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Load-Optimization in Reconfigurable Networks: Algorithms and - - PowerPoint PPT Presentation

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Load-Optimization in Reconfigurable Networks: Algorithms and Complexity of Flow Routing Wenkai Dai, Klaus-T. Foerster, David Fuchssteiner, Stefan Schmid (CT Group, University of Vienna) Motivation: Interconnecting Top of Rack in Datacenter

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Load-Optimization in Reconfigurable Networks: Algorithms and Complexity of Flow Routing

Wenkai Dai, Klaus-T. Foerster, David Fuchssteiner, Stefan Schmid (CT Group, University of Vienna)

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Motivation: Interconnecting Top of Rack in Datacenter

06.10.2020 Load-Optimization in Reconfigurable Networks: Algorithms and Complexity of Flow Routing (Performance 2020)

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Fat-Tree (Clos) Topology for Data Centers

  • Fat-Tree is good for all-to-all traffic

06.10.2020 Load-Optimization in Reconfigurable Networks: Algorithms and Complexity of Flow Routing (Performance 2020)

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  • However, DCN traffic is often not all-to-all

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Data Center Traffic ≠ Uniform

Traffic demands (normalized) between ToR switches. Halperin et al., SIGCOMM’11 Heatmap of rack to rack traffic. Color intensity is log-scale and normalized. Ghobadi et al., SIGCOMM’16

06.10.2020 Load-Optimization in Reconfigurable Networks: Algorithms and Complexity of Flow Routing (Performance 2020) “Data reveal that 46-99% of the rack pairs exchange no traffic at all”

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Circuit Switches vs Packet Switches

1. Circuit Switches: usually optical

  • Fast (high bandwidth)
  • Connection between ports can be adjusted dynamically

2. Packet Switches: usually electronic

  • Low bandwidth
  • The connections of links are fixed after deployment

https://www.laserfocusworld.com/optics/article/16556781/ma ny-approaches-taken-for-alloptical-switching (Hecht, 2001)

06.10.2020 Load-Optimization in Reconfigurable Networks: Algorithms and Complexity of Flow Routing (Performance 2020)

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  • Idea: implement “physical” connections

–

Difference: Not all-to-all switch

  • E.g. just 1 connection per node
  • A matching is selected to connect nodes

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Understand Circuit Switches Physical layer: It‘s a Match(ing)!

A C B D

Reconfigurable Switch

A C B D A C B D A C B D

(a) (b) (c)

06.10.2020 Load-Optimization in Reconfigurable Networks: Algorithms and Complexity of Flow Routing (Performance 2020)

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Hybrid Architecture for Datacenter (Helios, Farrington et al., SIGCOMM ‘10)

Helios

  • Adjust the topology dynamically for variant demands:
  • Elephant (big) flows → Circuit Switches
  • Mice (small) flows → Packet Switches

06.10.2020 Load-Optimization in Reconfigurable Networks: Algorithms and Complexity of Flow Routing (Performance 2020)

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Reconfigurable Data Center Networks (DCNs)

ProjecToR interconnect Ghobadi et al., SIGCOMM ‘16 Helios (core) Farrington et al., SIGCOMM ‘10 c-Through (HyPaC architecture) Wang et al., SIGCOMM ‘10 Rotornet (rotor switches) Mellette et al., SIGCOMM ‘17 Solstice (architecture & scheduling) Liu et al., CoNEXT ‘15 REACToR Liu et al., NSDI ‘15 … and many more … FireFly Hamedazimi et al., SIGCOMM ‘14

06.10.2020 Load-Optimization in Reconfigurable Networks: Algorithms and Complexity of Flow Routing (Performance 2020)

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Routing Models: Unsplittable vs Splittable

  • For each demand, e.g.,

A C E B D 10

Unsplittable

A C E B D

Unsplittable

A C E B D 4.5

Splittable

5.5 10

A→E: 10

06.10.2020 Load-Optimization in Reconfigurable Networks: Algorithms and Complexity of Flow Routing (Performance 2020)

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  • In a reconfigurable datacenter, for each demand:

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Routing Models: Segregated vs Nonsegregated

Segregated

A C E G B D F

Circuit Switch

A C E G B D F

Segregated

Circuit Switch

E.g., demand: A→E

A C E G B D F

Nonsegregated

Circuit Switch

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Four Routing Models in Reconfigurable Networks

Routing Models Segregation Model Nonsegregation Model Splittable Model

SS SN

Unsplittable Model

US UN

06.10.2020 Load-Optimization in Reconfigurable Networks: Algorithms and Complexity of Flow Routing (Performance 2020)

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Load-Optimization Reconfiguration Problem (Our Problem)

  • Given:

