Data Networks

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Data Networks

• Leo J.H. Hu
• InstructorDr. Y.S Lin
• Dept. Information Management
• National Taiwan University
• r90058_at_im.ntu.edu.tw

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Characterization of Optimal Routing

• cost function
• problemminimize
• subject to , for all
• , for all

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Characterization of Optimal Routing

• D(x) cost function
• first derivative length of path p
• Optimal path flow is positive only on paths with
  a minimum first derivative length

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Characterization of Optimal Routing

• Let is an optimal path flow vector
• If gt0 for some path p of an OD pair w , we
  must be able to shift a small amount from
  path p to any other path without improving the
  cost.
• must be nonnegative
• , for all

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Example

• D(x)D1(x1)D2(x2)
• Di(xi) x1 / Ci-xi , where for i1,2
• Ci is the capacity of link i , C1C2

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Example (contd)

• Case 1 x1r , x20

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Example (contd)

• Case 2 x1gt0 , x2gt0

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Example(contd)
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Characterization of Optimal Routing

• optimal solution can be implemented naturally
  even when the input rates are time varying
  -fractions
• for all
• circuit networks v.s datagram network
• routing varibale

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Feasible direction methods

• A set of path flows is strictly sub-optimal only
  if there is a positive amount of flow that
  travels on a non-MFDL path.
• Sub-optimal routing can be improved by shifting
  flow to an MFDL path from other paths for each OD
  pair.
• We should shift part of the flow of the other
  paths to the shortest path , otherwise it will
  oscillatory behavior resulting

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Feasible direction methodsw

• Solving the optimal routing problem by decreasing
  the cost function through incremental changes in
  path flows
• Choose a feasible flow vector and changing x
  along a direction ?x

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Feasible direction methods

• Two requirement
• ?x should be a feasible direction
• , x ?x is feasible
• for all for
  which

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Feasible direction methods

• ?x should be a descent direction
• inner product of and is negative
• The inner product is equal to the first
  derivative of the function G( ) D(x ) at
  0

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Feasible direction methods

• should satisfies the conservation of flow
  condition , such that
• for all paths p that are non- shortest in
  the sense
• , for some
• path of the same OD pair
• lt0 for at least one non-shortest path p

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Feasible direction methods

• We obtain a broad class of iterative algorithms
  for solving the optimal routing problem.
• Basic iteration is
• such that

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Flow Deviation Method

• This algorithm is a special case of so-called
  Frank-Wolfe method for solving general ,
  nonlinear programming problems with convex
  constraint sets.
• It can reduce the cost function to its minimum in
  the limit , but convergence rate near the optimum
  tends to be very slow.
• The flow is shifted from the non-shortest paths
  in equal proportion

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Frank-Wolfe (Flow Deviation) Method

• Notation
• be the vector of path flows if routed along
  the corresponding MFDL path
• is the stepsize that minimizes
• over all
• New set of path flows is

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Flow Deviation Method
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Example
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Example

• Constraints
• x1x2x31 , x1,x2,x3 0
• Cost function
• derived properties
• x3 has a large processing and propagation delay
• x1 x2

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Example(contd)

• In Analysis point of view
• two possible results
• x3 0 and x1x21/2
• x3i gt0 , and x1x2(1-i)/2
• Case b) is not possible 3i-0.1
• optimal solution
• (1/2,1/2,0)

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Example (contd)

• Applying F-W iteration
• shortest path is either 1 or 2 , corresponsively
  , (1,0,0) or (0,1,0)

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Example (contd)

• Deciding
• is obtained by line minimization over 0,1

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Example(contd)

• Since is a descent direction , we have
  .
• Therefore , the stepsize , which is the
  constrained minimum over 0,1 is given by

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Example(contd)

• Fig 5.66

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Example(contd)

• The rate of convergence becomes slow near the
  optimal solution.
• The directions of search tend to be-come
  orthogonal to the direction leading from the
  current iterate to final solution.
• The error ratio is always less than 1 , and
  converges to 1 as k goes infinite.
• This is sub-linear convergence rate , and is
  typical of the method.

