Routing and Congestion Problems in General Networks

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Routing and Congestion Problems in General
Networks

• Presented by Jun Zou
• CAS 744

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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

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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.

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2.1 Network Model

• Network a weighted graph G(V, E)
• Vn nodes and Em 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

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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

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2.3 Oblivious online routing algorithms

• Oblivious routing algorithm path selection for
  the i-th request si does not depend on routing
  decisions made for other requests sj
• Oblivious adversary The request sequence s is
  not allowed to depend on the selection of online
  algorithms

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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)

• Target For the same algorithm, find a small
  factor c for the graph G(V,E) , satisfying
• 
  (2)

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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 s, 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 s,
  an online algorithm which produces congestion Ct
  when it is processed on TG,, also produces
  congestion when it is processed on
  G.

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3.1 Construct a tree
?
A graph G(V,E)
Associated tree TG(Vt, Et)
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3.2 Definitions

• Vt a node in TG
• SVt the cluster in G corresponding to Vt
• Bandwidth between two sets

• Bandwidth of edges leaving set X
• The height of TG h(TG)
• Set of all level nodes

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3.2 Definitions (cont)

• 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

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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.

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4.2 Theorem 1

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

• Proof

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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

• Example

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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
  

where Ct is the congestion on TG, hh(TG),
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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

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5.2.2 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.du,v, , so the
  expected relative load at level l is at most
  Ct/q,
• Its expected relative load at all levels is

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5.2.3 capacity ratio

• q has a lower bound

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

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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))

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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.

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5.3.1 Proof and Chernoff bound

• Let X1, X2,Xn be independent 0-1random
  variables, i.e.
• Pr (Xi1)pi,
• mp1p2pn,
• SX1X2Xn, then

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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 ?(TG) O(log n), and
  maximum weight ratio d(TG)O(log n).
• The online routing algorithm is
  -competitive.

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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.

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7. Reference

• Full paper The paper published in the
  conference IEEE FOCS02 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.