3

4

2

Back from recursion

0

1

Running Example

ICNP 2010 - Oct 7th 317

5

3

4

2

Back from recursion

H10=

L10=

F10={(1,2),(1,4)}

0

1

Running Example

ICNP 2010 - Oct 7th 318

5

3

4

2

Back from recursion

H10=

L10=

F10={(1,2),(1,4)}

0

1

Running Example

ICNP 2010 - Oct 7th 319

5

3

4

2

Back from recursion

H10=

L10=

F10={(1,2),(1,4)}

0

1

Running Example

ICNP 2010 - Oct 7th 320

5

3

4

2

Back from recursion

H20={(2,1)}

L20={(2,4)}

F20={(2,5),(2,3)}

0

1

Running Example

ICNP 2010 - Oct 7th 321

5

3

4

2

Back from recursion

H20={(2,1)}

L20={(2,4)}

F20={(2,5),(2,3)}

all the edges in H20

directed towards 2

0

1

Running Example

ICNP 2010 - Oct 7th 322

5

3

4

2

Back from recursion

H20={(2,1)}

L20={(2,4)}

F20={(2,5),(2,3)}

0

1

Running Example

ICNP 2010 - Oct 7th 323

5

3

4

2

0

1

Running Example

ICNP 2010 - Oct 7th 324

5

3

4

2

assigning as relationships to satisfy the gao-rexford

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