Rewiring Review, Quantitative Analysis and Applications

1
Rewiring Review, Quantitative Analysis and
Applications

• Matthew Tang Wai Chung
• CUHK CSE MPhil
• 10/11/2003

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Agenda

• Review
• Introduction
• RAMBO, REWIRE
• RAMFIRE
• GBAW
• Quantitative Analysis
• Rewiring power
• Runtime
• Coverage
• Applications
• Circuit Partitioning
• FPGA LUT mapping

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Introduction What is Rewiring?

• Replacing a target wire by an alternative wire
  without changing the functionality of the
  circuit.
• An alternative wire is a redundant wire to the
  circuit.
• Its addition would make the target wire become
  redundant hence removable.
• Lets check out the classic example.

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Classic Example
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Classic Example Resultant Circuit
gates 9 ? 6 interconnect 15 ? 13
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How can we find AWs?

• RAMBO / REWIREATPG Mandatory assignments to
  find candidate nodes for AWs.
• RAMFIREFIRE Redundancy identification.ATPG
  connection type.
• GBAWGraph-based pattern matching.

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Automatic Test Pattern Generation (ATPG)

• We consider stuck-at-fault model.A wire is
  called stuck-at-1/0 (s-t-1/0) if the value at the
  wire is always 1/0.
• A wire is called redundant if its addition or
  removal to the circuit does not affect the
  functions at output.
• In ATPG, some stuck-at-faults for some wires are
  not testable ? these wires are redundant.

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ATPG by Example
0/1
g6 ? g7 s-t-1 fault testable ? g6 ? g7 not
redundant
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RAMBO Basic Notion

• Consider TW (ts, td) AW (as, ad)
• RAMBO considers absolute dominators td of as
  candidates for ad.
• Create conflicts in M.A. by choosing suitable
  as.e.g. if dominator is a AND gate, choose with
  node M.A. 0.

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RAMBO - Algorithm

• foreach wire w in the network N
• perform stuck-at-fault test on w
• build candidate set C for w
• Verify each wire in C by redundancy test
• C can be very large ? a lot of redundancy test
  needed.

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REWIRE

• Observation that some MA is forced, which means
  conflicts these would result in untestable
  stuck-at-fault. (observability MA)
• Some fast filters are proposed to prune the
  candidate set to avoid unnecessary time-costy
  implication.
• The flow of REWIRE is similar to that of RAMBO.

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REWIRE - Algorithm

• foreach node nd in the network N
• find OMA of nd
• construct array of wires with nd
• foreach wire wt of the array
• do stuck-at-fault test on wt
• build candidate set C
• apply filters to C
• Verify each wire in C

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RAMFIRE

• Inefficiency in finding redundancy is a big
  problem with RAMBO and REWIRE.
• FIRE a fast redundancy identification algorithm.
• No brute-force testing is needed, instead apply
  FIRE to get a set of alternative wires directly.

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FIRE - Example

Uncontrollability Unobservability





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RAMFIRE - Algorithm

• foreach wire w in the network N
• perform stuck-at-fault test on w (get MA)
• perform FIRE (1 0) on w
• Build a set of redundant wire R for w
• Form AW in R by creating conflicts in MA
• Only three logic implications are used for each
  wire.

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GBAW

• Based on pre-configured graph patterns.
• These patterns are extracted from result of
  RAMBO.
• There are around 20 25 patterns in the library.
• Each pattern correspond to a known TW / AW pair.
• Notion Pattern matching on the network to find
  AWs.

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GBAW Pattern Example

• Local 13 a champion pattern
• Actual logic transformation DeMorgans law

Forward
Backward
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Agenda

• Review
• Introduction
• RAMBO, REWIRE
• RAMFIRE
• GBAW
• Quantitative Analysis
• Rewiring power
• Runtime
• Coverage
• Applications
• Circuit Partitioning
• FPGA LUT mapping

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Analysis Basis

• Operation 1 Given a target wire, find a set of
  alternative wires.
• Operation 2 Given a redundant wire to be added,
  find a set of wires becoming redundant after this
  addition.

We compare the algorithms in handling operation
1. The algorithms of RAMBO and REWIRE are
modified accordingly. All the experimental
results are obtained on the same PC running
Redhat Linux 8.0. (Duron 1.3 GHz, 256 MB)
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Comparing Rewiring Power
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Comparing Rewiring Power
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Comparing Runtime (Theoretical)

• Consider a circuit n nodes and w wires.
• Let c and c be the sizes of the candidate set in
  RAMBO REWIRE respectively.
• Recursive Learning (rmax 1) O(w2)

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Comparing Runtime (Experimental)
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Comparing GBAW RAMBO (coverage)
Forward Pattern
PI side ? higher depth ?? lower depth ? PO side
Backward Pattern
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Comparing GBAW RAMBO (coverage)

• RAMBO is able to locate all forward patterns by
  GBAW.Why? Forward patterns are actually
  extracted from results of RAMBO.
• RAMBO cant find any backward patterns by
  GBAW.Why? The candidate destination node of the
  AW must be an absolute dominator in RAMBO.

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Comments

• GBAW RAMFIRE fast, strong in locating
  backward wires
• RAMBO REWIRE much stronger rewiring power
• Tradeoff CPU runtime ? Rewiring Power
• GBAW can be an efficient pre-process filter to
  skip time-costy redundancy test.

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Agenda

• Review
• Introduction
• RAMBO, REWIRE
• RAMFIRE
• GBAW
• Quantitative Analysis
• Rewiring power
• Runtime
• Coverage
• Applications
• Circuit Partitioning
• FPGA LUT mapping

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Circuit Partitioning

• Rewiring can be used in reducing the cut size.
• Consider the wires on the cut as target wires, we
  try to find alternative wires which can be placed
  within clusters ? smaller cut size.
• Experimental Results9 reduction in cut size
  with small penalty on the of literals. (5-way
  partition)

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FPGA LUT Mapping

• Given a Boolean network, map the nodes into fixed
  size lookup tables (LUT).
• The number of inputs to a LUT is fixed, say k. A
  k-LUT can implement any Boolean function with k
  variables.
• Objective
• Minimize the total number of LUT used.
• Minimize the depth of the mapping solution.

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LUT-based mapping Example

• k 3

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Motivation

• Most LUT-based mapping algorithms do not consider
  modifying the Boolean network during mapping.
• We would like to explore the possibility to apply
  logic transformation during mapping.
• Logic transformation our favourite rewiring
• We have picked Flowmap to work with due to its
  solutions depth optimality.

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Intuition Utilizing LUTs

• We found that quite a lot LUTs are not fully
  utilized, i.e. input used lt k.e.g. pcler8 k
  5 12 / 47 nodes has 5 inputs.
• Using rewiring, we can find backward wire to fill
  the unused input in LUTs.
• When some LUT is utilized further, we may save
  some LUTs in the whole network.

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Example
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Example (Cont)

• GBAW Local 111g6 ? g7 replaced by g6 ? g3

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Future Work

• Verify the intuition through more examples or
  experiments.
• Observe and find out more intuitions.
• Purpose a more complete scheme.

THANK YOU