Online%20Oblivious%20Routing

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Online Oblivious Routing

• Nikhil Bansal, Avrim Blum,
• Shuchi Chawla Adam Meyerson
• Carnegie Mellon University
• 6/7/2003

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The Routing Problem
Have An underlying network Requests arrive in
succession Want A distributed routing
algorithm minimizing congestion Congestion
maxe (flow on edge e)
Congestion 2
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The Routing Problem
Have An underlying network Requests arrive in
succession Want A distributed routing
algorithm minimizing congestion Congestion
maxe (flow on edge e)
In hindsight
Congestion 1
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The Routing Problem
Have An underlying network Requests arrive in
succession Want A distributed routing
algorithm minimizing congestion Congestion
maxe (flow on edge e)
Congestion 2
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Oblivious vs. Adaptive Routing

• Adaptive Routing
• For every request, pick a route based on all
  previously seen requests
• log-competitive centralized distributed algos
• Oblivious Routing
• Pick a route for every possible request in the
  beginning, and use it throughout
• Advantages
• - Easy to implement (hard-code routes)
• - Distributed

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Oblivious Routing Algorithms

• Borodin Hopcroft 85 Deterministic algorithms
  perform very poorly
• Solution Randomized algorithms
• For every request, output a probability
  distribution on paths ? a flow

Congestion 1.5
Henceforth, A Routing ? A collection of unit
flows (one for each request)
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Oblivious Routing Algorithms

• Until recently, good randomized algorithms known
  only for special graphs
• Lattices, Hypercubes ValiantBrebner81,
  Leighton92
• Racke02 For every graph, 9 an oblivious
  routing with congestion less than O(log3n) x OPT
• Can we find it?
• Azar et al03 Polytime algo to find an
  oblivious routing that has minimum worst case
  congestion
• In particular, achieve Rackes bound
• Min-Max Optimal Routing

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The Min-Max Optimal Routing

• Given graph G, and D the set of demands with
  optimal (hindsight) congestion 1
• Min-Max Optimal Routing rG
• argminr maxD (congestion of r on d)

• Azar et al find this routing in poly-time
• Racke shows that in every graph, congestion of rG
  O(log3n)

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Can we do better?

• Suppose we do not get the worst case demands

• - Do not want to optimize over the entire set D
• - In hindsight if the possible set of demands is
  D, we want r argminr maxD (cong. of r on d)

• Formally
• Every day we get demands dt
• Want to minimize the objective åt (cong. of r on
  dt)

i.e., want congestion to always be low, but only
on the observed demands dt not the entire set
D but, allow it to be high very occasionally
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Online Oblivious Routing

• Observe and serve demands every day
• Each morning, pick an Oblivious Routing of the
  day based on previous demands
• Based on observed trends, adjust the routing
  to better suit the observed traffic

• Still oblivious we pick the routing every day
  without knowing the future demands
• Cost(r) åt (cong. of r on dt)
• Cost(ALG) åt (cong. of rt on dt)

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The Static Optimal Routing

• Optimal Routing for the given set of demands
• r argminr Cost(r)
• (Not allowed to change over time, unlike the
  algorithm)
• We want ALG to be almost as good as the Static
  Optimal Routing
• Cost(ALG) (1e) Cost(r) (small factors)
• Note Min-Max Opt may not be competitive!
• Static Opt is at least as good as the Min-Max
  Opt, but can be much better!

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An online learning problem

• Picking a good routing is similar to playing a
  repeated game against an adversary that supplies
  demands
• Similar to machine learning problem of competing
  against the best expert
• Too many experts (feasible routings) traditional
  ML algorithms do not work
• Use the structure of the problem to design an
  efficient algorithm LP based

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Routing and Linear Programming

• For given demands d, and routing r,
• congestion on edge e d.r(e)
• We want minr2R maxe d.r(e) or min t td.r(e)
  8 e
• R is a convex polyhedron r should satisfy
  inflowoutflow at every node (except source and
  sink)
• Therefore, given demands, the best flow is given
  by a linear program

• How about Min-Max Opt flow?
• min t t d.r(e) 8e,d
• Azar et al Ellipsoid with a separation oracle
  compute worst case demands at every step

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Online Oblivious Routing and Online LP

• Online Linear Programming
• At every step, pick a point xt from convex set F
• Receive linear cost function gt
• cost at step t gt(xt)
• Online Oblivious Routing
• At every step, pick a point rt from R
• Receive demands dt
• cost at step t maxe dt.rt(e)

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Online Oblivious Routing and Online LP

• Online Linear Programming
• At every step, pick a point xt from convex set F
• Receive linear cost function gt
• cost at step t gt(xt)
• Online Oblivious Routing
• At every step, pick a point rt from R
• Receive demands dt and edge et
• cost at step t dt.rt(et)

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Online Oblivious Routing and Online LP

• Knowing the edge et only hurts us
• OPTs flow on edge et is smaller than its
  congestion
• ALGs flow on edge et is equal to its congestion
• OPTs cost decreases, while ALGs stays the same

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Solving the Online Linear Program

• Based on the work of Zinkevich03 and
  Kalai-Vempala02
• At every step
• Gradient Descent
• Move against the cost vector gradient yt1
  xt - h?ct
• (Natural Learning Strategy)
• Projection
• If yt1 is infeasible, orthogonally project it
  back to R xt1 argminx2R yt1-x
• (We use Semi-Definite Programming)

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Solving the Online Linear Program

• We get the following guarantee
• Cost(ALG) Cost(OPT) n3?T
• After n6/e2 steps,
• Cost(ALG) (1e) Cost(OPT)

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Extensions and Open Problems

• Change the routing every D days
• the rate of convergence slows down by factor of
  D
• Add extra constraints to the LP
• No individual days cost should exceed some
  prespecified value
• No individual days cost should cross twice the
  Min-Max Opt cost
• Avoid using the SDP Open!
• Purely combinatorial greedy algorithm

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• Questions?