Improving Matching algorithms for IQ switches

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Improving Matching algorithms for IQ switches

• Abhishek Das
• John J Kim

2
Motivation

• Results known
• With speedup 2, OQ switch can be emulated
  Shang-Tse Chuang et al
• Maximum Weight Matching (MWM) provides 100
  throughput McKeown et al
• 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 lt 2 sufficient to guarantee
  stability?

3
Fluid model equations for switch

• Queue-lengths represented by ?ij(t)
• ?ij(t) ?ij(0) Aij(t) - Dij(t) ? 0
• ?ij ?(t) ?ij(t) - Dij ?(t)
• Dij?(t) ? ijm 1?ij(t)gt0
• for any admissible load matrix ?, ??, ?(t)? ? W
  (using Birkhoff-von Neumann decomposition), where
  W is the weight of Maximum Weight Matching

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K-approximation to MWM

• Lyapunov function L(t) ??(t), ?(t)?
• L?(t) 2???(t), ?(t)?
• 2??(t), ?(t)? - ?D?(t), ?(t)?
• ? 2W - ?D?(t), ?(t)?
• Using speedup s, ?D?(t), ?(t)? ?s.? ijm , ?(t)?
• L?(t) ? 2W - ?s.? ijm , ?(t)?
• 2W - 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.

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K-approximation to MWM (contd.)

• Greedy iLQF is 2-approximation
• requires speedup s gt 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
• Cij(n) Xi(n) Yj(n) - ?ij(n)where Xi(n) ?j?
  ?ij?(n) and Yj(n) ?i? ?i?j(n)

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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)

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Matching size with speedup s

• Maximum size matching is not stable
• Requires speedup?
• L(t) ??(t), 1?
• L?(t) ???(t), 1?
• ??(t), 1? - ?D?(t), 1?
• ??(t), 1? ? N because ?i?ij ? 1 and ?j?ij ? 1
• ?D?(t), 1? ? ??ijm , 1? since D ij?(t) ?ijm
  1?ij(t)gt0
• L?(t) ? N - ?i?Mi?
• speedup s s matchings
• M is the size of the matching
• ? 0 if ?i?Mi? ? N, thus achieving 100
  throughput.
• Note that its total load (??(t), 1?) and not N

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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

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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 DallyTowles)

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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

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HIM1 vs PIM1 Matching Size
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HIM1 vs PIM1 Latency
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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?

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iSLIPHIM

• 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

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iSLIPHIM Size Matching
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iSLIPHIM Latency
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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

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Non-uniform Traffic Results
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Future Improvements

• Incorporate weights into HIM
• Have predetermined threshold on the size of the
  VOQ and use them as priorities

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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