1. Improving Matching algorithms for IQ switches Abhishek Das John J Kim

2. Motivation Results known: 1. With speedup 2, OQ switch can be emulated [Shang-Tse Chuang et al] 2. Maximum Weight Matching (MWM) provides 100% throughput [McKeown et al] 3. With speedup 2, any maximal matching can achieve 100% throughput [Dai and Prabhakar] What about maximal matching algorithms with weights within a factor of k of MWM (k- approximation)? Is speedup s < 2 sufficient to guarantee stability?

3. Fluid model equations for switch Queue-lengths represented by  ij (t) 1.  ij (t) =  ij (0) + A ij (t) - D ij (t)  0 2.  ij (t) = ij (t) - D ij (t) 3. D ij (t) =  ij m 1 {  ij (t)>0} for any admissible load matrix, ,  (t)   W * (using Birkhoff-von Neumann decomposition), where W * is the weight of Maximum Weight Matching

4. K-approximation to MWM Lyapunov function L(t) =  (t),  (t)  L(t) = 2  (t),  (t)  = 2[  (t),  (t)  -  D(t),  (t)  ]  2[W * -  D(t),  (t)  ] Using speedup s,  D(t),  (t)  =  s.  ij m,  (t)  L(t)  2[W * -  s.  ij m,  (t)  ] = 2[W * - s.w] (w is weight of the matching)  0 if W *  s.w Now for a k-approximation matching algorithm, k.w  W * If k  s, we satisfy W *  k.w  s.w and achieve 100% throughput.

5. K-approximation to MWM (contd.) Greedy iLQF is 2-approximation – requires speedup s >= 2 (but so does any maximal matching) Any other k-approximation to weighted matching? Weighted matching algorithms use min-cost max-flow algorithms Futile attempts at coming up with other Lyapunov functions – C ij (n) = X i (n) + Y j (n) -  ij (n) where X i (n) =  j  ij(n) and Y j (n) =  i  ij(n)

6. Why look at VOQ sizes? Disappointed souls looking at the bigger picture!! Weights are bad for hardware complexity – Requires multi-bit integer comparators LPF is already stable without speedup LPF vs LQF(MWM) – Ratio of matching weights vary from 1 (~70% load) to 20 (90% load)

7. Matching size with speedup s Maximum size matching is not stable – Requires speedup? L(t) =  (t), 1  =  (t), 1  -  D(t), 1   (t), 1   N because  i ij  1 and  j ij  1  D(t), 1    ij m, 1  since D ij (t) =  ij m 1 {  ij(t)>0} L(t)  N -  i |M i | – speedup s: s matchings – |M| is the size of the matching  0 if  i |M i |  N, thus achieving 100% throughput. – Note that its total load (  (t), 1  ) and not N

8. K-approximation vs Heuristics Many approximate matching algorithms known (linear and poly-logarithmic) Approximate matching algorithms compare to the maximum size matching (not to the load) Heuristics to improve the matching size of practical iterative matching algorithms – speedup

9. Holi-PIM (or HIM) Attempt to improve upon PIM by generating bigger size matching in each iteration Observation: poor matching occurs in PIM when inputs receive multiple grants Increase the size of matching by considering the number of requests from each input – equivalent to considering the number of HOL packets or the “fanout” from each input Similar to lonely output allocator (Interconnection Networks – Dally&Towles)

10. HIM implementation Requires log(N) control bits from each input Weights are assigned based on the fanout of each input How to break ties – Randomly – Round-robin manner Weighted probability vs strict weights

11. HIM1 vs PIM1 Matching Size

12. HIM1 vs PIM1 Latency

13. Problems with HIM HIM performs better than PIM but still does not give 100% throughput Fairness issue: HIM is not a fair algorithm as it will favor the shorter queues iSLIP1 is known to give 100% throughput on uniform traffic and has simple hardware complexity Can iSLIP take advantage of the HIM weights?

14. iSLIP+HIM Add the HIM weights to iSLIP The weight of each edge of the request is determined by combining the iSLIP weights (priority pointers) and the HIM weights At intermediate loads, HIM weight should improve the performance At high load, the HIM weights should be identical and iSLIP should dominate

15. iSLIP+HIM Size Matching

16. iSLIP+HIM Latency

17. Non-uniform Traffic iSLIP is known to behave poorly on non- uniform traffic pattern HIM does not significantly improve on non- uniform as it is an attempt of maximum size matching, not maximum weight

18. Non-uniform Traffic Results

19. Future Improvements Incorporate weights into HIM – Have predetermined threshold on the size of the VOQ and use them as priorities

20. Conclusions Showed required stability conditions for matching algorithms (with and without weight) Introduced and studied a new practical iterative matching algorithm: Holi-PIM under unform traffic