QoS Routing with Performance-Dependent Costs

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QoS Routing with Performance-Dependent Costs

• Funda Ergun Rakesh Sinha Lisa ZhangINFOCOM
  2000.Nineteenth Annual Joint Conference of the
  IEEE Computer and Communications
  Societies.Proceedings.IEEE Volume1,2000,Page(s)1
  37146 vol.1

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Outline

• Introduction
• Preliminaries
• Polynomial-time approximation
• Heuristics for partition
• Conclusion

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Abstract

• Present approximation algorithms guarantee to
  produce solutions that are within 1 of the
  optimal
• The running times are polynomial in the input
  size and 1/
• Present good approximations algorithms and
  heuristics which apply to general cost functions

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Introduction

• Todays Internet deploys best effort routing
  without any assurance of service quality
• Future Internet will support various QoS
  classes,and each class has its own set of service
  guarantee and associated costs
• Packets will be given higher priority with aim of
  satisfying performance requirements
• Stringent requirements should be charged a higher
  fee

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Introduction(cont.)

• QoS routing is to identify a routing path based
  on an applications Qos requirements and resource
  avalilability
• QoS requirements are specified either as
• Set of path constraints
• Set of link constraints
• A feasible path is a path with sufficient
  resources to satisfy QoS requirements
• Optimal criteria narrow the selection among
  feasible paths

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Introduction(cont.)

• This paper considers a model in which an
  application is charged a per link price depending
  on delay guarantee requested
• Service provider provides multiple service
  classes with different price

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Problems

• A network with n nodes and m links
• Each link has a cost function ce(d) to represent
  cost incurred by delay d on link e
• Given source s and destination t,and end-to-end
  delay constraint D needs to be satisfied
• There are following two problems
• Constrained minimum cost path (PATH)
• Constrained minimum cost partition (PARTITION)

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Problems(cont.)

• PATH problem
• Chose an s-t path and minimize sum of link costs
  along the path subject to delay constraint D
• PARTITION problem
• s-t path is already chosen
• Determine delay to be imposed on every link along
  path such that cost is minimized subject to the
  end-to-end delay constraint D

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Preliminaries

• Network N has n nodes and m links
• Given an s-t path P, let p be the number of links
  on the path
• Each link has associated cost function ce(d),and
  it is non-increasing
• Work with integral delays and costs
• Use de(c) to denote the inverse of ce(d) and it
  returns smallest delay that incurs cost at most c

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Preliminaries(cont.)

• Given link e and delay d,we can retrieve cost
  ce(d) in constant time
• Given link e and cost c,we can compute delay in
  logD time using binary search
• OPT denote the cost of the optimal s-t path
  subject to the delay constraint
• C denote the maximum possible cost on any link
  ,i.e. Cmaxece(1)

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Polynomial-time approximation

• Approximation algorithms for PATH and PARTITION
  are based on approximation algorithm for RSP
• RSP is a restricted version of PATH with each
  link has fixed cost and delay
• Theorem 1the result is given by Hassin
• RSP has an -approximation algorithm with
  running time O( mnloglog U)
• U is upper bound of OPT

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Polynomial-time approximation

• Theorem 2
• PATH has an -approximation algorithm with
  running time O(X mnloglog U),where X minD,
  logD, logD
• Theorem 3
• PARTITION has an -approximation algorithm with
  running time O(X p2loglog U),where XminD,
  logD, logD

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Algorithm 1

• Derive from approximation algorithm for RSP
• Transform network N(for PATH) into a network
  N1(for RSP) such that optimal RSP solution is
  equivalent to optimal PATH solution
• Replace each link e of N by D links e1,,eD
• Each link ei has cost cei(i) with delay i
• Apply Hassins approximation algorithm for RSP to
  N1
• Map resulting s-t path to a path in N
• Replace link ei with link e with delay i and cost
  ce(i)

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Algorithm 1(cont.)

