assigning as relationships to satisfy the gao-rexford
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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
3ICNP 2010 - Oct 7th
Provider
Customer
$
AS Relationships
4ICNP 2010 - Oct 7th
$Provider
Customer
$
AS Relationships
5ICNP 2010 - Oct 7th
$Provider
Customer
$
AS Relationships
6ICNP 2010 - Oct 7th
$ 10010
10010
01010
01010
Provider
Customer
AS Relationships
7ICNP 2010 - Oct 7th
Provider
Customer
PeerPeer
AS Relationships
8ICNP 2010 - Oct 7th
Provider
Customer
PeerPeer
AS Relationships
9ICNP 2010 - Oct 7th
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
ICNP 2010 - Oct 7th 11[1] L. Gao, J. Rexford. Stable Internet Routing without Global Coordination. SIGMETRICS, 2000
The Gao-Rexford[1] Conditions
Cust.
Prov.
PeerPeer
1
ICNP 2010 - Oct 7th 12[1] L. Gao, J. Rexford. Stable Internet Routing without Global Coordination. SIGMETRICS, 2000
The Gao-Rexford[1] Conditions
Cust.
Prov.
PeerPeer
1
ICNP 2010 - Oct 7th 13[1] L. Gao, J. Rexford. Stable Internet Routing without Global Coordination. SIGMETRICS, 2000
The Gao-Rexford[1] Conditions
Cust.
Prov.
PeerPeer
1
ICNP 2010 - Oct 7th 14[1] L. Gao, J. Rexford. Stable Internet Routing without Global Coordination. SIGMETRICS, 2000
The Gao-Rexford[1] Conditions
Cust.
Prov.
PeerPeer
1
ICNP 2010 - Oct 7th 15[1] L. Gao, J. Rexford. Stable Internet Routing without Global Coordination. SIGMETRICS, 2000
The Gao-Rexford[1] Conditions
Cust.
Prov.
PeerPeer
1
Acyclic
ICNP 2010 - Oct 7th 16[1] L. Gao, J. Rexford. Stable Internet Routing without Global Coordination. SIGMETRICS, 2000
The Gao-Rexford[1] Conditions
Cust.
Prov.
PeerPeer
1 2
Acyclic
ICNP 2010 - Oct 7th 17[1] L. Gao, J. Rexford. Stable Internet Routing without Global Coordination. SIGMETRICS, 2000
The Gao-Rexford[1] Conditions
Cust.
Prov.
PeerPeer
1 2
Acyclic
ICNP 2010 - Oct 7th 18[1] L. Gao, J. Rexford. Stable Internet Routing without Global Coordination. SIGMETRICS, 2000
The Gao-Rexford[1] Conditions
Cust.
Prov.
PeerPeer
1 2
Acyclic
ICNP 2010 - Oct 7th 19[1] L. Gao, J. Rexford. Stable Internet Routing without Global Coordination. SIGMETRICS, 2000
The Gao-Rexford[1] Conditions
Cust.
Prov.
PeerPeer
1 2
Acyclic
ICNP 2010 - Oct 7th 20[1] L. Gao, J. Rexford. Stable Internet Routing without Global Coordination. SIGMETRICS, 2000
The Gao-Rexford[1] Conditions
Cust.
Prov.
PeerPeer
1 2
Acyclic
ICNP 2010 - Oct 7th 21[1] L. Gao, J. Rexford. Stable Internet Routing without Global Coordination. SIGMETRICS, 2000
The Gao-Rexford[1] Conditions
Cust.
Prov.
PeerPeer
1 2
Acyclic Valley-free
ICNP 2010 - Oct 7th 22[1] L. Gao, J. Rexford. Stable Internet Routing without Global Coordination. SIGMETRICS, 2000
The Gao-Rexford[1] Conditions
Cust.
Prov.
PeerPeer
1 2 3
Acyclic Valley-free
ICNP 2010 - Oct 7th 23[1] L. Gao, J. Rexford. Stable Internet Routing without Global Coordination. SIGMETRICS, 2000
The Gao-Rexford[1] Conditions
Cust.
Prov.
PeerPeer
1 2 3
Acyclic Valley-free
ICNP 2010 - Oct 7th 24[1] L. Gao, J. Rexford. Stable Internet Routing without Global Coordination. SIGMETRICS, 2000
The Gao-Rexford[1] Conditions
Cust.