Demands Matrix 𝐸 A routing model 𝜐 ∈ {𝑇𝑇, 𝑇𝑂, 𝑉𝑇, 𝑉𝑂}

06.10.2020 Load-Optimization in Reconfigurable Networks: Algorithms and Complexity of Flow Routing (Performance 2020)

Static Network N = 𝑊, 𝐹, 𝐷 Circuit Switches

From: Al-Fares et al. 2008 From: calient.net

Set of reconfigurable links ℇ

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Load-Optimization Reconfiguration Problem (Our Problem)

  • Compute:
  • Objective: minimize the maximum link load in the hybrid network 𝑊, 𝐹ڂ 𝑁 , 𝐷

a matching from reconfigurable links;

06.10.2020 Load-Optimization in Reconfigurable Networks: Algorithms and Complexity of Flow Routing (Performance 2020)

and optimal routing schemes for demands Optimal routing schemes for demands in the hybrid network 𝑊, 𝐹ڂ 𝑁 , 𝐷

From: cisco.com

A routing model 𝜐 ∈ {𝑇𝑇, 𝑇𝑂, 𝑉𝑇, 𝑉𝑂}

+

Static Network N = 𝑊, 𝐹, 𝐷 Circuit Switches

From: Al-Fares et al. 2008 From: calient.net

Set of reconfigurable links ℇ Matching 𝑁 ⊂ ℇ

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An Example For Load-Optimization Reconfiguration Problem

A E D C B

06.10.2020 Load-Optimization in Reconfigurable Networks: Algorithms and Complexity of Flow Routing (Performance 2020)

Reconfigurable network

From: Al-Fares et al. 2008

Reconfigurable links

A E B D

C D B E A

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Example: Loads Depend on Reconfigurations

A E D C B

20 20 6 6

Maximum load 20

  • Consider demands D: A→B: 8, A→C: 6, C→B: 6, D→B: 6, A→E: 6
  • Goal: determine a matching in reconfigurable links to minimize the maximum load

A E D C B Compute flows for demands without reconfigurable links.

06.10.2020 Load-Optimization in Reconfigurable Networks: Algorithms and Complexity of Flow Routing (Performance 2020)

Reconfigurable network

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Example: Determine Matching by Greedy

  • Demands D: A→B: 8, A→C: 6, C→B: 6, D→B: 6, A→E: 6
  • Greedy chooses 𝐵, 𝐶 to serve A→B, then the matching is 𝐵, 𝐶 and 𝐸, 𝐹

A E D C B

20 20 6 6

Maximum load 20

Links: 𝐵, 𝐶 , 𝐸, 𝐹 configured

12 12 6 6

Greedy -> maximum load 12

A E D C B

8

06.10.2020 Load-Optimization in Reconfigurable Networks: Algorithms and Complexity of Flow Routing (Performance 2020)

A→B: 8

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Example: Optimal Matching

  • Demands D: A→B: 8, A→C: 6, C→B: 6, D→B: 6, A→E: 6
  • The optimal matching is 𝐸, 𝐶 and 𝐵, 𝐹

Optimal -> maximum load 10 A E D C B

20 20 6 6

Maximum load 20 Links: 𝐵, 𝐹 , 𝐸, 𝐶 configured A E D C B

10 10 4 4

06.10.2020 Load-Optimization in Reconfigurable Networks: Algorithms and Complexity of Flow Routing (Performance 2020)

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Complexity for Simple Trees

  • If the given static network is a tree with a height >=2, then
  • Reduction from 3-Partition problem
  • Especially, UN model is weakly NP-hard for star networks
  • Reduction from 2-Partition problem
  • Not hard anymore for small demands

Time Complexity Segregation Model Nonsegregation Model Splittable Model SS is strongly NP-hard SN is strongly NP-hard Unsplittable Model US is strongly NP-hard UN is strongly NP-hard

Height =2

06.10.2020 Load-Optimization in Reconfigurable Networks: Algorithms and Complexity of Flow Routing (Performance 2020)

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Non-Blocking Interconnects, e.g., Clos, Fat-Tree etc.

Non-Blocking Interconnections in above layers

06.10.2020 Load-Optimization in Reconfigurable Networks: Algorithms and Complexity of Flow Routing (Performance 2020)

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Simplified Problem defined by Non-Blocking Interconnections

Above layers abstracted as a packet switch.

06.10.2020 Load-Optimization in Reconfigurable Networks: Algorithms and Complexity of Flow Routing (Performance 2020)

(Mohammad Alizadeh et al. 2016).