• Running time includes two components
• Time of creating N1
• Time of applying approximation algorithm for RSP
• Time for creating N1 is mD
• Time for running Hassins algorithm is O( D
  mnloglog U)
• Lemma
• Algorithm 1 is an -approximation of PATH with
  running time O(mDD mnloglog U)

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Algorithm 2

• Key idea of to achieve weaker guarantee with far
  fewer links transformation
• Goal of algorithm 1 is to capture all possible
  choices of cost and delay assignments
• Subdivide range of cost 1,ce(1) into
  sub-ranges and pick one representative from
  each sub-range
• Reduce the linear blow-up to logarithmic
  blow-up
• Disadvantage is the approximation

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Algorithm 2(cont.)

• Precise description
• Consider a link e in N
• Each semi-open sub-range
  ,
• Find minimum delay di that incurs a cost within
  above range,and create a link in N2 with delay di
  and ce(di)

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Transformation example
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Algorithm 2(cont.)

• Each link in N is replaced by
  links
• Creating N2 requires at most m and
  computations of de(.) function,resulting in
  running time
• Lemma
• Algorithm 2 is an - approximation of PATH with
  running time

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Algorithm 3

• High-level idea
• Use TEST procedure to determine whether OPT is
  greater than some given value V
• Then start with some upper bound U and use EXACT
  procedure for exact value of OPT
• TEST procedure will determine whether OPT V or
  OPT (1 )V
• Maintain a range,L,(1 )U,for OPT
• To approximate OPT ,we repeatedly narrow the gap

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An Exact solution

• Let gv(c) be the minimum delay from s to v with
  total cost c
• Minimum delay path s-v goes through some
  intermediate node u
• Obtain gv(c) by minimizing all possible
  intermediate nodes u and all possible costs bltc
• OPT is the smallest cost of an s-t path with
  delay at most D
• gt(OPT) D and for any cltOPT, gt(c)gtD
• The smallest c such that gt(c) D is equal to
  OPT

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An Exact solution(cont.)
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An Exact solution(cont.)

• Running time for one iteration is O(mlogDmc)
• For loop has OPT iterations,so overall running
  time is O( mOPT logDmOPT2)

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The TEST procedure

• Approximately determine if a given value is
  greater than OPT
• TEST resembles EXACT,except
• Costs ce(d) are scaled down by a factor V / n
• For loop is executed for c1,2,,
• If a path of delay at most D is found for some c
  ,TEST outputs OPT (1 )V
• Scaled down cost and its inverse are denoted by
  and

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The TEST procedure(cont.)

• running time is

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The Approximation algorithm

• Total running time is

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Heuristics for PARTITION

• Greedy algorithms
• One unit of delay is initially assigned to each
  link along path P
• One unit of delay is added to the link that
  causes largest reduction in total cost

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Greedy algorithm(cont.)
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Heuristics

• Greedy algorithm always makes locally optimal
  choice
• For non-convex functions,make some locally
  non-optimal choices to reach globally optimal
• Two heuristics
• Greedy heuristic with Rollback
• Continues adding delay to one link
• Variable Step Size Heuristic
• Adding delay in chunks of various sizes

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Greedy Heuristic with Rollback

• Proceeds greedily by always adding one unit of
  delay to the link that offers largest cost
  reduction
• If cost reduction(g) due to current delay is
  greater than previous reduction,checks every link
  and performs a rollback
• Rollback consists of removing delay units from
  each link until a unit reach a unit whose
  reduction was at least g

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Greedy Heuristic with Rollback
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Variable step size heuristic

• Allow allocation of delay in sizes greater than
  single unit during a iteration
• Pick best link and best delay increment
• Motivation
• If a curve offers large per delay cost
  reduction,we are not stuck in earlier part of
  that curve
• Has additional advantage of making the heuristic
  run faster
• Two variants
• Delay allocations are all possible powers of 2
• All possible delay allocation between 1 and D

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Variable step size heuristic(cont.)
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Simulation results

• Experiments are run on a path of 30 links with
  D250
• Each group of experiments aims to test cost
  functions with different shapes
• Earlier groups tend to have more alternating
  convex/concave regions
• Actual cost functions are generated randomly
• Each number in table shows percentage error for
  corresponding heuristic
• Difference between initial cost and optimal cost

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Simulation results(cont.)
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Conclusion

• Present polynomial-time -approximations for
  PATH and PARTITION problems with general cost
  functions
• Applying results to more complicated structures
  such as multicast trees is left for future
  research