Prov.
PeerPeer
1 2 3
Acyclic Valley-free
ICNP 2010 - Oct 7th 25[1] L. Gao, J. Rexford. Stable Internet Routing without Global Coordination. SIGMETRICS, 2000
The Gao-Rexford[1] Conditions
Cust.
Prov.
PeerPeer
1 2 3
Acyclic Valley-free
ICNP 2010 - Oct 7th 26[1] L. Gao, J. Rexford. Stable Internet Routing without Global Coordination. SIGMETRICS, 2000
The Gao-Rexford[1] Conditions
Cust.
Prov.
PeerPeer
1 2 3
Acyclic Valley-freePrefer
customer
ICNP 2010 - Oct 7th 27[1] L. Gao, J. Rexford. Stable Internet Routing without Global Coordination. SIGMETRICS, 2000
The Gao-Rexford[1] Conditions
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.
Literature(and a bit of motivation)
Safety [2] is important......but hard to check [3], [4]
ICNP 2010 - Oct 7th 29
[2] T. Griffin, F. Shepherd, G. Wilfong. The Stable Paths Problem and Interdomain Routing. ToN, 2002.
[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.
Literature(and a bit of motivation)
Safety [2] is important......but hard to check [3], [4]Achieved by different approaches
ICNP 2010 - Oct 7th 30
[2] T. Griffin, F. Shepherd, G. Wilfong. The Stable Paths Problem and Interdomain Routing. ToN, 2002.
[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.
Literature(and a bit of motivation)
Safety [2] is important......but hard to check [3], [4]Achieved by different approaches
let oscillations occur, but dynamically resolve them [5], [6]
ICNP 2010 - Oct 7th 31
[2] T. Griffin, F. Shepherd, G. Wilfong. The Stable Paths Problem and Interdomain Routing. ToN, 2002.
[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.[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.
Literature(and a bit of motivation)
Safety [2] is important......but hard to check [3], [4]Achieved by different approaches
let oscillations occur, but dynamically resolve them [5], [6]limit policy expressiveness [7], [2]
ICNP 2010 - Oct 7th 32
[2] T. Griffin, F. Shepherd, G. Wilfong. The Stable Paths Problem and Interdomain Routing. ToN, 2002.
[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.[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.[7] N. Feamster, R. Johari, H. Balakrishnan. Implications of Autonomy for the Expressiveness of
Policy Routing. ToN, 2007.
Literature(and a bit of motivation)
Safety [2] is important......but hard to check [3], [4]Achieved by different approaches
let oscillations occur, but dynamically resolve them [5], [6]limit policy expressiveness [7], [2]GR
ICNP 2010 - Oct 7th 33
[2] T. Griffin, F. Shepherd, G. Wilfong. The Stable Paths Problem and Interdomain Routing. ToN, 2002.
[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.[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.[7] N. Feamster, R. Johari, H. Balakrishnan. Implications of Autonomy for the Expressiveness of
Policy Routing. ToN, 2007.
Literature(and a bit of motivation)
Safety [2] is important......but hard to check [3], [4]Achieved by different approaches
let oscillations occur, but dynamically resolve them [5], [6]limit policy expressiveness [7], [2]GR
ICNP 2010 - Oct 7th 34
[2] T. Griffin, F. Shepherd, G. Wilfong. The Stable Paths Problem and Interdomain Routing. ToN, 2002.
[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.[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.[7] N. Feamster, R. Johari, H. Balakrishnan. Implications of Autonomy for the Expressiveness of
Policy Routing. ToN, 2007.
Literature(and a bit of motivation)
Safety [2] is important......but hard to check [3], [4]Achieved by different approaches
let oscillations occur, but dynamically resolve them [5], [6]limit policy expressiveness [7], [2]GR
ICNP 2010 - Oct 7th 35
[2] T. Griffin, F. Shepherd, G. Wilfong. The Stable Paths Problem and Interdomain Routing. ToN, 2002.
[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.[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.[7] N. Feamster, R. Johari, H. Balakrishnan. Implications of Autonomy for the Expressiveness of
Policy Routing. ToN, 2007.
