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

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Presentation on theme: "1 Outline  Why Maximal and not Maximum  Definition and properties of Maximal Match  Parallel Iterative Matching (PIM)  iSLIP  Wavefront Arbiter (WFA)"— Presentation transcript:

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 A1 B C D E F Maximal Size Matching Maximum Size Matching A B C D E F

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: Requests  2: Grant  3: Accept/Match uar selection #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: Requests  2: Grant  3: Accept/Match #1 #2 Round-Robin Selection

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

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

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


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