A Reduced Complexity Algorithm for Minimizing N-Detect Tests

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A Reduced Complexity Algorithm for Minimizing
N-Detect Tests
Kalyana R. Kantipudi Vishwani D. Agrawal
Department of Electrical and Computer
Engineering Auburn University, AL 36849 USA 20th
Intl Conf. on VLSI Design, Bangalore, Jan 6-10th,
2006
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Motivation for This Work

• Ability of N-detect tests to improve the defect
  coverage.
• Easy assimilation of N-detect tests into the
  normal test generation strategy.
• Main limitation of N-detect tests is their size.
• Inability of ILP based method to produce a
  time-bound optimal solution.
• Inability of previous test minimization
  strategies in finding minimal tests for c6288
  benchmark.

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Outline

• ILP based N-detect test minimization
• Previous LP based methods
• Recursive rounding based approach
• The 3V3F example
• Minimal tests for c6288
• Results
• Conclusions

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ILP-Based N-Detect Test Minimization 1

• Use any N-detect test generation approach to
  obtain a set of k vectors which detect every
  fault at least N times.
• Use diagnostic fault simulation to get the vector
  subset Tj for each fault j.
• Assign integer variable ti to ith vector such
  that,
• ti 1 if ith vector is included in the minimal
  set.
• ti 0 if ith vector is not included.
• 1 K. R. Kantipudi and V. D. Agrawal, On the
  Size and Generation of Minimal N-Detection
  Tests, Proc. VLSI Design06.

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Objective and Constraints of ILP
Where, Nj is the multiplicity of detection for
the jth fault. Nj can be selected for each
individual fault based on some criticality
criteria or on the capability of the initial
vector set. ILP always generates an optimal
solution for the given set of test vectors.
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A Linear Programming Approach

• Though ILP guarantees an optimal solution, it
  takes exponential time to generate the solution.
• Time bounded ILP solutions deviate from
  optimality.
• LP takes polynomial time (sometimes in linear
  time) to generate a solution.
• Redefining the variables tis as real variables in
  the range 0.0,1.0 converts the ILP problem into
  a linear one.
• The problem now remains to convert it into an ILP
  solution.
• The optimal value of the relaxed-LP of the ILP
  minimization problem is a lower bound on the
  value of the optimal integer solution to the
  problem.

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Previous Solutions (Randomized rounding)

• The real variables are treated as probabilities.
• A random number xi uniformly distributed over the
  range 0.0,1.0 is generated for each variable
  ti.
• If ti xi then ti is rounded to 1, otherwise
  rounded to 0.
• If the rounded variables satisfy the constraints,
  then the rounded solution is accepted.
• Otherwise, rounding is again performed starting
  from the original LP solution.

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Limitations of Randomized Rounding

• Consider three faults f1,f2 and f3, and three
  vectors.
• We assign a real variable ti to vector i.
• Now the single detection problem is specified as
• Minimize t1 t2 t3
• Subject to constraints,
• f1 t1 t2 1
• f2 t2 t3 1
• f3 t3 t1 1
• The number of tests is much larger
• than the size of the minimal test set.
• The randomized rounding becomes a random search.

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Recursive Rounding (New Method)

• Step 1 Obtain an LP solution.
• Stop if each ti is either 0.0 or 1.0
• Step 2 Round the largest ti and fix its value to
  1.0
• If several tis have the largest value,
  arbitrarily set only one to 1.0. Go
  to Step 1.
• Maximum number of LP runs is bounded by the final
  minimized test set size.
• Final set is guaranteed to cover all faults.
• This method takes polynomial time even in the
  worst case.
• LP provides a lower bound on solution.
• Lower Bound exact ILP solution recursive LP
  solution
• Absolute optimality is not guaranteed.

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The 3V3F Example

• Step 1
• LP gives t1 t2 t3 0.5
• Step 2
• We arbitrarily set t1 1.0
• Step 1
• Gives t2 1, t3 0
• or t2 0, t3 1
• or t2 t3 0.5
• Step 2 (last case)
• We arbitrarily set t2 1.0
• Step 1 Gives t3 0

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Minimal Tests for Array Multipliers

• There exists a huge difference between its
  theoretical lower bound of six and its
  practically achieved test set of size 12.
• A 15 x 16 matrix of full-adders (FA) and
  half-adders (HA).
• To make use of its recursive
• structure and apply
• linear programming
• techniques.

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Tests for c6288 16-Bit Multiplier

• Known results (Hamzaoglu and Patel, IEEE-TCAD,
  2000)
• Theoretical lower bound 6 vectors
• Smallest known set 12 vectors, 306 CPU s
• Our results
• Up to four-bit multipliers need six vectors
• Five-bit multiplier requires seven vectors
• c6288
• 900 vectors constructed from optimized vector
  sets of smaller multipliers
• ILP, 10 vectors in two days of CPU time
• Recursive LP, lower bound 7, optimized set
  12, in 301 CPU s

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Comparison of ILP and Recursive LP
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Sizes of 5-Detect Tests for ISCAS85 Circuits
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CPU Time to Minimize 5-Detect Tests
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Optimized 15-detect Tests
Circuit Name Unopti. Vecs LP/recursive Rounding LP/recursive Rounding ILP 1 ILP 1 Heuristic2 Heuristic2 L.B.
Circuit Name Unopti. Vecs Vect. CPU s Vect. CPU s Vect. CPU s L.B.
c432 14882 430 83.5 430 444.8 505 292.1 405
c499 1850 780 17.8 780 24.9 793 153.2 780
c880 4976 322 94.5 321 521.4 338 229.6 195
c1355 2341 1260 41.2 1260 52.1 1274 5674.6 1260
c1908 6609 1590 150.4 1590 191 1648 1563.9 1590
c2670 8767 1248 380.6 1248 607.8 962 9357.6 660
c3540 4782 1407 239.6 1411 1223.7 - - 1200
c5315 4318 924 494.3 924 1368.4 - - 555
c6288 731 134 250.5 134 1206.3 144 1813.8 90
c7552 6995 2371 359.1 2370 346.1 - - 975
1 K R Kantipudi and V D Agrawal, Proc. VLSI
Design, 2006 2 Lee, Cobb, Dworak, Grimaila and
Mercer, Proc. DATE, 2002
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Conclusion

• Single and N-detection tests can be efficiently
  minimized by the new procedure.
• The quality of the result from recursive rounding
  LP is close to that of ILP.
• The 10 vector test set for c6288 signifies the
  shortcomings of present test set minimization
  techniques.
• The recursive rounding LP method has numerous
  other applications where ILP is traditionally
  used and is found to be expensive.

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Thank You . . .