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Traffic Engineering with Equal-Cost-MultiPath: An Algorithmic Perspective Marco Chiesa s joint work with        Guy Kindler               Michael Schapira Israel Networking Day 2014                   April 24th                              Infocom 2014

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Page 1: th Infocom 2014 Traffic Engineering with EqualCost ...€¦ · Traffic Engineering with EqualCostMultiPath: An Algorithmic Perspective Marco Chiesa s joint work with Guy Kindler Michael

Traffic Engineering withEqual­Cost­MultiPath:

An Algorithmic Perspective

Marco Chiesa s

joint work with

       Guy Kindler                Michael Schapira

Israel Networking Day 2014                   April 24th 

                             Infocom 2014

Page 2: th Infocom 2014 Traffic Engineering with EqualCost ...€¦ · Traffic Engineering with EqualCostMultiPath: An Algorithmic Perspective Marco Chiesa s joint work with Guy Kindler Michael

traffic­engineering (TE)

network operators' goal: ● provide best possible service● minimize costs

 how?● fully exploit network resources

  → route flows of traffic along the “best” paths

Page 3: th Infocom 2014 Traffic Engineering with EqualCost ...€¦ · Traffic Engineering with EqualCostMultiPath: An Algorithmic Perspective Marco Chiesa s joint work with Guy Kindler Michael

traditional TE tools

ECMP (Equal­Cost­MultiPath)● the most widely deployed TE mechanism● load­balancing tool● very simple mechanism

Page 4: th Infocom 2014 Traffic Engineering with EqualCost ...€¦ · Traffic Engineering with EqualCostMultiPath: An Algorithmic Perspective Marco Chiesa s joint work with Guy Kindler Michael

contributions

ECMP and arbitrary topologies       no reasonable approximation is possible→

ECMP and datacenter topologies:● hypercubes vs folded Clos networks ● large flows in folded Clos networks 

:(

:):)

Page 5: th Infocom 2014 Traffic Engineering with EqualCost ...€¦ · Traffic Engineering with EqualCostMultiPath: An Algorithmic Perspective Marco Chiesa s joint work with Guy Kindler Michael

traditional TE tools

ECMP (Equal­Cost­MultiPath)● operators set link weights● traffic is routed along shortest­paths

   

s

t

Page 6: th Infocom 2014 Traffic Engineering with EqualCost ...€¦ · Traffic Engineering with EqualCostMultiPath: An Algorithmic Perspective Marco Chiesa s joint work with Guy Kindler Michael

traditional TE tools

ECMP (Equal­Cost­MultiPath)● operators set link weights● traffic is routed along shortest­paths

   

s3

   3

2

199

t

Page 7: th Infocom 2014 Traffic Engineering with EqualCost ...€¦ · Traffic Engineering with EqualCostMultiPath: An Algorithmic Perspective Marco Chiesa s joint work with Guy Kindler Michael

traditional TE tools

when multiple shortest paths are available:● per­packet level equal split● per­flow level hash­based split

 → equal split for many small flows

3

199

   3

2

s

t

4

22

2

fst=4

Page 8: th Infocom 2014 Traffic Engineering with EqualCost ...€¦ · Traffic Engineering with EqualCostMultiPath: An Algorithmic Perspective Marco Chiesa s joint work with Guy Kindler Michael

traditional TE tools

when multiple shortest paths are available:● per­packet level equal split● per­flow level hash­based split

 → equal split for many small flows 

3

99 4

   3

2

s

t

2

11

3

2

fst=4

1

Page 9: th Infocom 2014 Traffic Engineering with EqualCost ...€¦ · Traffic Engineering with EqualCostMultiPath: An Algorithmic Perspective Marco Chiesa s joint work with Guy Kindler Michael

traditional TE tools

when multiple shortest paths are available:● per­packet level equal split● per­flow level hash­based split

 → equal split for many small flows 

3

1 99

   3

2

s

t

2

2

2

2

fst=4

99 4

Page 10: th Infocom 2014 Traffic Engineering with EqualCost ...€¦ · Traffic Engineering with EqualCostMultiPath: An Algorithmic Perspective Marco Chiesa s joint work with Guy Kindler Michael

traditional TE tools

OSPF + ECMP (Equal­Cost­MultiPath)● operators set link weights

how? heuristic approaches: ● local search [Fortz,Thorup, 2000][Sundaresan et al, 2010]

● memetic algorithms [Buriol et al, 2002]

● genetic algorithms [Ericsson et al, 2002]

● branch­and­cut for mixed­ILP [Palmar et al, 2006] 

wanted: algorithm with provable guarantees?

