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Outline
Why Maximal and not Maximum Definition and properties of Maximal
Match Parallel Iterative Matching (PIM) iSLIP Wavefront Arbiter (WFA)
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Why doesn’t maximizing instantaneous throughput give 100% throughput for non-
uniform traffic?
2/1
2/1
2/1
32
21
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100%). t(throughpu stable not is switch 0.0358 if so And
But
most at is served is 1 input which at rate total The
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both and and , time at that Assume
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-δ// - -λ
//
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nQnQ n, L nn, L
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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.
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Example of Maximal Size Matching
A 1
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D
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F
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Maximal Size Matching
Maximum Size Matching
A
B
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D
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F
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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.
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Outline
Definition and properties of Maximal Match
Parallel Iterative Matching (PIM) iSLIP Wavefront Arbiter (WFA)
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Parallel Iterative Matching
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1: Requests
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42: Grant
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43: Accept/Match
uar selection
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uar selection
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#1
#2
Itera
tion
:
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Parallel Iterative MatchingConvergence Time
E C Nlog
E U i N2
4i------- C # of iterations required to resolve connections=
N # of ports =
U i # of unresolved connections after iteration i=
Number of iterations to converge:
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 resolvedB. grant rejected – n-k are resolved
At most k(1-k/n) are unresolved n/4
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16x16 switch
Parallel Iterative Matching
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Parallel Iterative Matching
PIM with a single iteration
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Parallel Iterative Matching
PIM with 4 iterations
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PIM Fairness Problems:(under inadmissible load )
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Outline
Definition and properties of Maximal Match
Parallel Iterative Matching (PIM) iSLIP Wavefront Arbiter (WFA)
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iSLIP
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1: Requests
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42: Grant
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43: Accept/Match
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#1
#2
Round-Robin Selection
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Round-Robin Selection
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SLIP vs. Round Robin
1. Request: each input send a request to every output i, |VOQi|>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.
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SLIP vs. Round Robin
1. Request: each input send a request to every output i, |VOQi|>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.
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iSLIP vs. Round Robin
1. Request: each input send a request to every output i, |VOQi|>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 1st iteration.
in 1st iteration
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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: 1st: 1-1 is matched 2nd: 2-2 is matched
1
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2 2
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iSLIPProperties
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
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16x16 switch
iSLIP
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iSLIP
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iSLIP Match Size
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iSLIPImplementation
Grant
Grant
Grant
Accept
Accept
Accept
1
2
N
1
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N
State
N
N
N
Decision
log2N
log2N
log2N
ProgrammablePriority Encoder
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iSLIP Variations
L priority levels replace each pointer by L pointers
threshold SLIP
Weighted SLIP
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Outline
Definition and properties of Maximal Match
Parallel Iterative Matching (PIM) iSLIP Wavefront Arbiter (WFA)
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Wave Front Arbiter(Tamir)
Requests Match
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Wave Front Arbiter
Requests Match
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Wave Front ArbiterImplementation
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
Simple combinational logic blocks
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Wave Front ArbiterWrapped WFA (WWFA)
Requests Match
N steps instead of2N-1