Optimization and Control of Networks

Path Algebra

Lijun Chen 02/16/2016

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Algebra framework for network routing

❒  Goal q  Convergence properties of dynamic routing protocols q  Characteristics of the paths the protocol converges to

❒  Approach q  Formulation in an algebraic framework q  Different protocols can be seen as different instantiations

of this algebra

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Shortest path routing as a path vector protocol

❒  Given a weighted graph G=(V,E,w), find the shortest paths to a destination.

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d!

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Shortest path routing as a path vector protocol

❒  Abstractions

q  Each link is assigned a weight (label)

q  Each path has an aggregate weight (signature)

q  Path extension: path weight is the summation of weights of all its links (composition operation)

q  Preference/selection rule: choose the minimum weight path (ordering/preference)

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Shortest path algebra

Set of labels: L

Set of signatures: Σ

),( ≺WTotally ordered set of weights:

Σ→Σ×⊕ L:Composition:

Wf →Σ:Weighing function:

Special signature: φ

R+

{ } ),( ≤∞+∪+R

+

Identity

{ }∞+∪+R

∞+

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Path vector algebra

❒  Defined as a seven-tuple ❒  Labels model link data-structure, amounts to

applying edge policy to the path data-structure when the path is extended

❒  Signatures model the path data-structure, contain enough information to determine a path’s weight using

❒  Weight ordering defines preference relation, heavier paths are less preferred

( )fLw ,,,,,, ⊕Σ φ≺⊕

f

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Path Vector protocol

❒  Instantiating an algebra produces a protocol ❒  A node knows a path to when having a

signature for (either for trivial path, or for a path extending a neighbor’s path)

❒  The best path to is a path with lowest weight ❒  To advertise a path to node , is sent

along signaling edge with some associated label , and is the signature of the imported, extended path at

)()( PsluPs ⊕=

v P dP )(ds )(Ps

P u )(Ps

),( uv

lu

d

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What we care

❒  Convergence is the most important goal ❒  Characteristics of the paths the protocol

converges to (optimal? In what sense?) ❒  SP routing converges and is optimal. Let’s first

check what properties SP algebra has, and then abstract them to general algebra and see whether they are necessary and/or sufficient for the convergence and optimality.

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Maximality and Absorption

❒  Usable path has finite weight - Maximality

❒  Cannot extend an un-usable path to a usable one - Absorption

{ } )()( φαφα ff ≺−Σ∈∀

φφ =⊕∀∈ lLl

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Monotonicity

)α()α(α signature and labelevery For -

⊕lffl

≺

1 2 0

than lessnot weigh does 21 PP!

P, s(P)=αl

❒  The path weight is non-decreasing along the path

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Isotonicity

)β(α)()β()α(β and α signatures and labelevery For -

⊕⊕⇒ lflfffl

≺≺

1 2

QPQP!! 21 morenot weigh does 21

then , than morenot weigh does If

0 l

P, s(P)=α

Q, s(Q)=β

❒  Path ordering is kept when extended onto common link

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Some Definitions

❒  An optimal path from to is a usable path with minimum weight among all usable paths from to

❒  An optimal path in-tree rooted at node is an in-tree routed at satisfying, for every node that belongs to the in-tree, the only path in the in-tree is an optimal path

dudu

dd u

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u!

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❒  The local-optimal path from to exists only with respect to a set of paths, which are advertised to u by its neighbors. ❒  Given in-tree rooted at , define as the set of in-tree paths which has an out-neighbor of for origin ❒  In-tree is local-optimal-paths in-tree satisfying, for each

of its node , the only path in the in-tree is a local-optimal path with respect to

du

dT d)( du TV

u

dTu

)( du TV

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Freeness

given a cycle and a set of paths with origins at the nodes

of the cycle, all with the same weight, at least one of these paths will see its weight change as it extends into the cycle.

for example, in shortest-path routing, the network is free,

since the algebra is strictly monotone

( ){ }( ) ( ) wuulfwf

nifWwuuu

ii

ncycle

≠⊕⇒=

Σ∈∀≤<∃−∈∀∀

− αα

αφ

)( 0

1

01!

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Convergence

If the algebra is monotone, then the path vector protocol can be made to converge to local-optimal in-tree.

Network is free Convergence to local-optimal in-tree

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Failure of monotonicity

)α()α( ⊕lff ≻

0

l

lP

s(P)=αQ

s(Q)=α

1 2

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Isotonicity and optimality

If the signature/weight depends on some global property of the path, we may say something about it by this theorem. For example, SP routing is optimal.

Local-optimal in-trees Optimal in-trees

Algebra is isotone =

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Failure of isotonicity

0 l

P, s(P)=α

Q, s(Q)=β

)β(α)(but )β()α( ⊕⊕ lflfff ≻≺

A local-optimal-path in-tree that is not optimal

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Gao-Rexford: import/export rules

Provider

Peer

Customer

X

Customer routes Peer routes Provider routes

q  Prefer customer routes to peer routes, and then to provider routes q  Export only customer routes to peers/providers, and export all kinds of routes to customers

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Algebra: labels

❒  Each link carries a label q  Customer link, c q  Provider link, p q  Peer link, r

c

c c c

c c

p

p p p

p p

r

r

r

r

r r

r

r 2 3

0

3 is a provider of 0; 0 is a customer of 3

3 and 4 are peers

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Algebra: signatures

❒  Each path has a signature q  Trivial path, εq  Unusable path, φq  Customer path, c q  Provider path, p q  Peer path, r

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A node exports to its peers routes learned from its customers

A node does not export to a provider a route learned from another provider

Algebra: composition

ε c r p φ

c c c φ φ φ

r r r φ φ φ

p p p p p φ

Signature

Labe

l

⊕

c

c

r 5 6

4

1

2 3

0

p c

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Algebra: weights

0 1 2 3

ε c r p φ

c c c φ φ φ

r r r φ φ φ

p p p p p φ

weights

Prefer customer routes to peer routes and then to provider routes

⊕

∞+f

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Algebra: properties 0 1 2 3

ε c r p φ

c c c φ φ φ

r r r φ φ φ

p p p p p φ

⊕

∞+f

q  The algebra is q  Monotone q  Isotone

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Network: freeness

❒  L0 is empty, since ❒  L1={c}, since ❒  L2 is empty, since ❒  L3={p}, since

)()(0 εε ⊕= lff ≺

)()(1 ccfcf ⊕==

)()(2 rlfrf ⊕≠=

)()(3 ppfpf ⊕==

q  Non-free cycles: q  Only customer links q  Only provider links

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Gao-Rexford: convergence

❒  The protocol converges, independent of whatever preference given to the paths with the same weight, if the network is free.

❒  Means that the ordering within one class (customer, provider) is not important for convergence. Any preference given to them will result in convergence

❒  Since the algebra is also isotone, the path is optimal. But this says nothing about global property of the path, since the weight is only decided by the first link of the paths.

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References

❒  “Stable Internet routing without global coordination,” L. Gao, J. Rexford, ToN 2001

❒  “The stable paths problem and interdomain routing,” T. Griffin, F. Shepherd, G. Wilfong, ToN 2002

❒  “Algebra and algorithms for QoS path computation and hop-by-hop routing in the Internet,” J. Sobrinho ToN 2002

❒  “An algebraic theory of dynamic network routing,” J. Sobrinho, ToN 2005.