Minimizing Congestion in General Networks Harald Rcke Presented - - PowerPoint PPT Presentation

Minimizing Congestion in General Networks

Harald Räcke Presented By- Gaurav Gupta

Definition

c – Competitive

CA(σ) ≤ c . Copt(σ) + k

G(V,E)

b : E -> R+

T(V,E)

bt : Et -> R+

Previous Work

Maggs et al. gave (log n) competitive

algorithm for meshes.(1997)

Aspnes et al. gave (log n) competitive

algorithm for centralized networks.(1993).

Outline of the Framework

It is a three step algorithm

Step 1. Simulate tree TG using G.

Ct ≤ C

Step 2. Find routing is the tree. Step 3. Simulate G using TG

C’ ≤ c.Ct ≤ c.C

The Decomposition Tree

Laminar set. Example 1. {{v1}, {v2},{v1,v2}} Find edges in tree.

• Define h(TG) = height of TG

ut <-> Sut , out(X) = Cap(X, X’) Vl

G = set of vertices at level l.

Define bandwidth of edges in tree.

,

( , ) ( , )

x X y Y

Cap X Y b x y

∈ ∈

= ∑

Contd...

• Take case l+1. Intuitively it measures the
• utflow from X.

X=X1 υ X2

=> wl(X) = wl(X1) + wl(X2)

Take special cases e.g. take X = Sut. Now simulate G on TG.

( ) ( , ) ( , )

l t

l vt vt vt V

w X Cap X V Cap X S S

∈

= − ∩

∑

Define

• ( )

{ } ( , \ )

max

vt

vt U S vt

• ut U

cap U S U λ

⊂

=

1( )

{ } ( , \ )

max

vt

l vt U S vt

w U cap U S U δ

+ ⊂

=

( ) max { }, ( ) max { },

t t

G vt V vt G vt V vt

T δ δ δ λ λ

∈ ∈

= =

Simulate TG on G

Choose v € Svt with probability- Consider l -1 node ut with some children

(one is vi).

Define a CMCF-Problem.

• q = sol’n of solution fraction.

1 1

( ) ( )

l l vt

w v w S

+ +

1 , 1

( ) ( ) . ( ). ( ) ( )

l l u v vi l vi l ut

w v w u d

• ut S

w S w S

+ +

=

E(L(e)) = O(h.Ct/q)

First prove E(Ll(e)) = O(Ct/q). ut level l-1. vi one of the children. Absolute load in tree edge =

Ct.bt(ut,vi)=Ct.wl(Svi).

Find expected load between (u,v) and

find congestion in it.

Voila..

This number equals Ct.du,v. When du,v demand congestion = 1/q.

When Ct.du,v congestion = Ct/q.

For all levels Congestion= O(h.Ct/q.) We don’t know h and q.

Find q.

q=Ω(Φ/log n).

Φ = value of the sparsest cut.

• Thus Congestion=

How do we measure the goodness of

decomposition.

(max{ , }) O φ δ λ ≥

( .max{ , }.log( ). )

t

O h n C δ λ

The Graph Decomposition

There exist a decomposition tree with Combining all these we get competitive

ration of (log n)3.

But finding a decomposition tree is an NP-C

problem.

(log ), (log ), (log ) h O n O n O n δ λ = = =

Work done after this paper.

Azar et al. gave polynomial time routing

algorithm.

Represent each flow in terms of linear equation.

• subjected to congestion Z.

Formulate LP and solve using ellipsoid or

karmarkar algorithm.

, ,

( , , ) * ( )

i j i j ij

flow e f D D f e =∑

Thank you