Motivation(and a bit of literature)
A GR-compliant network......preserves autonomy of each AS in configuring local policies
ICNP 2010 - Oct 7th 36
Motivation(and a bit of literature)
A GR-compliant network......preserves autonomy of each AS in configuring local policies
...is safe and robust [8]
ICNP 2010 - Oct 7th 37
[8] L. Gao, T. Griffin, J. Rexford. Inherently Safe Backup Routing with BGP. INFOCOM 2001
Motivation(and a bit of literature)
A GR-compliant network......preserves autonomy of each AS in configuring local policies
...is safe and robust [8]
...has a convergence time that is roughly bounded by a constant [9]
ICNP 2010 - Oct 7th 38
[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
Motivation(and a bit of literature)
A GR-compliant network......preserves autonomy of each AS in configuring local policies
...is safe and robust [8]
...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]
ICNP 2010 - Oct 7th 39
[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
ICNP 2010 - Oct 7th 40
[10] L. Gao. On Inferring Autonomous System Relationships in the Internet. ToN, 2001.
Other “Grail Seekers”
[10]: relationship inference heuristic
[11]: a valley-free assignment can be achieved efficiently
ICNP 2010 - Oct 7th 41
[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.
Other “Grail Seekers”
[10]: relationship inference heuristic
[11]: a valley-free assignment can be achieved efficiently
[12]: a valley-free+acyclic assignment can be achieved efficiently
ICNP 2010 - Oct 7th 42
[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] S. Kosub, M. G. Maaß, H. Täubig. Acyclic Type-of-Relationship Problems on the Internet.
CAAN 2006.
Other “Grail Seekers”
[10]: relationship inference heuristic
[11]: a valley-free assignment can be achieved efficiently
[12]: a valley-free+acyclic assignment can be achieved efficiently
[13]: distributed detection of the GR conditions (with known relationships)
ICNP 2010 - Oct 7th 43
[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] 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
ProblemGAO-REXFORD-CHECK
ICNP 2010 - Oct 7th
Instance: (model of) a BGP configuration
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
ProblemGAO-REXFORD-CHECK
ICNP 2010 - Oct 7th
Instance: (model of) a BGP configuration
47
ProblemGAO-REXFORD-CHECK
ICNP 2010 - Oct 7th
Instance: (model of) a BGP configuration
48
ProblemGAO-REXFORD-CHECK
ICNP 2010 - Oct 7th
Instance: (model of) a BGP configuration
49
ProblemGAO-REXFORD-CHECK
ICNP 2010 - Oct 7th
Instance: (model of) a BGP configuration
50
ProblemGAO-REXFORD-CHECK
ICNP 2010 - Oct 7th
Instance: (model of) a BGP configuration
Question: Can the network be partially oriented to a customer-provider graph that is GR-compliant?
Results
ICNP 2010 - Oct 7th 51
Results
1. Polynomial algorithm forGAO-REXFORD-CHECK
ICNP 2010 - Oct 7th 52
Results
1. Polynomial algorithm forGAO-REXFORD-CHECK
GAO-REXFORD-STRICT-CHECK:
same as GAO-REXFORD-CHECK, but peers are preferred to providers
ICNP 2010 - Oct 7th 53
Results
1. Polynomial algorithm forGAO-REXFORD-CHECK
GAO-REXFORD-STRICT-CHECK:
same as GAO-REXFORD-CHECK, but peers are preferred to providers
ICNP 2010 - Oct 7th 54
Results
1. Polynomial algorithm forGAO-REXFORD-CHECK
GAO-REXFORD-STRICT-CHECK:
same as GAO-REXFORD-CHECK, but peers are preferred to providers
ICNP 2010 - Oct 7th 55
Results
1. Polynomial algorithm forGAO-REXFORD-CHECK
GAO-REXFORD-STRICT-CHECK:
same as GAO-REXFORD-CHECK, but peers are preferred to providers
ICNP 2010 - Oct 7th 56
Results
1. Polynomial algorithm forGAO-REXFORD-CHECK
GAO-REXFORD-STRICT-CHECK:
same as GAO-REXFORD-CHECK, but peers are preferred to providers
ICNP 2010 - Oct 7th 57
Results
1. Polynomial algorithm forGAO-REXFORD-CHECK
GAO-REXFORD-STRICT-CHECK:
same as GAO-REXFORD-CHECK, but peers are preferred to providers
2. NP-hardness ofGAO-REXFORD-STRICT-CHECK
ICNP 2010 - Oct 7th 58
Models (briefly)
ICNP 2010 - Oct 7th 59
1 2
Models (briefly)
ICNP 2010 - Oct 7th 60
3 4
21
0
1 2
Models (briefly)
ICNP 2010 - Oct 7th 61
3 4
21
0
1 2
Models (briefly)
ICNP 2010 - Oct 7th 62
3 4
21
0
420430
21020
13010
30
1 2
Models (briefly)
ICNP 2010 - Oct 7th 63
3 4
21
0
420430
21020
13010
30
1 2
Models (briefly)
ICNP 2010 - Oct 7th 64
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)
ICNP 2010 - Oct 7th 65
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.