Page 11: th Infocom 2014 Traffic Engineering with EqualCost ...€¦ · Traffic Engineering with EqualCostMultiPath: An Algorithmic Perspective Marco Chiesa s joint work with Guy Kindler Michael

TE model:multi commidity flow

input:●  capacitated graph G=(V,E)●  demand matrix D={dij}

constraints:● flows cannot exceed links capacities ● flows are equally split among all shortest­paths

 optimization functions:● maximize total throughput (flow)● minimize congestion● minimize sum of link­costs

Page 12: th Infocom 2014 Traffic Engineering with EqualCost ...€¦ · Traffic Engineering with EqualCostMultiPath: An Algorithmic Perspective Marco Chiesa s joint work with Guy Kindler Michael

known results:inapproximability for max­flow

Theorem [FortzThorup,2000]. Given an instance I=(G,D), it is NP­hard to distinguish whether: 

OPT(I) = 1 or OPT(I) = 2/3 k,

for any k>0

           → no efficient algorithm can provably route              at least a fraction         of OPT(I)

23

23+e

Page 13: th Infocom 2014 Traffic Engineering with EqualCost ...€¦ · Traffic Engineering with EqualCostMultiPath: An Algorithmic Perspective Marco Chiesa s joint work with Guy Kindler Michael

our first results:inapproximability for max­flow

Theorem [FortzThorup,2000]. Given an instance I=(G,D), with a single entry in D, it is NP­hard to distinguish whether:

OPT(I) = 1 or OPT(I) = 2/ k,

for any k>0

23

Page 14: th Infocom 2014 Traffic Engineering with EqualCost ...€¦ · Traffic Engineering with EqualCostMultiPath: An Algorithmic Perspective Marco Chiesa s joint work with Guy Kindler Michael

our first results:no inapproximability 

for max­flowTheorem [FortzThorup,2000]. Given an instance I=(G,D), with a single entry in D, it is NP­hard to distinguish whether: 

OPT(I) = 1 or OPT(I) = 2/3q,

for any q>0

 → no algorithm can provably route at least a fraction k of OPT(G) in polynomial time

23

Page 15: th Infocom 2014 Traffic Engineering with EqualCost ...€¦ · Traffic Engineering with EqualCostMultiPath: An Algorithmic Perspective Marco Chiesa s joint work with Guy Kindler Michael

Theorem [FortzThorup,2000]. Given an instance I=(G,D), with a single entry in D, it is NP­hard to distinguish whether: 

OPT(I) = 1 or OPT(I) = 2/3q,

for any q>0

           → no efficient algorithm can provably route              at least a fraction q of OPT(I)

our first results:any constant inapproximability 

for max­flow

23

Page 16: th Infocom 2014 Traffic Engineering with EqualCost ...€¦ · Traffic Engineering with EqualCostMultiPath: An Algorithmic Perspective Marco Chiesa s joint work with Guy Kindler Michael

key tool:amplification operator X

operator X: instance I   → instance Inew

  such that

OPT(Inew) = OPT(I) 2

OPTA(A(G)) = (OPTG)4

...

