A Practical Algorithm for Constructing Oblivious Routing Schemes

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A Practical Algorithm for Constructing Oblivious
Routing Schemes

• Marcin Bienkowski
• Miroslaw Korzeniowski
• Harald Räcke

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Problem Definition
Virtual Circuit Routing Problem

• given a graph describing a
  computer network
• input sequence
  of routing requests
• select a system of paths between and for
  each request
• each request routed fractionally over the paths
  chosen for it
• cost-measure congestion, i.e., the maximum load
  of an edge

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Problem Definition

• online algorithm
• online algorithm must specify each path-system
  without knowingfuture requests
• compare congestion of the online-algorithm to the
  optimal offline congestion for
• oblivious algorithm
• the system of paths chosen for a request cannot
  depend on any other request

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Related Work

• algorithms for specific networks
• Maggs, Meyer auf der Heide, Vöcking and
  Westermann 1997Bartal, Leonardi 1997
• lower bound of for the competitive
  ratio in the mesh
• logarithmic upper bounds for many specific
  networks as e.g., hypercubic networks, meshes...
• non-polynomial algorithm for general networks
• Räcke 2002
• polylogarithmic ( ) upper bound
  for general networks

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Related Work

• optimal algorithm
• Azar, Cohen, Fiat, Kaplan, Räcke 2003
• optimal competitive ratio for oblivious algorithm
• based on linear programming and the Ellipsoid
  algorithm with a separation oracle

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Results

• Improvement on Räckes result from 2002
• Simplified proof for the upper
  bound on the competitive ratio
• competitive polynomial-time
  algorithm for general graphs
• competitive polynomial-time
  algorithm for planar graphs

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Related Work

• a better result
• Harrelson, Hildrum, Rao 2003
• a competitive polynomial-time
  algorithm based on decomposition

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Hierarchical Decomposition
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Hierarchical Routing Scheme
?
if offline has congestion 1 we only send messages
along this virtual edge
t
s
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Questions

• Decomposition
• How is the partitioning done?
• Routing Scheme
• How are intermediate nodes chosen?
• How are routing paths between intermediate nodes
  chosen?

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Choosing Intermediate Nodes (1)

• Probability distribution for choosing blue
  intermediate node of cluster
• weight function is the bandwidth of
  edges connecting to nodes outside of

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Choosing Intermediate Nodes (2)

• Probability distribution for choosing white
  intermediate node of cluster , whose
  subclustering is
• weight function is the bandwidth of
  edges connecting to nodes in other
  sub-clusters

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Choosing Routing Paths (1)
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Choosing Routing Paths (2)
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Choosing Routing Paths (3)

• The decomposition has to fulfill
• tree height is
• throughput property for each cluster and its
  subclusters
• for CMCF-problem with demands
• the solution of the flow must satisfy each
  demand and produce a congestion of at most

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A Bad-Case Example

• No clustering of the following example can
  fulfill the throughput property

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Precondition

• A set fulfills the precondition iff for
  each
• such that

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Decomposition Theorem

• Theorem
• We can partition any cluster that fulfills
  the precondition into subclusters such that
  
• fulfills the throughput property
• for each we have
• each fulfills the precondition

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Illustration of the Algorithm

• We divide the cluster into single-node
  subclusters
• If there is no solution for the flow we find a
  witness for this fact
• we merge it
• .... and cut it to fulfill the
  precondition
• We can round an ugly set losing logarithmic
  factor

capacity of edges bet- ween different
clusters decreases
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Precondition Lemma

• Lemma
• A cluster can be partitioned into
  subclusters such that
• each fulfills the precondition
• A partitioning
• exists for
• can be computed in polynomial time for

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Future Work

• Is there a class of networks for which adaptive
  online algorithms are asymptotically better than
  oblivious algorithms?
• How low can we make the competitive ratio?
  ?
• How to repair the structure of the tree quickly
  if the graph changes?

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A Practical Algorithm for Constructing Oblivious
Routing Schemes

• Thank you for your attention

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Flows and Cuts

• Concurrent Multicommodity Flow and Sparsest Cut

General graphs
Existence
Planar graphs
CMCF
SparsestCut
Computed Cut
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CMCF, SparsestCut - definition

• Concurrent MultiCommodity Flow problem
• Deliver some fraction of each demand
• Respect the edges capacities
• To maximize the smallest delivered fraction of
  a demand
• If each demand satisfied with ratio then we
  can route with congestion

• Sparsity of a cut

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Fulfilling the precondition

• .
  
• amortize the new created capacity
  against edges in
• an edge is used at most times for
  amortization since