1 0
2
3
4
210
10134...
342...310
421...
Models (briefly)
ICNP 2010 - Oct 7th 66
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.
Succinct SPP(SSPP)
1 0
2
3
4
210
10134...
342...310
421...
Models (briefly)
ICNP 2010 - Oct 7th 67
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.
Succinct SPP(SSPP)
1 0
2
3
4
210
10134...
342...310
421...
Models (briefly)
ICNP 2010 - Oct 7th 68
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.
Succinct SPP(SSPP)
1 0
2
3
4
210
10134...
342...310
421...
Models (briefly)
ICNP 2010 - Oct 7th 69
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.
Succinct SPP(SSPP)
1 0
2
3
4
210
10134...
342...310
421...
Models (briefly)
ICNP 2010 - Oct 7th 70
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.
Succinct SPP(SSPP)
1 0
2
3
4
210
10134...
342...310
421...
Models (briefly)
ICNP 2010 - Oct 7th 71
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.
Succinct SPP(SSPP)
1 0
2
3
4
210
10134...
342...310
421...
Models (briefly)
ICNP 2010 - Oct 7th 72
3 4
21
0
420430
21020
13010
30
1 2
Stable Paths Problem(SPP) [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)
ICNP 2010 - Oct 7th 73
1 2
Stable Paths Problem(SPP) [2]
Succinct SPP(SSPP)
Size: polynomial in |V|Close to real configurations
Size: exponential in |V|Highly expressive
Models (briefly)
ICNP 2010 - Oct 7th 74
1 2
Stable Paths Problem(SPP) [2]
Succinct SPP(SSPP)
Size: polynomial in |V|Close to real configurations
unique mapping
by chainingpath fragments
Size: exponential in |V|Highly expressive
Models (briefly)
ICNP 2010 - Oct 7th 75
1 2
Stable Paths Problem(SPP) [2]
Succinct SPP(SSPP)
Size: polynomial in |V|Close to real configurations
Our results hold in both models
unique mapping
by chainingpath fragments
Size: exponential in |V|Highly expressive
A Polynomial Time Algorithmfor GAO-REXFORD-CHECK
Approach
Input: instance of (S)SPP
ICNP 2010 - Oct 7th 77
Approach
Input: instance of (S)SPP
Consider relationiff prefers some path starting
with to some path starting with
take the transitive closure
ICNP 2010 - Oct 7th 78
Approach
Input: instance of (S)SPP
Consider relationiff prefers some path starting
with to some path starting with
take the transitive closure
ICNP 2010 - Oct 7th 79
uvxuyuwz
v
u
x
y
w z
Approach
Input: instance of (S)SPP
Consider relationiff prefers some path starting
with to some path starting with
take the transitive closure
interpretation: reads
ICNP 2010 - Oct 7th 80
Approach
Input: instance of (S)SPP
Consider relationiff prefers some path starting
with to some path starting with
take the transitive closure
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?
