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

Transcription

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

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

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

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]

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]

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

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!

8 Diameter 33% 66% Diameter of DRG(N, r) log r N r

9 Average Shortest Path Length 14% Average k shortest path length for DRG(N, r) j=1 h 1 jr j + hr k(n 1)

10 Average k-shortest Path Length Average k shortest path length for DRG N, r = LK(N, r, k) j=1 h 1 jr j + hr k(n 1)

11 Path Distribution Network Degree=30, N=901 min number of paths of length H r + r r H N 1

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

13 Load Balancing max link load for DRG N, r = ML(N, r) LK N, r, k (N 1) r

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?

15 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 GDBG(6,2) 7 Lemmas. Some More Math! node i connects to nodes: i d, i d + 1,, i + 1 d 1 mod N

16 GDBG: A Near Optimal r-regular Topology

17 GDBG: A Near Optimal r-regular Topology

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

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

20 Node Level Throughput N=150, Network Degree=8

21 Switch Level Throughput

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

23 Thanks for your time Questions