Yashar Ganjali

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Minimum-delay Routing

• Yashar Ganjali
• High Performance Networking Group
• Stanford University
• September 17, 2003

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Outline

• Network and flow model
• Delay model
• Problem statement
• Previous work
• New algorithm
• A simple example
• Outline of the optimality proof

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Network Flow Model

• Network G(V,E)
• N nodes
• M links
• K commodities
• Source si
• Destination ti
• Demand di

t2
s3
t3
s2
t1
s1
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Flow Constraints

• Conservation of flow constraint
• For any node v and commodity i
• di(v)?uvfi(uv)- ?vufi(vu) 0
• Capacity constraint
• For any link uv and commodity i
• ?i fi(uv) lt Cuv

f1(vx)
f1(uv)
v
f1(vy)
f1(wv)
f1(vz)
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Delay (cost) Model

• Delay at each link
• Duv fuv/(Cuv-fuv)
• Increasing
• Convex

u
V
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Problem Statement

• Goal Minimizing the total delay in the network.
• Total delay ?uv Duv(fuv)
• Problem How to divide flows at each node of the
  network, i.e. finding routing tables.

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Previous Results

• Cantor 74 Linear Programming
• Centralized
• Gallager 77 Distributed algorithm
• Network dependent
• Bertsekas et al. 97 Distributed Fast
  Approximation
• Single commodity
• Plotkin et al. 95 Distributed Multicommodity
  Flow Algorithm
• Linear cost function
• Our method
• Distributed
• Fast convergence
• Multicommodity

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Relaxing Conservation of Flow Constraint

• We relax the conservation of flow constraint
• di(v)?uvfi(uv)- ?vufi(vu) gi(f,v)
• We call gi(f,v) the excess of commodity i at node
  v and flow f is called a pre-flow.
• We will use this quantity to find points of high
  pressure in the network.

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Minimum-delay RoutingPotential Function ?

• We define ?1 ?uv,iexp(?gi(f,uv)/di) and ?2
  ?uv Duv(fuv )/B
• ?1 measures how close the current pre-flow f is
  to a flow.
• ?2 measures how close the cost of the current
  flow is to the budget B.
• We let ? ?1 ? ?1 x?2

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Our Algorithm

• Minimum-delay algorithm Starting from zero flows
  our goals is to minimize ?
• O(?-1log(m?-1)) phases
• O(?-1) iterations
• Increase capacities by a factor of ?
• Increase demands by a factor of ?
• Update the amount of excess at each node
• BALANCE EXCESSES
• Rescale capacities and demands
• Update ? if needed

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Balancing Excesses

• Each node divides its excess evenly among
  adjacent links.
• Each link locally minimize ?.

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Example
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Example
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Example
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Example
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Outline of the Proof

1. If ? is small enough we are close to the optimal
   solution.
2. In each ITERATION the increase in ? is small.
3. At the end of each PHASE the amount of ? is
   divided by two.

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Small ? ? Close to Optimal

• We know ? ?1 x ?2
• ? is small means both ?1 and ?2 are small.
• ?1 is small means conservation of flows is almost
  satisfied.
• ?2 is small means cost is close to optimal.
• We can show that if ? lt (1?)? at least (1-?) of
  each demand is satisfied and our cost is at most
  (1?) times the optimal cost

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Increase in ? is small

• Consider an optimal flow f
• We have f lt C and ?f lt ?C
• Therefore, in each iteration we can have ?f ?f
  
• We can show the increase in ? is small in this
  case (cost function is convex and has a bounded
  derivative).
• Therefore, if we minimize ? the amount of
  increase is small.

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In each phase ? decreases by a factor lt 1

• At the end of each phase we divide all demands
  and flows by two.
• Therefore excesses and delays are reduced.
• We can show this reduces ? by a factor which is
  less than 1.

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Complexity of the Algorithm

• Original ? is bounded.
• We know ? is multiplied by a factor less than one
  in each phase.
• We can conclude that the algorithm has
  O(?-1log(m?-1)) phases.
• Each phase consists of O(?-1) iterations.
• Running Time O(?-2?-2KM2N)
• Improved running time O(?-3?-3KMN2)

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Future Work

• Sensitivity and stability analysis
• How sensitive the algorithm is to perturbations?
• Cost function
• Realistic cost function
• Implementations issues
• Where?
• How to incorporate with existing routing
  protocols?

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Thank you!