Page 17: th Infocom 2014 Traffic Engineering with EqualCost ...€¦ · Traffic Engineering with EqualCostMultiPath: An Algorithmic Perspective Marco Chiesa s joint work with Guy Kindler Michael

amplifying the gap

OPT(I) = 1              or             OPT(I) =  it is NP­hard to distinguish between 1 and ~0.6

OPT(X(G)) = 1        or        OPT(X(H)) =  it is NP­hard to distinguish between 1 and ~0.4

OPT(X2(G)) = 1       or      OPT(X2(H)) = it is NP­hard to distinguish between 1 and ~0.2 

… 

23

Page 18: th Infocom 2014 Traffic Engineering with EqualCost ...€¦ · Traffic Engineering with EqualCostMultiPath: An Algorithmic Perspective Marco Chiesa s joint work with Guy Kindler Michael

amplifying the gap

OPT(I) = 1              or             OPT(I) =  it is NP­hard to distinguish between 1 and ~0.6

OPT(X(I)) = 1        or        OPT(X(I)) =  it is NP­hard to distinguish between 1 and ~0.4

OPT(X2(G)) = 1       or      OPT(X2(H)) = it is NP­hard to distinguish between 1 and ~0.2 

… 

49

23

Page 19: th Infocom 2014 Traffic Engineering with EqualCost ...€¦ · Traffic Engineering with EqualCostMultiPath: An Algorithmic Perspective Marco Chiesa s joint work with Guy Kindler Michael

amplifying the gap

OPT(I) = 1              or             OPT(I) =  it is NP­hard to distinguish between 1 and ~0.6

OPT(X(I)) = 1        or        OPT(X(I)) =  it is NP­hard to distinguish between 1 and ~0.4

OPT(X2(I)) = 1       or      OPT(X2(I)) = it is NP­hard to distinguish between 1 and ~0.2 

… 

23

49

1681

Page 20: th Infocom 2014 Traffic Engineering with EqualCost ...€¦ · Traffic Engineering with EqualCostMultiPath: An Algorithmic Perspective Marco Chiesa s joint work with Guy Kindler Michael

amplification gap technique:graph G

kc1

kc3

kc2

kc4

kc5

s

t

● source s● target t● capacitated edges

Page 21: th Infocom 2014 Traffic Engineering with EqualCost ...€¦ · Traffic Engineering with EqualCostMultiPath: An Algorithmic Perspective Marco Chiesa s joint work with Guy Kindler Michael

amplification gap technique:graph G

kc1

kc3

kc2

kc4

kc5

s

t

● source s● target t● capacitated edges

t

max­flow (s   t) = → OPT

Page 22: th Infocom 2014 Traffic Engineering with EqualCost ...€¦ · Traffic Engineering with EqualCostMultiPath: An Algorithmic Perspective Marco Chiesa s joint work with Guy Kindler Michael

  amplification gap technique:            recursive replacement

kc1

kc3

kc2

kc4

kc5

s

t

c2

c2c2c2c1

c2c3

c2c5c2c4

s'

t'

Page 23: th Infocom 2014 Traffic Engineering with EqualCost ...€¦ · Traffic Engineering with EqualCostMultiPath: An Algorithmic Perspective Marco Chiesa s joint work with Guy Kindler Michael

  amplification gap technique:            recursive replacement

kc1

kc3

kc2

kc4

kc5

s

t

c2

c2c2c2c1

c2c3

c2c5c2c4

s'

t'

max­flow(s'   → t') = c2OPT

Page 24: th Infocom 2014 Traffic Engineering with EqualCost ...€¦ · Traffic Engineering with EqualCostMultiPath: An Algorithmic Perspective Marco Chiesa s joint work with Guy Kindler Michael

  amplification gap technique:            recursive replacement

kc1

kc3

kc2

kc4

kc5

s

t

c2

c2c2c2c1

c2c3

c2c5c2c4

s'

t'

max­flow(s'   → t') = c2OPT

Page 25: th Infocom 2014 Traffic Engineering with EqualCost ...€¦ · Traffic Engineering with EqualCostMultiPath: An Algorithmic Perspective Marco Chiesa s joint work with Guy Kindler Michael

  amplification gap technique:            recursive replacement

c2O

PT

s

t

c2

c2c2c2c1

c2c3

c2c5c2c4

s'

t'

