STABLE PATH PROBLEM

Presented by:Sangeetha A. J.

Based on The Stable Path Problem and Interdomain Routing Timothy G. Griffin, Bruce Shepherd, Gordon Wilfong

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OverviewBGP basics

Simple Path Vector Protocol

Stable Path Problem

Dispute Wheels

3

Border Gateway ProtocolThe only interdomain routing protocol used in

the internet today.

Not based on shortest path.

Path vector protocol.

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How is BGP different from shortest path algorithms?Policy based metrics

Independent routing policies at each AS

These policies could be conflicting

BGP can be considered a distributed algorithm to solve the Stable Path Problem.

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BGP route selection

Order of preference

1. Highest local preference

2. Shortest AS path

3. Lowest IP address for next hop

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Peering transformationComposition of three transformations

1. Export policies

r1 = export (u w, r)

2. Path vector transformation r2 = PVT (u w, r1)

3. Import policies

r3 = import (u w, r2)

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Peering transformationpt (u w, r) =

import (u w, PVT(u w, export(u w, r)))

Suppose u0 is originating AS

P = uk uk-1 …..u1 u0

Route record received is

pt( uk uk-1, pt( uk-1

uk-2, ……..pt ( u1 u0 )))..))

Ranking function λ uk (P)

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Simple Path Vector Protocolprocess spvp(u)

{

receive P from w

{

rib-in(uw) := P

if rib(u) != best(u)

{

rib(u) := best(u)

for each v in peers(u)

{

send rib(u) to v

}

}

}

}

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Stable path problemMathematical formulation that captures the

semantics of any path vector protocol.

Does every node have a stable route to one destination node?

Node 0 is assumed to be origin or destination of all routes.

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SHORTEST - 1

1

3

4

2

0

10130

20210

430420

30

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SHORTEST 1- Path assignment

NodeTime

1 2 3 4

1 10 20 30 -

2 10 20 30 430

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SHORTEST 1

1

3

4

2

0

10130

20210

430420

30

13

Stable path problemUndirected graph G = (V, E) V- set of nodes , E - set of edges

Path is either empty path or a sequence of nodes

(vk vk -1…..v1 v0 )

Pv - set of permitted paths from v to origin

If P1 , P2 ϵ Pv and λ v (P1) < λ v (P2)

then P2 is preferred over P1.

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Stable path problemAn instance of stable path problem, S = (G,P,Λ) Graph together with permitted paths and ranking function for each

node.

Some propertiesP0 = {(0)} for all v ϵ V – {0}

◦ ϵ ϵ Pv (null path is always present in the set)

◦ λ v (ϵ) =0 , λ v (P) > 0 for P ≠ ϵ (null path has lowest preference)

◦ P ϵ Pv then P is a simple path (no repeated nodes).

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Stable path problemA path assignment is a function π function that maps a

node to a permitted path.

π(u) ϵ Pu

All possible permitted paths at u , choices (π, u) =

{ (u v) π(v) | {u,v} ϵ E } ∩ Pu (u ≠ 0)

{ (0)} otherwise

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Stable path problemW is subset of Pu such that each path in W has a distinct

next hop.

Best path in W, best(W, u)

{P ϵ W with maximal λ u (P) W ≠ Φ

ϵ otherwise }

Path assignment π is stable at node u if

π(u) = best (choices (π, u) ,u)

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Stable path problemπ is stable, if it is stable at each node u

Path assignment vector (P1 , P2 ,….., Pn ) where π(u)=Pu

If π is stable and π(u)= (u w) P, then π(w)= P.

Thus stable path implies tree rooted at the origin.

The stable path problem, S = (G,P,Λ) is solvable if there is a stable path assignment for S.

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SHORTEST 2

1

3

4

2

0

10130

20210

420430

30

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SHORTEST 2

1

3

4

2

0

10130

20210

420430

30

20

Good Gadget

1

3

4

2

0

13010

21020

430420

30

21

Good Gadget

NodeTime

1 2 3 4

1 10 20 30 -

2 130 210 30 430

3 130 20 30 430

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Good Gadget

1

3

4

2

0

13010

21020

430420

30

23

Bad Gadget

1

3

4

2

0

13010

21020

420430

342030

24

Bad Gadget NodeTime

1 2 3 4

1 10 20 30 -

2 130 210 30 420

3 130 20 3420 430

4 10 20 30 420

5 130 210 30 420

25

Naughty Gadget

1

3

4

2

0

13010

21020

430420

342030

26

Naughty Gadget NodeTime

1 2 3 4

1 10 20 30 -

2 130 210 30 420

3 130 20 3420 430

4 10 20 30 420

5 130 210 3420 430

6 10 20 30 -

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Multiple solutions possible

1

1 2

2

0

12010

21020

00

1 2

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Observations about SPPA stable path problem is safe if SPVP always converges.

SPP solvability does not imply safety.◦ Naughty gadget has single solution but SPVP may diverge.

Oscillations are possible.

It always diverges if there is no solution.

Multiple stable solutions are possible for a SPP.

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Dispute WheelsA sequence of nodes , ui

.

A sequence of non-empty paths, Qi and Ri.

u i+1

u1

u2

u i

u k-1

u0

v

Qi

R1

R i

R k-1

Q0

Q1

Q2

R0

Qi+1

Qk-1

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Dispute Wheels A circular set of dependencies between routing policies that

cannot be simultaneously satisfied.

Ri is a path from ui to ui+1

Qi ϵ Pui

Ri Qi+1 ϵ Pui

λ ui (Qi) < λ ui (Ri Qi+1 )

Preferred path for every node goes around the wheel instead of following the spoke.

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Dispute Wheel of Bad Gadget

1

3

4

2

0

13010

21020

420430

342030

R0

R1

R2

Q0 Q2

Q1

u0u2

u1

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Dispute Wheel of Naughty Gadget

3 420

4304203420

30

R0

R1

Q0

Q1

u0u1

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Properties of dispute wheel

No dispute wheel implies

Solvability

Unique solution

Safety

Robustness

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Dispute wheel Construct a sequence of subsets of V, V0 , V1…

Corresponding path assignments π0, π1….

Direct path P = (u1 u2) P2 P2= π(u2)

Add nodes to the set, if its highest rank consistent with πi is direct path.

If you are unable to find nodes whose paths are consistent with existing path assignments, you get a dispute wheel.

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Good Gadget

1

3

4

2

0

13010

21020

430420

30

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Good Gadget

NodeTime

0 1 2 3 4

1 (0) - - - -

2 (0) - - 30 -

3 (0) 130 - 30 430

4 (0) 130 210 30 430

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Bad Gadget

1

3

4

2

0

13010

21020

420430

342030

38

Bad Gadget

NodeTime

0 1 2 3 4

1 (0) - - - -

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Solvability

Q0

R0

v1

u0u1

R1

u2

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Unique solution

u0

R0

u1

R1

u2u3R2

R3

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Robustness If stable path problem, S, has no dispute wheel then S is

robust.

S is fragile if S is solvable but there exist some E’ (subset of E) such that S/E’ is not solvable.

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Bad Backup- Fragile

1

3

4

2

0

13010

21020

40420430

342030

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ConclusionA network with unique solution can have a dispute

wheel, if transient oscillations are present. This is a rigid condition.

Even in networks with multiple stable states, we get dispute wheels.

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Complementary approachesOperational guidelines

Static analysis of routing policies

Dynamic detection

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THANK YOU