Optimal Oblivious Routing in Polynomial Time

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Optimal Oblivious Routing in Polynomial Time
Yossi Azar Amos Fiat Haim Kaplan Tel-Aviv
University

• Harald Räcke
• Paderborn

Edith Cohen ATT Labs-Research
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Routing, Demands, Flow, Congestion

• Routing a unit s-t flow for each
  origin-destination pair
• fab(i,j) gt 0 routing for OD pair a,b on
  edge (i,j)
• Demands Dab gt 0 for each OD pair a,b
• Flow on edge e(i,j) when routing D with f
• flow(e,f,D)Sab fab(i,j) Dab
• Congestion on edge e(i,j) when routing D with f
  cong(e,f,D)flow(e,f,d)/capacity(e)

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Congestion, Oblivious Routing

• Congestion of demands D with routing f
  cong(f,D) maxe cong(e,f,D)
• Optimal routing for D min possible congestion
  opt(D) minf cong(f,D)
• Oblivious ratio of f
• obliv(f) maxD cong(f,D)/opt(D)
• Optimal Oblivious Ratio of G
• obliv-opt(G)minf obliv(f)

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Example
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1 2 3 4
1 1 1 1 1
2 1 1 1 1
3 1 1 1 1
4 1 1 1 1
1
3
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Routing f Route each OD pair on direct
edge Demands D unit demand for all
pairs cong(e,f,D)2 for all edges Thus,
cong(f,D)2 (f is optimal for D)
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Example
1 2 3 4
1 0 0 1 0
2 0 0 0 0
3 0 0 0 0
4 0 0 0 0
Routing f Route each OD pair on direct
edge Demands D unit demand for ONE
pair cong(e,f,D)1 for used edge, 0
otherwise. Thus, cong(f,D)1 (f is
NOT optimal for D)
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Example
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1 2 3 4
1 0 0 1 0
2 0 0 0 0
3 0 0 0 0
4 0 0 0 0
1
3
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Routing f Route each OD pair on the 3 1,2 hop
paths Demands D unit demand for one
pair cong(e,f,D)1/3 for used edges cong(f,D)1/3
direct routing has oblivious ratio gt 3
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Example
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1 2 3 4
1 1 1 1 1
2 1 1 1 1
3 1 1 1 1
4 1 1 1 1
1
3
4
Routing f Route each OD pair on the 3 1,2 hop
paths Demands D unit demand for all
pairs cong(e,f,D)10/3 for all edges (10 pairs
use each edge) cong(f,D)10/3 (f is NOT optimal
for D) 2-hop routing has oblivious ratio gt 5/3
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Optimal oblivious routing

• Balances performance across all demand matrices.
• Why is it interesting?
• Demands are dynamic
• Changes to routing are hard
• Sometimes we dont know the demands

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History

• Specific networks, VC routing
• Raghavan/Thompson 87Aspnes et al 93
• Valiant/Brebner 81 Hypercubes
• Räcke 02
• Any undirected network has an oblivious
  routing with ratio O(log3 n)!!
• Questions
• Poly time algorithm.
• Get an optimal routing.
• Directed networks?

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LP for Optimal Oblivious Ratio

• Minimize r s.t.
• fab(i,j) is a routing (1-flow for every a,b)
• For all demands Dab gt 0 which can be routed
  with congestion 1
• For all edges e(i,j) (cong(e,f,D) lt r)
• Sab fab(i,j) Dab/capacity(e) lt r

But Infinite number of constraints ? use
Ellipsoid
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Separation Oracle

• Given a routing fab(i,j), find its oblivious
  ratio and a demand matrix D which maximizes the
  ratio (the worst demands for f).
• For each edge e(i,j) solve the LP (and then
  take the maximum over these LPs)

• Maximize Sab fab(i,j) Dab/capacity(e)
• gab(i,j) is a flow of demand Dab gt 0
• For all edges h, S gab(h) lt capacity(h)

Need to insure that the numbers dont grow too
much
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Directed Networks (Asymmetric link capacities)

• Our algorithm computes optimal oblivious routing
  for undirected and directed networks.
• Räckes O(log3 n) bound applies only to
  undirected networks.
• We show that some directed networks have optimal
  oblivious ratio of W(sqrt(n)).

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Extensions

• Subset of OD pair demands
• Ranges of demands
• Node congestion
• Limiting dilation

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Follow up/subsequent work

• Polytime construction of a Räcke-like
  decomposition
• (two SPAA 03 papers Harrelson/Hildrum/Rao
  Bienkowski/Korzeniowski/Räcke)
• More efficient polynomial time algorithm
  (Applegate/Cohen SIGCOMM 03)
• Oblivious routing on ISP topologies
• (Applegate/Cohen SIGCOMM 03)
• Online oblivious routing
• (Bansal/Blum/Chawla/Meyerson SPAA 03)

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Open Problems

• Tighten Räckes bound
• O(log3 n) ?? W(log n)
• (Currently, O(log2 n log log n) by
• Harrelson/Hildrum/Rao 03)
• Single source demands
• Is there a constant optimal oblivious ratio ?