kc1

kc3

kc4

kc5

max­flow(s'   → t') = c2OPT

Page 26: th Infocom 2014 Traffic Engineering with EqualCost ...€¦ · Traffic Engineering with EqualCostMultiPath: An Algorithmic Perspective Marco Chiesa s joint work with Guy Kindler Michael

  amplification gap technique:            recursive replacement

c1OPT

c3OPT

c2O

PT

c4O

PT

c5OPT

s

t

c2

c2c2c2c1

c2c3

c2c5c2c4

s'

t'

max­flow(s'   → t') = c2OPT

Page 27: th Infocom 2014 Traffic Engineering with EqualCost ...€¦ · Traffic Engineering with EqualCostMultiPath: An Algorithmic Perspective Marco Chiesa s joint work with Guy Kindler Michael

amplification gap technique:graph X(G)

c1OPT

c3OPT

c2O

PT

c4O

PT

c5OPT

s

t

max­flow(s   t,G') = →OPT   ...⋅

Page 28: th Infocom 2014 Traffic Engineering with EqualCost ...€¦ · Traffic Engineering with EqualCostMultiPath: An Algorithmic Perspective Marco Chiesa s joint work with Guy Kindler Michael

amplification gap technique:graph X(G)

c1OPT

c3OPT

c2O

PT

c4O

PT

c5OPT

s

t

max­flow(s   t,G') = →OPT   ⋅ OPT = OPT 2

Page 29: th Infocom 2014 Traffic Engineering with EqualCost ...€¦ · Traffic Engineering with EqualCostMultiPath: An Algorithmic Perspective Marco Chiesa s joint work with Guy Kindler Michael

data­center topologies and ECMP

topology constraints:●  d­dimensional hypercubes (e.g., bCube­like)

 → NP­hard●  l­layers folded Clos networks (e.g., VL2­like)

 → easy●  random regular graphs (e.g., Jellyfish)

 → future work

Page 30: th Infocom 2014 Traffic Engineering with EqualCost ...€¦ · Traffic Engineering with EqualCostMultiPath: An Algorithmic Perspective Marco Chiesa s joint work with Guy Kindler Michael

data­center topologies and ECMP

topology constraints:●  d­dimensional hypercubes (e.g., bCube) Clos networks (e.g., VL2) network

d=4

Page 31: th Infocom 2014 Traffic Engineering with EqualCost ...€¦ · Traffic Engineering with EqualCostMultiPath: An Algorithmic Perspective Marco Chiesa s joint work with Guy Kindler Michael

data­center topologies and ECMP

topology constraints:●  d­dimensional hypercubes (e.g., bCube) Clos networks (e.g., VL2) network

d=4

computing the best weight assignment is computationally intractable

Page 32: th Infocom 2014 Traffic Engineering with EqualCost ...€¦ · Traffic Engineering with EqualCostMultiPath: An Algorithmic Perspective Marco Chiesa s joint work with Guy Kindler Michael

data­center topologies and ECMP

topology constraints:●  l­layers folded Clos networks (e.g.,VL2) 

l=2

servers

l=2

Page 33: th Infocom 2014 Traffic Engineering with EqualCost ...€¦ · Traffic Engineering with EqualCostMultiPath: An Algorithmic Perspective Marco Chiesa s joint work with Guy Kindler Michael

data­center topologies and ECMP

topology constraints:●  l­layers folded Clos networks (e.g.,VL2) 

l=3

servers

Page 34: th Infocom 2014 Traffic Engineering with EqualCost ...€¦ · Traffic Engineering with EqualCostMultiPath: An Algorithmic Perspective Marco Chiesa s joint work with Guy Kindler Michael

data­center topologies and ECMP

topology constraints:●  l­layers folded Clos networks (e.g.,VL2) 

l=3

servers

Page 35: th Infocom 2014 Traffic Engineering with EqualCost ...€¦ · Traffic Engineering with EqualCostMultiPath: An Algorithmic Perspective Marco Chiesa s joint work with Guy Kindler Michael

data­center topologies and ECMP

topology constraints:●  l­layers folded Clos networks (e.g.,VL2) 

