Random Regular Graph & Generalized De Bruijn Graph with k - - PowerPoint PPT Presentation

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Random Regular Graph & Generalized De Bruijn Graph with k - - PowerPoint PPT Presentation

Feb 19, 2024 285 likes β€’525 views

Random Regular Graph & Generalized De Bruijn Graph with k -shortest Path Routing Peyman Faizian , Md Atiqul Mollah, Xin Yuan Scott Pakin, Michael Lang Florida State University Los Alamos National Laboratory Where it All Started Random

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SLIDE 1

Random Regular Graph & Generalized De Bruijn Graph with k-shortest Path Routing

Peyman Faizian, Md Atiqul Mollah, Xin Yuan

Florida State University

Scott Pakin, Michael Lang

Los Alamos National Laboratory

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SLIDE 2

Where it All Started

  • Random Regular topologies(Jellyfish) have been proposed for data

centers and HPC clusters

  • They are known to have some good topological properties
  • They achieve higher performance in compare to Fat Tree under

certain traffic patterns

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SLIDE 3

We are Going to do this the Hard Way

  • Derive theoretical bounds for the following key metrics on any DRG
  • Diameter
  • Average k-shortest path length
  • Load balancing property
  • Evaluate RRG based on these criteria
  • Introduce a near-optimal DRG and compare it to RRG through

simulation

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SLIDE 4

What is a Random Regular Topology

  • Random graph between ToR switches
  • Each switch has k ports
  • r ports connected to other switches
  • k-r ports connected to processing nodes
  • r-regular random interconnect

topology(RRG)

[Singla et al; NSDI’12]

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SLIDE 5

What We Know About RRG

  • High bisection bandwidth
  • High network capacity
  • Sufficient short path diversity
  • k-shortest path routing is preferred

[Singla et al; NSDI’12]

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SLIDE 6

What is Needed for K-Path Routing

  • High bisection bandwidth
  • Minimal resource usage
  • Low diameter
  • Low average k-shortest path length
  • Good load balancing
  • Even distribution of paths over SD pairs
  • Even distribution of load over all network links
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SLIDE 7

What Should We Compare RRG to

  • RRG has been compared to one of the best available designs
  • Fat Tree
  • We want to compare it to the best possible design
  • Random Regular Graphs are a special case of Directed Regular Graphs
  • We derived the optimal bounds for all discussed properties on Regular Graphs

4 Lemmas. Some Graph Theory!

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SLIDE 8

Diameter

𝐸𝑗𝑏𝑛𝑓𝑒𝑓𝑠 𝑝𝑔 𝐸𝑆𝐻(𝑂, 𝑠) β‰₯ π‘šπ‘π‘•π‘  𝑂 𝑠 βˆ’ 1 + 1 βˆ’ 1 33% 66%

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SLIDE 9

Average Shortest Path Length

𝐡𝑀𝑓𝑠𝑏𝑕𝑓 𝑙 π‘‘β„Žπ‘π‘ π‘’π‘“π‘‘π‘’ π‘žπ‘π‘’β„Ž π‘šπ‘“π‘œπ‘•π‘’β„Ž 𝑔𝑝𝑠 𝐸𝑆𝐻(𝑂, 𝑠) β‰₯ π‘˜=1

β„Žβˆ’1 π‘˜π‘ π‘˜ + β„Žπ‘†

𝑙(𝑂 βˆ’ 1) 14%

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SLIDE 10

Average k-Shortest Path Length

𝐡𝑀𝑓𝑠𝑏𝑕𝑓 𝑙 π‘‘β„Žπ‘π‘ π‘’π‘“π‘‘π‘’ π‘žπ‘π‘’β„Ž π‘šπ‘“π‘œπ‘•π‘’β„Ž 𝑔𝑝𝑠 𝐸𝑆𝐻 𝑂, 𝑠 = 𝑀𝐿(𝑂, 𝑠, 𝑙) β‰₯ π‘˜=1

β„Žβˆ’1 π‘˜π‘ π‘˜ + β„Žπ‘†

𝑙(𝑂 βˆ’ 1)

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SLIDE 11

Path Distribution

Network Degree=30, N=901 min π‘œπ‘£π‘›π‘π‘“π‘  𝑝𝑔 π‘žπ‘π‘’β„Žπ‘‘ 𝑝𝑔 π‘šπ‘“π‘œπ‘•π‘’β„Ž ≀ 𝐼 ∢ 𝑠 + 𝑠2 + β‹― + 𝑠𝐼 𝑂 βˆ’ 1

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SLIDE 12

Load Balancing

  • All-to-All communication pattern
  • Use K-shortest paths between each SD pair
  • Monitor the number of flows passing each link
  • Pick the load value on the most loaded link in the network
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SLIDE 13

Load Balancing

max π‘šπ‘—π‘œπ‘™ π‘šπ‘π‘π‘’ 𝑔𝑝𝑠 𝐸𝑆𝐻 𝑂, 𝑠 = 𝑁𝑀(𝑂, 𝑠) β‰₯ 𝑀𝐿 𝑂, 𝑠, 𝑙 Γ— (𝑂 βˆ’ 1) 𝑠

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SLIDE 14

A Quick Recap

  • We compared these topological properties of Jellyfish with optimal:
  • Diameter
  • Average k-shortest path length
  • Path distribution
  • Load distribution
  • In some of the cases Jellyfish is far from theoretical bounds
  • But is there any near optimal r-regular topology?
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SLIDE 15

𝐻𝐸𝐢𝐻(6,2) π‘œπ‘π‘’π‘“ 𝑗 π‘‘π‘π‘œπ‘œπ‘“π‘‘π‘’π‘‘ 𝑒𝑝 π‘œπ‘π‘’π‘“π‘‘:

𝑗 βˆ— 𝑒, 𝑗 βˆ— 𝑒 + 1, … , 𝑗 + 1 βˆ— 𝑒 βˆ’ 1 𝑛𝑝𝑒 𝑂

Generalized de Bruijn Graph

  • We proved that GDBG has:
  • Near optimal diameter
  • Near optimal average

k-shortest path length

  • Near optimal path distribution
  • Near optimal load distribution

7 Lemmas. Some More Math!

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SLIDE 16

GDBG: A Near Optimal r-Regular Topology

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SLIDE 17

GDBG: A Near Optimal r-Regular Topology

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SLIDE 18

There is more…

  • GDBG is a near optimal r-regular topology
  • It can be used as a simulation benchmark to evaluate other

topologies in this family

  • We are going to evaluate RRG
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SLIDE 19

Simulation

  • Topology
  • Generalized de Bruijn Graph(almost optimal)
  • Jellyfish
  • Routing
  • Hop limited all path routing (GDBG)
  • k-shortest path routing (Jellyfish)
  • Traffic Pattern
  • Node level random permutation
  • Node level shift
  • Switch level random permutation
  • Switch level shift
  • Metric
  • Aggregate throughput
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SLIDE 20

Node Level Throughput

N=150, Network Degree=8

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SLIDE 21

Switch Level Throughput

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SLIDE 22

Conclusion

  • Derived theoretical bounds for the following key metrics on any DRG
  • Average k-shortest path length
  • Load balancing property
  • RRG is near optimal in terms of average k-shortest path length
  • RRG is far from optimal for all other metrics
  • GDBG was found near optimal for all metrics
  • GDBG was used as a simulation benchmark to evaluate RRG
  • Depending on traffic pattern, RRG is not always near optimal
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SLIDE 23

Thanks for your time

Questions