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  • Consider a decision problem
  • Assume the optimized maximum load: 𝜄
  • Let 𝑇 be the set of possible values for 𝜄
  • 𝑇 contains the load for each static link before reconfiguration
  • Next, we show how to compute the set 𝑇

06.10.2020 Load-Optimization in Reconfigurable Networks: Algorithms and Complexity of Flow Routing (Performance 2020) Page 21

Optimal Algorithms for Simplified Problem (Notations)

The set 𝑇 A E D C B

20 20 6 6

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Useful Observations

06.10.2020 Load-Optimization in Reconfigurable Networks: Algorithms and Complexity of Flow Routing (Performance 2020)

A E D C B A E D C B

Any demand : X→Y, one node in the triangle, the other not in, then flows always go through the center

A E D C B

E.g., demand : B→E

  • For each reconfigurable link 𝑌, 𝑍 , focus on its triangle.
  • E.g., the triangle 𝐵, 𝐹, 𝐷
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  • For each reconfigurable link 𝑌, 𝑍 , in the triangle 𝑌, 𝑍, 𝐷 :
  • Compute local demands, and find optimal load for the local demands

06.10.2020 Load-Optimization in Reconfigurable Networks: Algorithms and Complexity of Flow Routing (Performance 2020) Page 23

Local Optimization For Each Triangle

A E D C B

Local demands :D’(C→E)=D(B→E)+D(D→E) D’(E→C)=D(E→B)+D(E→D)

A E D C B A E C

  • Find optimal routing in 𝑃(1)
  • Let the maximum load be Δ𝑗
  • Put Δ𝑗 into the set 𝑇

Local demands D’

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Optimal Algorithm: Mark Target Nodes

  • Binary search in the set 𝑇 to find the actual 𝜄 (optimized maximum load) within 𝑃(log |𝑊|)
  • For a specific 𝜄:
  • Mark each node “target” (𝑊𝑠 ⊆ 𝑊 ) if its link load is larger than 𝜄 before reconfiguration

A E D C B

20 20 6 6

A E D C B

20 20 6 6

𝜄=10

06.10.2020 Load-Optimization in Reconfigurable Networks: Algorithms and Complexity of Flow Routing (Performance 2020)

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Optimal Algorithm: Compute Useful Reconfigurable Links

  • For a specific 𝜄:
  • Define a set ℰ′: useful reconfigurable links, where ℰ′ ⊆ ℰ
  • For each triangle, if its maximum load Δ𝑗 ≤ 𝜄, put its reconfigurable link ℰ′

06.10.2020 Load-Optimization in Reconfigurable Networks: Algorithms and Complexity of Flow Routing (Performance 2020)

A E C

  • Find optimal routing in 𝑃(1)
  • Let the maximum load be Δ𝑗
  • If Δ𝑗 ≤ 𝜄, put 𝐵, 𝐹 in the set ℰ′
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Optimal Algorithm: Red-Target Matching and Binary Search

  • For each specific 𝜄: (𝑊𝑠 and ℰ′ computed )
  • Obtain a new graph G′ = (𝑊, ℰ′)
  • Find a matching 𝑁 in G′ to cover all target nodes 𝑊𝑠 (by maximum weight matching)
  • Total run-time cost: 𝑃(log |𝑊| ∗ 𝑈), and 𝑈 is the run-time of maximum weight matching

A E D C B

20 20 6 6

A E D C B

20 20 6 6

𝜄=10 Cover all red nodes

06.10.2020 Load-Optimization in Reconfigurable Networks: Algorithms and Complexity of Flow Routing (Performance 2020)

A E D C B

10 10 4 4

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Theoretical Analysis of Performance

  • Lower bound: the maximum load decreased by 50% by adding reconfigurable links
  • Why: at most two paths between any two nodes
  • Our optimal algorithm achieves the lower bound
  • Maximum matching works badly:
  • For some cases, maximum matching can only

decrease the maximum load by an arbitrarily small value 𝜁

06.10.2020 Load-Optimization in Reconfigurable Networks: Algorithms and Complexity of Flow Routing (Performance 2020)

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  • Traces from

Load-Optimization in Reconfigurable Networks: Algorithms and Complexity of Flow Routing (Performance 2020) 29

Evaluation: Minimize Maximum Link Load

←Static (normalized) ←Max. Matching ←Our Algorithm

Better

performance 2x, similar run time

Topology

+

Greedy (Firefly)

06.10.2020

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  • Theoretical Running Time:
  • Greedy: 𝑃(|𝑊|)
  • Maximum Matching (Blossom Alg.): 𝑃(|𝐹||𝑊|2)
  • Our Algorithm: 𝑃 log 𝑊 ∗ 𝐹 𝑊 2
  • The experiments match our theoretical analysis

Load-Optimization in Reconfigurable Networks: Algorithms and Complexity of Flow Routing (Performance 2020) 30

Evaluation: Comparing Time Costs

  • Max. Matching

Our Algorithm SN Greedy (Firefly) Our Algorithm US

06.10.2020

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Load-Optimization in Reconfigurable Networks: Algorithms and Complexity of Flow Routing

Wenkai Dai, Klaus-T. Foerster, David Fuchssteiner, Stefan Schmid (CT Group, University of Vienna)

Thank you! ☺