l=3

servers

Page 36: th Infocom 2014 Traffic Engineering with EqualCost ...€¦ · Traffic Engineering with EqualCostMultiPath: An Algorithmic Perspective Marco Chiesa s joint work with Guy Kindler Michael

data­center topologies and ECMP

topology constraints:●  l­layers folded Clos networks (e.g.,VL2) 

l=3

servers

Page 37: th Infocom 2014 Traffic Engineering with EqualCost ...€¦ · Traffic Engineering with EqualCostMultiPath: An Algorithmic Perspective Marco Chiesa s joint work with Guy Kindler Michael

data­center topologies and ECMP

topology constraints:●  l­layers folded Clos networks (e.g.,VL2) 

l=4

servers

Page 38: th Infocom 2014 Traffic Engineering with EqualCost ...€¦ · Traffic Engineering with EqualCostMultiPath: An Algorithmic Perspective Marco Chiesa s joint work with Guy Kindler Michael

data­center topologies and ECMP

topology constraints:●  l­layers folded Clos networks (e.g.,VL2) 

l=4

servers

Page 39: th Infocom 2014 Traffic Engineering with EqualCost ...€¦ · Traffic Engineering with EqualCostMultiPath: An Algorithmic Perspective Marco Chiesa s joint work with Guy Kindler Michael

data­center topologies and ECMP

topology constraints:●  l­layers folded Clos networks (e.g.,VL2) 

l=4

servers

Page 40: th Infocom 2014 Traffic Engineering with EqualCost ...€¦ · Traffic Engineering with EqualCostMultiPath: An Algorithmic Perspective Marco Chiesa s joint work with Guy Kindler Michael

data­center topologies and ECMP

topology constraints:●  l­layers folded Clos networks (e.g.,VL2) 

l=4

servers

Page 41: th Infocom 2014 Traffic Engineering with EqualCost ...€¦ · Traffic Engineering with EqualCostMultiPath: An Algorithmic Perspective Marco Chiesa s joint work with Guy Kindler Michael

data­center topologies and ECMP

topology constraints:●  l­layers folded Clos networks (e.g.,VL2) 

l=4

servers

Page 42: th Infocom 2014 Traffic Engineering with EqualCost ...€¦ · Traffic Engineering with EqualCostMultiPath: An Algorithmic Perspective Marco Chiesa s joint work with Guy Kindler Michael

data­center topologies and ECMP

topology constraints:●  l­layers folded Clos networks (e.g.,VL2) 

l=4

setting all link weights to 1 is optimal

Page 43: th Infocom 2014 Traffic Engineering with EqualCost ...€¦ · Traffic Engineering with EqualCostMultiPath: An Algorithmic Perspective Marco Chiesa s joint work with Guy Kindler Michael

optimality proof sketch

topology constraints:●  l­layers folded Clos networks (e.g.,VL2) 

l=2

servers

Page 44: th Infocom 2014 Traffic Engineering with EqualCost ...€¦ · Traffic Engineering with EqualCostMultiPath: An Algorithmic Perspective Marco Chiesa s joint work with Guy Kindler Michael

topology constraints:●  l­layers folded Clos networks (e.g.,VL2) 

l=4

optimality proof sketch

servers

Page 45: th Infocom 2014 Traffic Engineering with EqualCost ...€¦ · Traffic Engineering with EqualCostMultiPath: An Algorithmic Perspective Marco Chiesa s joint work with Guy Kindler Michael

topology constraints:●  l­layers folded Clos networks (e.g.,VL2) 

l=4

optimality proof sketch

servers

Page 46: th Infocom 2014 Traffic Engineering with EqualCost ...€¦ · Traffic Engineering with EqualCostMultiPath: An Algorithmic Perspective Marco Chiesa s joint work with Guy Kindler Michael

topology constraints:●  l­layers folded Clos networks (e.g.,VL2) 

l=4

optimality proof sketch

servers

Page 47: th Infocom 2014 Traffic Engineering with EqualCost ...€¦ · Traffic Engineering with EqualCostMultiPath: An Algorithmic Perspective Marco Chiesa s joint work with Guy Kindler Michael

topology constraints:●  l­layers folded Clos networks (e.g.,VL2) 

l=4

optimality proof sketch

servers

Page 48: th Infocom 2014 Traffic Engineering with EqualCost ...€¦ · Traffic Engineering with EqualCostMultiPath: An Algorithmic Perspective Marco Chiesa s joint work with Guy Kindler Michael

