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 � C A ( σ ) ≤ c . C opt ( σ ) + k � G(V,E) � b : E -> R + � T(V,E) � b t : E t -> 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 T G using G. � C t ≤ C � Step 2. Find routing is the tree. � Step 3. Simulate G using T G � C ’ ≤ c.C t ≤ c.C

The Decomposition Tree � Laminar set. � Example 1. {{v 1 }, {v 2 },{v 1 ,v 2 }} � Find edges in tree. = ∑ � Cap X Y ( , ) b x y ( , ) ∈ ∈ x X y Y , � Define h(T G ) = height of T G � u t <-> S ut , out(X) = Cap(X, X ’ ) � V l G = set of vertices at level l. � Define bandwidth of edges in tree.

Contd... ∑ = − ∩ w X ( ) Cap X V ( , ) Cap X ( S , S ) � l vt vt ∈ l vt V t � Take case l+1. Intuitively it measures the outflow from X. � X=X 1 υ X 2 => w l (X) = w l (X 1 ) + w l (X 2 ) � Take special cases e.g. take X = S ut . � Now simulate G on T G .

Define out U ( ) λ = � max { } vt cap U S ( , \ U ) ⊂ U S vt vt w 1 ( ) U δ = + l max � { } vt cap U S ( , \ U ) ⊂ U S vt vt δ = δ δ λ = λ � ( T ) max { }, ( ) max { }, ∈ ∈ G vt V vt G vt V vt t t

Simulate T G on G w ( ) v + l 1 � Choose v € S vt with probability- w ( S ) + l 1 vt � Consider l -1 node u t with some children (one is v i ). � Define a CMCF-Problem. w ( ) v w u ( ) = + l 1 l d . out S ( ). u v , vi w ( S ) w S ( ) + l 1 vi l ut q = sol’n of solution fraction. �

E(L(e)) = O(h.C t /q) � First prove E(L l (e)) = O(C t /q). � u t level l-1. v i one of the children. � Absolute load in tree edge = � C t .b t (u t ,v i )=C t .w l (S vi ). � Find expected load between (u,v) and find congestion in it.

Voila.. � This number equals C t .d u,v . � When d u,v demand congestion = 1/q. � When C t .d u,v congestion = C t /q. � For all levels Congestion= O(h.C t /q.) � We don’t know h and q.

Find q. � q= Ω ( Φ /log n). � Φ = value of the sparsest cut. φ ≥ δ λ O (max{ , }) Thus Congestion= � δ λ O h ( .max{ , }.log( ). n C ) t � How do we measure the goodness of decomposition.

The Graph Decomposition � There exist a decomposition tree with = δ = λ = h O (log ), n O (log ), n O (log ) n � Combining all these we get competitive ration of (log n) 3 . � But finding a decomposition tree is an NP-C problem.

Work done after this paper. � Azar et al. gave polynomial time routing algorithm. � Represent each flow in terms of linear equation. = ∑ flow e f D ( , , ) D * f ( ) e i j , i j , ij subjected to congestion Z. � � Formulate LP and solve using ellipsoid or karmarkar algorithm.

Thank you