Routing and Congestion Problems in General Networks Presented by Jun Zou CAS 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

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 σ i does not depend on routing decisions made for other requests σ j 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 c t -competitive online algorithm for the tree T G =(V t, E t ) associated with a graph G=(V,E) : (1) Target: For the same algorithm, find a small factor c for the graph G=(V,E), satisfying (2)

8 2.5 Three steps to achieve it 1 st step: Find a method to construct an associated tree which satisfies the following conditions: 2 nd step: A tree T G 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 T G. 3 rd step: Prove that for any request sequence σ, an online algorithm which produces congestion C t when it is processed on T G,, also produces congestion when it is processed on G.

9 3.1 Construct a tree → A graph G=(V,E): Associated tree T G =(V t, E t )

Definitions V t : a node in T G S Vt : the cluster in G corresponding to V t Bandwidth between two sets Bandwidth of edges leaving set X: The height of T G : h(T G ) Set of all level nodes:

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: Additive Example Bandwidth-ratio: weight-ratio:

12 4.1How to Simulate G on T G A node is simulated by a node v t in T G corresponding to cluster S vt = {v}. So, a message sent between node u and v in G is sent along one unique path connecting the respective counterparts in T G. Example Our goal is to states that this simulation does not increase the congestion.

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 T G yields congestion Proof:

14 5.1How to Simulate T G on G A level node v t of T G is simulated by a random node of the corresponding cluster S vt with the probability: Example

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 where C t is the congestion on T G, h=h(T G ),

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

absolute/relative load on G Expected number of messages have to be routed between u and v is The minimum capacity of edge is q.d u,v,, so the expected relative load at level l is at most C t /q, Its expected relative load at all levels is

capacity ratio q has a lower bound ： Therefore, the expected relative load on G has a upper bound:

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))

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.

Proof and Chernoff bound Let X 1, X 2,…X n be independent 0-1random variables, i.e. Pr (X i =1)=p i, m=p 1 +p 2 +…+p n, S=X 1 +X 2 +…+X n, then

Improvement to Theorem 3 Theorem 4: Give a graph G, there exists a associated tree that has height h(T G )=O(log n), maximum bandwidth ratio λ (T G )= O(log n), and maximum weight ratio δ (T G )=O(log n). The online routing algorithm is - competitive.

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.

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.