ICNP 2010 - Oct 7th 81
Approach
Inspired by [12]
ICNP 2010 - Oct 7th 82
[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
ICNP 2010 - Oct 7th 83
[12] S. Kosub, M. G. Maaß, H. Täubig. Acyclic Type-of-Relationship Problems on the Internet. CAAN 2006
v
Approach
Inspired by [12]Find a that• never appears as an internal node in any paths
ICNP 2010 - Oct 7th 84
[12] S. Kosub, M. G. Maaß, H. Täubig. Acyclic Type-of-Relationship Problems on the Internet. CAAN 2006
v w x y
Approach
Inspired by [12]Find a that• never appears as an internal node in any paths
• does not have incoming edges
ICNP 2010 - Oct 7th 85
[12] S. Kosub, M. G. Maaß, H. Täubig. Acyclic Type-of-Relationship Problems on the Internet. CAAN 2006
v w x y
Approach
Inspired by [12]Find a that• never appears as an internal node in any paths
• does not have incoming edges– one must exist in any GR-compliant orientation
ICNP 2010 - Oct 7th 86
[12] S. Kosub, M. G. Maaß, H. Täubig. Acyclic Type-of-Relationship Problems on the Internet. CAAN 2006
v w x y
Approach
Inspired by [12]Find a that• never appears as an internal node in any paths
• does not have incoming edges– one must exist in any GR-compliant orientation
Orient edges away from
ICNP 2010 - Oct 7th 87
[12] S. Kosub, M. G. Maaß, H. Täubig. Acyclic Type-of-Relationship Problems on the Internet. CAAN 2006
v w x y
Approach
Inspired by [12]Find a that• never appears as an internal node in any paths
• does not have incoming edges– one must exist in any GR-compliant orientation
Orient edges away from
ICNP 2010 - Oct 7th 88
[12] S. Kosub, M. G. Maaß, H. Täubig. Acyclic Type-of-Relationship Problems on the Internet. CAAN 2006
v w x y
Approach
Inspired by [12]Find a that• never appears as an internal node in any paths
• does not have incoming edges– one must exist in any GR-compliant orientation
Orient edges away from
ICNP 2010 - Oct 7th 89
[12] S. Kosub, M. G. Maaß, H. Täubig. Acyclic Type-of-Relationship Problems on the Internet. CAAN 2006
v w x y
Approach
Inspired by [12]Find a that• never appears as an internal node in any paths
• does not have incoming edges– one must exist in any GR-compliant orientation
Orient edges away from
ICNP 2010 - Oct 7th 90
[12] S. Kosub, M. G. Maaß, H. Täubig. Acyclic Type-of-Relationship Problems on the Internet. CAAN 2006
v w x y
Approach
Inspired by [12]Find a that• never appears as an internal node in any paths
• does not have incoming edges– one must exist in any GR-compliant orientation
Orient edges away from
ICNP 2010 - Oct 7th 91
[12] S. Kosub, M. G. Maaß, H. Täubig. Acyclic Type-of-Relationship Problems on the Internet. CAAN 2006
v w x y
Approach
Inspired by [12]Find a that• never appears as an internal node in any paths
• does not have incoming edges– one must exist in any GR-compliant orientation
Orient edges away from
Recursive call
ICNP 2010 - Oct 7th 92
[12] S. Kosub, M. G. Maaß, H. Täubig. Acyclic Type-of-Relationship Problems on the Internet. CAAN 2006
v w x y
Approach
Inspired by [12]Find a that• never appears as an internal node in any paths
• does not have incoming edges– one must exist in any GR-compliant orientation
Orient edges away from
Recursive call
ICNP 2010 - Oct 7th 93
[12] S. Kosub, M. G. Maaß, H. Täubig. Acyclic Type-of-Relationship Problems on the Internet. CAAN 2006
w x y
Approach
Inspired by [12]Find a that• never appears as an internal node in any paths
• does not have incoming edges– one must exist in any GR-compliant orientation
Orient edges away from
Recursive call
Not that easy due to constraints...
ICNP 2010 - Oct 7th 94
[12] S. Kosub, M. G. Maaß, H. Täubig. Acyclic Type-of-Relationship Problems on the Internet. CAAN 2006
The Algorithm
ICNP 2010 - Oct 7th 95
v
The Algorithm
ICNP 2010 - Oct 7th 96
v
The Algorithm
ICNP 2010 - Oct 7th 97
v
The Algorithm
ICNP 2010 - Oct 7th 98
v
u
The Algorithm
ICNP 2010 - Oct 7th 99
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
Unconstrained Orientations
ICNP 2010 - Oct 7th 130
c
b
a
v
d
e
fu
Unconstrained Orientations
ICNP 2010 - Oct 7th 131
c
b
a
v
d
e
fu
Unconstrained Orientations
ICNP 2010 - Oct 7th 132
c
b
a
v
d
e
fu
Unconstrained Orientations
ICNP 2010 - Oct 7th 133
c
b
a
v
d
e
fu
Unconstrained Orientations
ICNP 2010 - Oct 7th 134
c
b
a
v
d
e
fu
Unconstrained Orientations
ICNP 2010 - Oct 7th 135
c
b
a
v
d
e
fu
Unconstrained Orientations
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
Solves GAO-REXFORD-CHECK with constraints
edges are oriented only if...
