1. CAS 721 Course Project Minimum Weighted Clique Cover of Test Set By Wei He (0041415)

2. Contents Background of problem Background of problem Algorithm Implementation Algorithm Implementation Combinatorial Computation Combinatorial Computation Algorithm Constraints Algorithm Constraints Summary Summary

3. How to Test Chips? Source: Essentials of Electronic Testing, Kluwer 2000 … 11 … 00 ….. … 01 Input Patterns Very Large Scale Digital Circuit Output Patterns 10 … 00 … ….. 01 … COMPARATOR STORED CORRECT RESPONSE Test Result

4. Test Patterns Generation Functional Test Functional Test 1) Generate a complete set of test-patterns 1) Generate a complete set of test-patterns 2) Apply to small circuits 2) Apply to small circuits Structural Test Structural Test 1) Discard equivalent stuck-at faults and the number 1) Discard equivalent stuck-at faults and the number of test patterns is decreased dramatically of test patterns is decreased dramatically 2) Apply to VLSI testing mostly 2) Apply to VLSI testing mostly ATPG Algorithm ATPG Algorithm 1) Inject faulty Bits in test pattern 1) Inject faulty Bits in test pattern 2) High fault coverage 2) High fault coverage Source: Essentials of Electronic Testing, Kluwer 2000

5. Problem Formulation Given original Test Data : 64 bits long Given original Test Data : 64 bits long 1xxx1001x10xxx0xxx011x0x0xxx1xx0x00xx0x010xxx0x1xxx1xxx01xxx0xxx 1xxx1001x10xxx0xxx011x0x0xxx1xx0x00xx0x010xxx0x1xxx1xxx01xxx0xxx How to remove the redundant bits? How to remove the redundant bits? Which way to achieve the most reduction? Which way to achieve the most reduction? Divide to 8 test patterns with 8 bits long each pattern 0: 1 x x x 1 0 0 1 pattern 1: x 1 0 x x x 0 x pattern 2: x x 0 1 1 x 0 x pattern 3: 0 x x x 1 x x 0 pattern 4: x 0 0 x x 0 x 0 pattern 5: 1 0 x x x 0 x 1 pattern 6: x x x 1 x x x 0 pattern 7: 1 x x x 0 x x x

6. Problem Refinement Combine test patterns by using “don’t care” bits Combine test patterns by using “don’t care” bits Consider each pattern as a single node of a clique Consider each pattern as a single node of a clique Problem refined to achieve the lowest weight Problem refined to achieve the lowest weight Complete Clique Cover

7. Algorithm Implementation match_count = 8 if two patterns i and k are exactly compatible match_count = 8 if two patterns i and k are exactly compatible Edge[ i ] [ k ] = 1 if patterns i and k are exactly compatible otherwise Edge [ i ] [ k ] = 0 Edge[ i ] [ k ] = 1 if patterns i and k are exactly compatible otherwise Edge [ i ] [ k ] = 0 To create a clique, check the Edge[ i ] [ k ] To create a clique, check the Edge[ i ] [ k ] Weight of edge measured by the compatibility of patterns Weight of edge measured by the compatibility of patterns weight [ i ] [ k ] = 8 such as 1 1 0 x 1 0 0 1 weight [ i ] [ k ] = 8 such as 1 1 0 x 1 0 0 1 weight [ i ] [ k ] = 4 such as 1 0 x x 1 0 0 1 weight [ i ] [ k ] = 4 such as 1 0 x x 1 0 0 1

8. Combinatorial Computing Choose the clique with lowest weight Choose the clique with lowest weight Solution to given test data Solution to given test data P 025 : 1 0 0 1 1 0 0 1 P 167 : 1 1 0 1 0 0 0 0 P 34 : 0 0 0 x 1 0 x 0 P 025 : 1 0 0 1 1 0 0 1 P 467 : 1 0 0 1 0 0 x 0 P 13 : 0 0 0 x 1 0 x 0

9. Combinatorial Computing Clique 012 found with weight 16! Clique 012 found with weight 16! Clique 025 found with weight 12! Clique 025 found with weight 12! Clique 123 found with weight 20! Clique 123 found with weight 20! Clique 1236 found with weight 44! Clique 1236 found with weight 44! Clique 126 found with weight 20! Clique 126 found with weight 20! Clique 136 found with weight 24! Clique 136 found with weight 24! Clique 167 found with weight 24! Clique 167 found with weight 24! Clique 234 found with weight 24! Clique 234 found with weight 24! Clique 2346 found with weight 48! Clique 2346 found with weight 48! Clique 236 found with weight 24! Clique 236 found with weight 24! Clique 246 found with weight 24! Clique 246 found with weight 24! Clique 346 found with weight 24! Clique 346 found with weight 24! Clique 467 found with weight 24! Clique 467 found with weight 24!

10. Algorithm Constraints To apply this algorithm, require:  A feasible set of all the possible combinations beforehand  Feasible set always considered as a whole entity  Test data length not oversize  Dimension of test pattern matrix not fixed  Selected cliques to cover all the nodes

11. Algorithm Refinement Long Test Data should be partitioned based on the specification Long Test Data should be partitioned based on the specification Develop to search for the complete clique cover Develop to search for the complete clique cover Multiple solutions occur Multiple solutions occur Optimal solution not unique Optimal solution not unique Cliques increased tremendously Cliques increased tremendously Long time computing Long time computing Get rid of redundant cliques Get rid of redundant cliques

12. Summary Manufacturing defect inevitable, chips require test Manufacturing defect inevitable, chips require test Reduce the testing time by test data compression Reduce the testing time by test data compression By this algorithm: Test data matrix reduced from 8 x 8 to 3 x 8 Test data matrix reduced from 8 x 8 to 3 x 8 Optimal solution easily reached for short test data Optimal solution easily reached for short test data but usually not unique but usually not unique Refinement required for long test data Refinement required for long test data