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Example(contd)
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Simpler method

• Make a second-order Taylor approxi-mation of G(a)
  around a0
• minimize with respect to a over the
  interval 0,1

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Simpler method(contd)

• For the stepsize , it converges to the optimal
  set of total link flows if the starting set of
  total link is sufficiently close to the optimal.
• For the cost functions used in routing problems ,
  it appears it leads to convergence even when the
  starting total link flows are far from optimal.

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Frank-Wolfe methodinsightful geometrical
interpretation
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Simpler method(contd)

• It is well suited for distributed implementation.
  But appears inferior to the projection method ,
  because it is essential for the network nodes to
  synchronize their calculations in order to obtain
  the stepsize.

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Advantage of Frank-Wolfe method

• It can be implemented in a way that only the
  current total link flows together with the
  current shortest paths for all OD pairs are
  maintained in memory at each iteration.
• The amount of storage required is small , and
  allowing to solve a large network problem.

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Projection methods for optimal routing

• An increment of flow change is calculated for
  each path on the basis of the relative magnitudes
  of the path lengths and the second derivatives of
  the cost function.
• If the increment is so large that the path flow
  is negative , the path flow is simply set to zero.

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Projection methods for optimal routing

• Unconstrained Nonlinear Optimization
• steepest descent
• Nonlinear Optimization over the Positive Orthant
• Application to Optimal Routing

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Steepest descent method

• f is a twice differentiable function of the
  n-dimensional vector x with a gradient and
  Hessian matrix at any x denoted

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Steepest descent method

• are the minimizing stepsize determined by
• and a constant positive stepsize
• , for all k

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Steepest descent method

• If f is a positive definite quadratic function ,
• Where , and M,m are the largest and
  smallest eigenvalues of
• The speed of convergence can be quite slow , if
  the ratio M/m is large ( this corresponds to the
  equal-cost surface being very elongated )

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Steepest descent method

• Improve the convergence rate by premultiplying
  the gradient by a suitable positive definite
  scaling matrix.
• By Newtons method

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Steepest descent method

• The excellent convergence rate is achieved at the
  expence of the overhead associated with
  computation
• Alternative approximate optimal

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gradient projection method

• the convergence results mentioned earlier also
  hold true , so obtaining

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gradient projection method
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Application on Optimal Routing

• We can eliminate the equality constraints
  by substituting
• , then obtaining a problem
• minimize
• subject to for all
• where is the vector of all not MFDL path
  flows

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Application on Optimal Routing

• calculating the derivatives needed
• ,for all
• ,for all
• Set of links belonging to either p ,or
• the corresponding MFDL path , not both

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Application on Optimal Routing

• from above,the iteration takes the form
• We refer to the iteration as the projection
  algorithm.

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Observations

• It may also be viewed as a generalization of the
  adaptive routing method based on shortest paths.
• Those non-shortest path flows that are zero will
  stay at zero.
• Only paths that carried positive flow at the
  start or were MFDL paths at some previous
  iteration can carry positive flow at the
  beginning of an iteration.

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Determining ak

• Keeping ak constant is well suited for
  distributed implementation
• Let ak equal to 1 is a good choice.
• When the iteration is carried out one OD pair at
  a time , even better performance is usually
  obtained.
• gt dropping the off-diagonal terms of the
  Hessian matrix.

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Convergence

• It yields rapid convergence to neighborhood of an
  optimal solution , but slows down near a
  solution.
• It is often satisfactory near a solution and
  usually far better than Frank-Wolfe method.
• Taking off-diagonal terms of the Hessian matrix
  into account will obtain faster convergence near
  a optimal solution.

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Comparison
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Example

• Paths of OD pair (1,5)
• p1(1)1,4,5
• p2(1)1,3,4,5
• Paths of OD pair (2.5)
• p1(2)2,4,5
• p2(2)2,3,4,5
• Dij(Fij)1/2 (Fij)2

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Example
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Example
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Example
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Example(contd)

• Let x1(1),x2(1),x1(2),x2(2) denote the flows
  along the paths p1(1) , p2(1) , p2(1),p2(2),the
  corresponding length are

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Example(contd)
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Example(contd)