ICNP 2010 - Oct 7th 146
About the Algorithm
Solves GAO-REXFORD-CHECK with constraints
edges are oriented only if...
• ...constrained
ICNP 2010 - Oct 7th 147
About the Algorithm
Solves GAO-REXFORD-CHECK with constraints
edges are oriented only if...
• ...constrained
• ...this does not introduce conflicts
ICNP 2010 - Oct 7th 148
About the Algorithm
Solves GAO-REXFORD-CHECK with constraints
edges are oriented only if...
• ...constrained
• ...this does not introduce conflicts
Solves GAO-REXFORD-CHECK
ICNP 2010 - Oct 7th 149
About the Algorithm
Solves GAO-REXFORD-CHECK with constraints
edges are oriented only if...
• ...constrained
• ...this does not introduce conflicts
Solves GAO-REXFORD-CHECK
Polynomial
ICNP 2010 - Oct 7th 150
About the Algorithm
Solves GAO-REXFORD-CHECK with constraints
edges are oriented only if...
• ...constrained
• ...this does not introduce conflicts
Solves GAO-REXFORD-CHECK
Polynomial
ICNP 2010 - Oct 7th 151
steps before recursion
steps after recursion
About the Algorithm
Solves GAO-REXFORD-CHECK with constraints
edges are oriented only if...
• ...constrained
• ...this does not introduce conflicts
Solves GAO-REXFORD-CHECK
Polynomial
ICNP 2010 - Oct 7th 152
steps before recursion
steps after recursion
one vertex removed at each call
About the Algorithm
Solves GAO-REXFORD-CHECK with constraints
edges are oriented only if...
• ...constrained
• ...this does not introduce conflicts
Solves GAO-REXFORD-CHECK
Polynomial
Works pretty much the same in the succinct model
ICNP 2010 - Oct 7th 153
steps before recursion
steps after recursion
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
3SAT GAO-REXFORD-STRICT-CHECK
satisfying assignment Gao-Rexford-Strict-compliant orientation
ICNP 2010 - Oct 7th 156
Proof Outline
3SAT GAO-REXFORD-STRICT-CHECK
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
== =
x1
x2
x3
u1 u2 u3
v3v2v1
== =
x1
x2
x3
u1 u2 u3
v3v2v1
== =
x1
x2
x3
u1 u2 u3
v3v2v1
== =
x1
x2
x3
u1 u2 u3
v3v2v1
== =
x1
x2
x3
u1 u2 u3
v3v2v1
== =
x1
x2
x3
u1 u2 u3
v3v2v1
== =
x1
x2
x3
u1 u2 u3
v3v2v1
== =
x1
x2
x3
u1 u2 u3
v3v2v1
=
v3v2v1
= =
x1
x2
x3
u1 u2 u3
=
v3v2v1
= =
x1
x2
x3
u1 u2 u3
The Forcing Configuration
ICNP 2010 - Oct 7th 175
u
w x
v
The Forcing Configuration
ICNP 2010 - Oct 7th 176
u
w x
v
0 0
The Forcing Configuration
ICNP 2010 - Oct 7th 177
u
w x
v
0 0
The Forcing Configuration
ICNP 2010 - Oct 7th 178
u
w x
v
+
+
+
0
+
+
+
0
The Forcing Configuration
ICNP 2010 - Oct 7th 179
u
w x
v
+
+
+
0
+
+
+
0
The Forcing Configuration
ICNP 2010 - Oct 7th 180
u
w x
v
+
+
+
0
+
+
+
0
The Forcing Configuration
ICNP 2010 - Oct 7th 181
u
w x
v
+
+
+
0
+
+
+
0
The Forcing Configuration
ICNP 2010 - Oct 7th 182
u
w x
v
+
+
+
0
+
+
+
0
The Forcing Configuration
ICNP 2010 - Oct 7th 183
u
w x
v
+
+
+
0
+
+
+
0
peer-to-peer
