Routing and Congestion Problems in General Networks

Presented by Jun ZouCAS 744

2

Outline

• 1. Introduction to Routing and Congestion

• 2. Network Model and Objective• 3. Construction of Tree• 4. Simulation Graph on Tree• 5. Simulation Tree on Graph• 6. Conclusion and Application• 7. References: its full paper and

improved one

3

1. Introduction to Routing

• The function of routing is to find a ‘best’ path from source to destination for each incoming packet.

• What is ‘best’? : Minimum hop count, minimum delay, security, etc

• In this paper, our goal is to minimize the congestion of the whole network links.

4

2.1 Network Model

• Network: a weighted graph G=(V, E)• |V|=n nodes and |E|=m edges• Bandwidth: a function b(e): E→ R+

• Absolute load: amount of data transmitted on a edge e

• Relative load L(e): Absolute load/bandwidth• Congestion C: Maximum over the relative

load of all links in the network

R

EeeLC

))(max(

5

2.2 Two approaches to solve routing problems

• Traffic modeling and simulation: Simplify the traffic model (such as M/M/1 model), simulate the routing protocols and analyze results by using queuing theory

• Simulation graph on a tree: Combine a tree solution of an online problem and tree representation of the network

6

2.3 Oblivious online routing algorithms

• Oblivious routing algorithm: path selection for the i-th request σi does not depend on routing decisions made for other requests σj

• Oblivious adversary: The request sequence {σ} is not allowed to depend on the selection of online algorithms

7

2.4 Assumption and target

• Assumption: There is a ct-competitive online algorithm for the tree TG=(Vt, Et) associated with a graph G=(V,E):

(1)

tGopttGonl kTCcTC )()(

• Target: For the same algorithm, find a small factor c for the graph G=(V,E) , satisfying

(2)

kkGCccGC topttonl )()(

8

2.5 Three steps to achieve it

• 1st step: Find a method to construct an associated tree which satisfies the following conditions:

• 2nd step: A tree TG can simulate the network G, i.e. for any request sequence σ, an algorithm which produces congestion C when it is processed on graph G,, also produces congestion when it is processed on the tree TG.

• 3rd step: Prove that for any request sequence σ, an online algorithm which produces congestion Ct

when it is processed on TG,, also produces congestion when it is processed on G.

CCt

tCcC

9

3.1 Construct a tree

→

A graph G=(V,E): Associated tree TG=(Vt, Et)

10

3.2 Definitions

• Vt: a node in TG

• SVt: the cluster in G corresponding to Vt

• Bandwidth between two sets

YyXx

yxbYXcap,

)),((:),(

• Bandwidth of edges leaving set X:

• The height of TG: h(TG)

• Set of all level nodes:

),(:)( XXcapXout

l l

tV

11

3.2 Definitions (con’t)• Weight function:

For a subset X, the bandwidth of all edges that are adjacent to nodes in X and that do not connect nodes of the same cluster to .

One important property: AdditiveExample

l

tt

tt

VVVVl SSXcapVXcapXW ),(),(:)(

l

tV

• Bandwidth-ratio:

)\,(

)(max:

2/

,, USUcap

Uout

t

tV

tVt

VSU

SUV

• weight-ratio:

)\,(

)(max: 1

2/

,, USUcap

Uw

t

tV

tVt

V

l

SU

SUV

12

4.1How to Simulate G on TG

• A node is simulated by a node vt in TG corresponding to cluster Svt = {v}.

• So, a message sent between node u and v in G is sent along one unique path connecting the respective counterparts in TG.

• Example

• Our goal is to states that this simulation does not increase the congestion.

Vv

13

4.2 Theorem 1

• Theorem 1: For any request σ for an routing problem on network G that can be processed with congestion C, its simulation on TG yields congestion CCt

• Proof:

14

5.1How to Simulate TG on G

• A level node vt of TG is simulated by a random node of the corresponding cluster Svt with the probability:

l

)(

)(

1

1

tvl

l

SW

vW

• Example

15

5.2 Theorem 2

• Theorem 2: The expectation of the relative load L(e) of an edge in graph G, due to the simulation of a tree strategy on G is bounded by

)},max{(log))(( tChnOeLE where Ct is the congestion on TG, h=h(TG),

}{maxttt vVv }{max

ttt vVv

16

5.2.1 CMCF Problem

• Concurrent Multi-commodity Flow Problem (CMCF): Each commodity fu,v has a flow size q.du,v, where q is the maximized minimal throughput fraction over all commodities, and

)()(

)(

)(

)(

1

1

1

1, v

ul

l

vl

lvu Sout

SW

uW

SW

vWd

ti

17

5.2.2 absolute/relative load on G

• Expected number of messages have to be routed between u and v is

vutvtul

l

vl

l dCSoutCSW

uW

SW

vWi

ti

,1

1

1

1 )()(

)(

)(

)(

• The minimum capacity of edge is q.du,v, , so the expected relative load at level l is at most Ct/q,

• Its expected relative load at all levels isq/Ch t

18

5.2.3 capacity ratio

• q has a lower bound：

},max{log

1

n

)},max{(log))(( tChnOeLE

• Therefore, the expected relative load on G has a upper bound:

19

5.2.4 Next target

• So far, we show that a path of tree can be simulated by a path in graph such that the expected relative load of this path on the graph has a upper bound.

• Our goal is to show that the congestion in graph also has a upper bound compared to that in tree, i.e, to satisfy the lemma 2. So we should extend the expected value to ‘true’ value, that means show:

L(e)=O(L(e))

20

5.3 Theorem 3

• Theorem 3: Give a graph G and an associated tree, there exists an oblivious online routing algorithm, which is

-competitive with respect to the congestion.

}),max{(log hnO

21

5.3.1 Proof and Chernoff bound

• Let X1, X2,…Xn be independent 0-1random variables, i.e.

Pr (Xi=1)=pi,

m=p1+p2+…+pn,

S=X1+X2+…+Xn, then

3/2

))1(Pr( memS

22

5.3.2 Improvement to Theorem 3

• Theorem 4: Give a graph G, there exists a associated tree that has height h(TG)=O(log n), maximum bandwidth ratio λ(TG)= O(log n), and maximum weight ratio δ(TG)=O(log n).

• The online routing algorithm is -competitive.

)(log3 nO

23

6. Conclusion and Application

• The paper proposes a method to construct a associated tree regarding to a general network and proves that the congestion on the network is only a small factor c= larger than the congestion on the tree.

• Since the tree topology is much simpler than graph, we can study the routing algorithm on a tree and also can get a ‘good’ competitive algorithm on the general network. It is a very useful tool for research on routing problems on general networks.

)(log3 nO

24

7. Reference

• Full paper: The paper published in the conference IEEE FOCS’02 skips some proofs due to space limitation. I contacted the author and got the full paper. It is available to everyone If you are interested.

• Improved paper: The author improve the results in the following paper:

“A Practical Algorithm for Constructing Oblivious Routing Schemes”, published at fifteenth annual ACM symposium on Parallel algorithms and architectures.