1. 1 Outline  Why Maximal and not Maximum  Definition and properties of Maximal Match  Parallel Iterative Matching (PIM)  iSLIP  Wavefront Arbiter (WFA)

2. 2 Why doesn’t maximizing instantaneous throughput give 100% throughput for non- uniform traffic? Three possible matches, S (n):

3. 3 Maximal Matching  A maximal matching is one in which each edge is added one at a time, and is not later removed from the matching.  i.e. no augmenting paths allowed (they remove edges added earlier).  No input and output are left unnecessarily idle.

4. 4 Example of Maximal Size Matching A1 B C D E F 2 3 4 5 6 A1 B C D E F 2 3 4 5 6 Maximal Size Matching Maximum Size Matching A B C D E F 1 2 3 4 5 6

5. 5 Maximal Matchings  In general, maximal matching is simpler to implement, and has a faster running time.  A maximal size matching is at least half the size of a maximum size matching.  A maximal weight matching is defined in the obvious way.  A maximal weight matching is at least half the weight of a maximum weight matching.

6. 6 Outline  Definition and properties of Maximal Match  Parallel Iterative Matching (PIM)  iSLIP  Wavefront Arbiter (WFA)

7. 7 Parallel Iterative Matching 1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4  1: Requests 1 2 3 4 1 2 3 4  2: Grant 1 2 3 4 1 2 3 4  3: Accept/Match uar selection 1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4 #1 #2 Iteration:

8. 8 Parallel Iterative Matching Convergence Time Number of iterations to converge: Q k inputs with no other grant n-k inputs with grants from others with prob. 1 all n inputs are resolved A.grant is accepted – all are resolved B.grant rejected – n-k are resolved At most (n-k)  (1-k/n) are unresolved  n/4

9. 9 Parallel Iterative Matching

10. 10 Parallel Iterative Matching PIM with a single iteration

11. 11 Parallel Iterative Matching PIM with 4 iterations

12. 12 PIM Fairness Problems: (under inadmissible load )

13. 13 Outline  Definition and properties of Maximal Match  Parallel Iterative Matching (PIM)  iSLIP  Wavefront Arbiter (WFA)

14. 14 iSLIP 1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4  1: Requests 1 2 3 4 1 2 3 4  2: Grant 1 2 3 4 1 2 3 4  3: Accept/Match 1 2 3 4 1 2 3 4 #1 #2 Round-Robin Selection 1 2 3 4 1 2 3 4

15. 15 SLIP vs. Round Robin 1. Request: each input send a request to every output i, |VOQ i |>0 2. Grant: chose a request next in RR order and advance pointer beyond it. 3. Accept:chose the among the grants the one after the pointer and advance the pointer beyond.

16. 16 SLIP vs. Round Robin 1. Request: each input send a request to every output i, |VOQ i |>0 2. Grant: chose a request next in RR order and advance pointer beyond it if accepted. 3. Accept:chose the among the grants the one after the pointer and advance the pointer beyond.

17. 17 iSLIP vs. Round Robin 1. Request: each input send a request to every output i, |VOQ i |>0 2. Grant: chose a request next in RR order and advance pointer beyond it if accepted. 3. Accept:chose the among the grants the one after the pointer and advance the pointer beyond only if matched in 1 st iteration. in 1 st iteration

18. 18 why update pointers only in the 1 st round?  assume all pointers point at 1.  time 1:  1 st : 1-1 is matched  2 nd : 2-2 is matched  time 2  1 st : 1-3 & 3-2 are matched  time 3:  1 st : 1-1 is matched  2 nd : 2-2 is matched 1 3 22 3 1

19. 19 iSLIP Properties  Random under low load  TDM under high load  Lowest priority to MRU  1 iteration: fair to outputs  Converges in at most N iterations. On average < log 2 N  Implementation: N priority encoders  Up to 100% throughput for uniform i.i.d. traffic

20. 20 iSLIP

21. 21 iSLIP

22. 22 iSLIP Implementation Grant Accept 1 2 N 1 2 N State N N N Decision log 2 N Programmable Priority Encoder

23. 23 iSLIP Variations  L priority levels  replace each pointer by L pointers  threshold SLIP  Weighted SLIP

24. 24 Outline  Definition and properties of Maximal Match  Parallel Iterative Matching (PIM)  iSLIP  Wavefront Arbiter (WFA)

25. 25 Wave Front Arbiter (Tamir) RequestsMatch 1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4

26. 26 Wave Front Arbiter RequestsMatch

27. 27 Wave Front Arbiter Implementation 1,11,21,31,42,12,22,32,43,13,23,33,44,14,24,34,4 Simple combinational logic blocks

28. 28 Wave Front Arbiter Wrapped WFA (WWFA) Requests Match N steps instead of 2N-1