The Forcing Configuration
ICNP 2010 - Oct 7th 184
u
w x
v
+
+
+
0
+
+
+
0
peer-to-peer
The Forcing Configuration
ICNP 2010 - Oct 7th 185
u
w x
v
+
+
+
0
+
+
+
0
The Forcing Configuration
ICNP 2010 - Oct 7th 186
u
w x
v
+
+
+
0
+
+
+
0
== =
x1
x2
x3
v3v2v1
u1 u2 u3
== =
x1
x2
x3
v3v2v1
u1 u2 u3
The Symmetric Configuration
ICNP 2010 - Oct 7th 189
u v w
The Symmetric Configuration
ICNP 2010 - Oct 7th 190
u v
00 0
0
w
The Symmetric Configuration
ICNP 2010 - Oct 7th 191
u v
+
0
+
0
+
0+
0
w
The Symmetric Configuration
ICNP 2010 - Oct 7th 192
u v
+
0
+
0
+
0+
0
w
The Symmetric Configuration
ICNP 2010 - Oct 7th 193
u v
+
0
P1
+
0
+
0+
0
P3
P2 P4w
The Symmetric Configuration
ICNP 2010 - Oct 7th 194
u v
+
0
P1
+
0
+
0+
0
P3
overall rank:P1
P3
P4
P2
P2 P4w
The Symmetric Configuration
ICNP 2010 - Oct 7th 195
u v
+
0
P1
+
0
+
0+
0
P3
overall rank:P1
P3
P4
P2
P2 P4w
The Symmetric Configuration
ICNP 2010 - Oct 7th 196
u v
+
0
P1
+
0
+
0+
0
P3
overall rank:P1
P3
P4
P2
P2 P4w
The Symmetric Configuration
ICNP 2010 - Oct 7th 197
u v
+
0
P1
+
0
+
0+
0
P3
overall rank:P1
P3
P4
P2
P2 P4w
The Symmetric Configuration
ICNP 2010 - Oct 7th 198
u v
+
0
P1
+
0
+
0+
0
P3
overall rank:P1
P3
P4
P2
P2 P4w
The Symmetric Configuration
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
The De-Coupling Configuration
ICNP 2010 - Oct 7th 201
u
+
0
P1
+
0
+
0+
0
P3
P2 P4xv w
v w
The De-Coupling Configuration
ICNP 2010 - Oct 7th 202
u
+
0
P1
+
0
+
0+
0
P3
P2 P4x
+
0
+
0
+
0+
0
v w
v w
The De-Coupling Configuration
ICNP 2010 - Oct 7th 203
u
+
0
P1
+
0
+
0+
0
P3
P2 P4x
+
0
+
0
+
0+
0
v w
v w
The De-Coupling Configuration
ICNP 2010 - Oct 7th 204
u
+
0
P1
+
0
+
0+
0
P3
P2 P4x
+
0
+
0
+
0+
0
v w
v w
The De-Coupling Configuration
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
The De-Coupling Configuration
ICNP 2010 - Oct 7th 206
u
+
0
P1
+
0
+
0+
0
P3
P2 P4x
+
0
+
0
+
0+
0
u x
v w
v w
The De-Coupling Configuration
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
== =
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
+
“true/false” path
The Variable Gadget
ICNP 2010 - Oct 7th 216
0
u x=v w
+
“true/false” path
The Variable Gadget
ICNP 2010 - Oct 7th 217
0
u x=v w
+
“true/false” path
The Variable Gadget
ICNP 2010 - Oct 7th 218
0
u x=v w
+
xi=false
“true/false” path
The Variable Gadget
ICNP 2010 - Oct 7th 219
0
u x=v w
+
“true/false” path
The Variable Gadget
ICNP 2010 - Oct 7th 220
0
u x=v w
+
“true/false” path
The Variable Gadget
ICNP 2010 - Oct 7th 221
0
u x=v w
+
xi=true
“true/false” path
The Variable Gadget
ICNP 2010 - Oct 7th 222
0
u x=v w
+
peer-to-peer prevented by valley-freeness
“true/false” path
== =
x1
x2
x3
v3v2v1
u1 u2 u3
== =
x1
x2
x3
v3v2v1
u1 u2 u3
The Tap Configuration
ICNP 2010 - Oct 7th 225
=
v
u
The Tap Configuration
ICNP 2010 - Oct 7th 226
