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

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

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

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

7. Parallel Iterative Matching1 2 3 4 #1 #2
Iteration:
uar selection
uar selection 1 2 3 4 1 2 3 4
f2: Grant 1 2 3 4
f3: Accept/Match
f1: Requests 1 2 3 4 1 2 3 4

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

9. Parallel Iterative Matching
16x16 switch

10. Parallel Iterative Matching
PIM with a single iteration

11. Parallel Iterative Matching
PIM with 4 iterations

12. PIM Fairness Problems: (under inadmissible load )

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

14. iSLIP
Round-Robin Selection
Round-Robin Selection 1 2 3 4 1 2 3 4
F2: Grant 1 2 3 4
F3: Accept/Match 1 2 3 4 #1 #2
F1: Requests 1 2 3 4 1 2 3 4

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

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

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

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

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 < log2N
Implementation: N priority encoders
Up to 100% throughput for uniform i.i.d. traffic

20. iSLIP
16x16 switch

21. iSLIP

22. iSLIP Match Size

23. iSLIP Implementation Programmable Priority Encoder State Decision1 1
log2N
Decision
Grant
Accept 2 2 N
Grant
Accept
log2N N N N
Grant
Accept
log2N

24. iSLIP Variations L priority levels Weighted SLIP
replace each pointer by L pointers
threshold SLIP
Weighted SLIP

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

26. Wave Front Arbiter (Tamir)
Requests
Match 1 1 1 1 2 2 2 2 3 3 3 3 4 4 4 4

27. Wave Front Arbiter
Requests
Match

28. Wave Front Arbiter Implementation
Simple combinational logic blocks
1,1
1,2
1,3
1,4
2,1
2,2
2,3
2,4
3,1
3,2
3,3
3,4
4,1
4,2
4,3
4,4

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