1. Fast and Efficient Static Compaction of Test Sequences Based on Greedy Algorithms Jaan Raik, Artur Jutman, Raimund Ubar Tallinn Technical University, Estonia

2. Motivation for test set compaction Compaction techniques Bipartite graph representations Greedy algorithm for test set compaction Experimental results Conclusions Fast and Efficient Static Compaction of Test Sequences Based on Greedy Algorithms

3. Motivation Test set minimization is an essential problem for the chip manufacturer, who faces the test of millions of units per annum. The time required to test a chip by the ATE is proportional with the length of the test sequence.

4. Compaction techniques Static versus dynamic With and without iterative fault simulation GA-based, deterministic and greedy methods

5. Minimization. With or without fault simulation? With iterative fault simulation –Optimal results –VERY time-consuming Without iterative fault simulation –Possibly non-optimal result –Fast minimization –No need for circuit model data, nor for a fault simulation tool.

6. State-of-the-Art Static compaction approaches with iterative fault simulation impractical for large circuits Fast approach by Hsiao et al based on considering traversed states –Potential loss of fault coverage Another approach by Corno et al exploiting test sets divided into independent sequences –No lower bound estimation

7. Matrix vs bipartite graph representations

8. Compaction algorithm Detecting essential patterns Greedy minimization Reducing search space Putting it all together Determining lower bounds and global minima

9. Test set as a matrix Test set T consisting of n faults and m test sequences can be viewed as a matrix, where t si,fj is equal to k if sequence s i covers fault f j at the k-th vector and zero if sequence s i does not cover fault f j.

10. Detecting essential patterns If fault f j is detected by the k-th vector of test sequence s i and is not detected by any other sequence then k first vectors of sequence s i are called essential.

11. Greedy optimization Selection function: selects Maxrange vectors from s i

12. Search space redustion (collapsing equivalent faults) column f a will be removed from matrix T if there exists another column f b, where In other words, if we have multiple identical columns we will unite them into a single one.

13. Search space redustion (removing redundant sequences) Sequence s b is redundant if there exists sequence s a such that

14. Compaction algorithm Select essential vectors. Remove the faults covered by these vectors. While exist uncovered faults { Remove redundant sequences. Collaps equivalent faults. If new essential vectors appeared then Select essential vectors. Else Select vectors by greedy selection. Endif Remove the faults covered by selected vectors. }

15. Determining lower bounds and global minima Lower bound = # of vectors selected up to the first greedy selection (included) If the number of patterns in the minimized test set equals to the lower bound, the optimal result (global minimum) for the task has been found.

16. Experimental results (average compaction)

17. Experimental results (speed-up compared to Corno et al)

18. Branch-and-Bound Algorithm

19. If we do not consider equivalent search states the total number of decisions to be considered will be n!(1/0! + 1/1! +1/2! + … + 1/(n-1)!), which approaches asymptotically e  n!. However, by identifying equivalent search states we have to consider 2 n -1 decisions in the worst case.

20. Experimental Results

21. Comparison of static compaction methods Genetic algorithm [3] Greedy search [13] Branch-and- bound Average compaction 49.86 %50.06 %50.27 % Best compaction results 323040 Proved optimal results N/A3040

22. Matrices could be partitioned into independent subsets Is there an efficient Boolean matrice representation for this problem? Future Work