ECMP and large flowsper­flow level hash­based split

 → when there are a few large flows, traffic may not be properly load­balanced

l=4

servers

Page 49: th Infocom 2014 Traffic Engineering with EqualCost ...€¦ · Traffic Engineering with EqualCostMultiPath: An Algorithmic Perspective Marco Chiesa s joint work with Guy Kindler Michael

ECMP and large flowsper­flow level hash­based split

 → when there are a few large flows, traffic may not be properly load­balanced

l=4

servers

Page 50: th Infocom 2014 Traffic Engineering with EqualCost ...€¦ · Traffic Engineering with EqualCostMultiPath: An Algorithmic Perspective Marco Chiesa s joint work with Guy Kindler Michael

ECMP and large flowsper­flow level hash­based split

 → when there are a few large flows, traffic may not be properly load­balanced

l=4

servers

Page 51: th Infocom 2014 Traffic Engineering with EqualCost ...€¦ · Traffic Engineering with EqualCostMultiPath: An Algorithmic Perspective Marco Chiesa s joint work with Guy Kindler Michael

ECMP and large flowsper­flow level hash­based split

 → when there are a few large flows, traffic may not be properly load­balanced

l=4

severe performance degradation [Al Fares et al, 2010]:30% of the bandwidth in a datacenter is wasted

Page 52: th Infocom 2014 Traffic Engineering with EqualCost ...€¦ · Traffic Engineering with EqualCostMultiPath: An Algorithmic Perspective Marco Chiesa s joint work with Guy Kindler Michael

ECMP, Clos networks, and large flows

proposed solution [Al­Fares et al, 2010]:● route small flows (mice) using ECMP● route large flows (elephant) using a greedy 

algorithm 

our results:● 2 ­inapproximability● greedy is a    ­approximation algorithm

●   ­approximation if all flows have equal size

12

15

14

Page 53: th Infocom 2014 Traffic Engineering with EqualCost ...€¦ · Traffic Engineering with EqualCostMultiPath: An Algorithmic Perspective Marco Chiesa s joint work with Guy Kindler Michael

conclusions

● in general, no efficient algorithm exists to assign the best link weights

datacenter topologies:

hypercubes   still hard to find the best weights→

folded Clos networks   set all weights to 1→

greedy algorithm for routing large flows is a     ­approximation in a 3­layers Folded Clos network (VL2­like)

Page 54: th Infocom 2014 Traffic Engineering with EqualCost ...€¦ · Traffic Engineering with EqualCostMultiPath: An Algorithmic Perspective Marco Chiesa s joint work with Guy Kindler Michael

conclusions

● in general, no efficient algorithm exists to assign the best reasonable link weights

datacenter topologies:

hypercubes   hard to find the best weights→

folded Clos networks   set all weights to 1→

greedy algorithm for routing large flows is a     ­approximation in a 3­layers Folded Clos network (VL2­like)

Page 55: th Infocom 2014 Traffic Engineering with EqualCost ...€¦ · Traffic Engineering with EqualCostMultiPath: An Algorithmic Perspective Marco Chiesa s joint work with Guy Kindler Michael

conclusions

● in general, no efficient algorithm exists to assign the best reasonable link weights

● datacenter topologies:● hypercubes   hard to find the best weights→● folded Clos networks   set all weights to 1→

– greedy algorithm for routing large flows is a     ­approximation in a 3­layers Folded Clos network (VL2­like)

15

Page 56: th Infocom 2014 Traffic Engineering with EqualCost ...€¦ · Traffic Engineering with EqualCostMultiPath: An Algorithmic Perspective Marco Chiesa s joint work with Guy Kindler Michael

thank you