=
v
u
+
0
The Tap Configuration
ICNP 2010 - Oct 7th 227
=
v
u
+
0
+
0
The Tap Configuration
ICNP 2010 - Oct 7th 228
=
v
u
+
0
+
0
The Tap Configuration
ICNP 2010 - Oct 7th 229
=
v
u
+
0
+
0
The Tap Configuration
ICNP 2010 - Oct 7th 230
=
v
u
+
0
+
0
The Tap Configuration
ICNP 2010 - Oct 7th 231
=
v
u
+
0
+
0
The Tap Configuration
ICNP 2010 - Oct 7th 232
=
v
u
+
0
+
0
The Tap Configuration
ICNP 2010 - Oct 7th 233
=
v
u
The Tap Configuration
ICNP 2010 - Oct 7th 234
=
v
u0
+
+
The Tap Configuration
ICNP 2010 - Oct 7th 235
=
v
u0
+
+
+
0
The Tap Configuration
ICNP 2010 - Oct 7th 236
=
v
u0
+
+
+
0
The Tap Configuration
ICNP 2010 - Oct 7th 237
=
v
u0
+
+
+
0
The Tap Configuration
ICNP 2010 - Oct 7th 238
=
v
u0
+
+
+
0
The Tap Configuration
ICNP 2010 - Oct 7th 239
=
v
u0
+
+
+
0
== =
x1
x2
x3
u1 u2 u3
v3v2v1
0
+
+
0
+
0
+
0
+
+
0
+
+
0
== =
x1
x2
x3
u1 u2 u3
v3v2v1
0
+
+
0
+
0
+
0
+
+
0
+
+
0
== =
x1
x2
x3
u1 u2 u3
v3v2v1
0
+
+
0
+
0
+
0
+
+
0
+
+
0
== =
x1
x2
x3
u1 u2 u3
v3v2v1
0
+
+
0
+
0
+
0
+
+
0
+
+
0
== =
x1
x2
x3
u1 u2 u3
v3v2v1
0
+
+
0
+
0
+
0
+
+
0
+
+
0
== =
x1
x2
x3
u1 u2 u3
v3v2v1
0
+
+
0
+
0
+
0
+
+
0
+
+
0
== =
x1
x2
x3
u1 u2 u3
v3v2v1
0
+
+
0
+
0
+
0
+
+
0
+
+
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
Would not work with the original Gao-Rexford conditions
NP-hardness Proof
251
u v
+
0
P1
+
0
+
0+
0
P3
P1
P3
P4
P2
P2 P4w x
Would not work with the original Gao-Rexford conditions
NP-hardness Proof
252
u v
+
0
P1
+
0
+
0+
0
P3
P1
P3
P4
P2
P2 P4w x
Would not work with the original Gao-Rexford conditions
NP-hardness Proof
253
u v
+
0
P1
+
0
+
0+
0
P3
P1
P3
P4
P2
P2 P4w x
Would not work with the original Gao-Rexford conditions
NP-hardness Proof
254
u v
+
0
P1
+
0
+
0+
0
P3
P1
P3
P4
P2
P2 P4w x
xi=false
Would not work with the original Gao-Rexford conditions
NP-hardness Proof
255
u v
+
0
P1
+
0
+
0+
0
P3
P1
P3
P4
P2
P2 P4w x
xi=false
Would not work with the original Gao-Rexford conditions
NP-hardness Proof
256
u v
+
0
P1
+
0
+
0+
0
P3
P1
P3
P4
P2
P2 P4w x
xi=false
Would not work with the original Gao-Rexford conditions
NP-hardness Proof
257
u v
+
0
P1
+
0
+
0+
0
P3
P1
P3
P4
P2
P2 P4w x
xi=true
Would not work with the original Gao-Rexford conditions
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
Our contribution:feasibility of checking GR• relevant for routing stability
Applicability:
Open Problems:
262
(in 4 words):
(hints):
Concluding Remarks
Our contribution:feasibility of checking GR• relevant for routing stability
Applicability:network simulators
iBGP, confederations
Open Problems:
263
(in 4 words):
(hints):
Concluding Remarks
Our contribution:feasibility of checking GR• relevant for routing stability
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
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