1. Improved Path Clustering for Adaptive Path-Delay Testing Tuck-Boon Chan* and Prof. Andrew B. Kahng*# UC San Diego ECE* & CSE # Departments

2. 2 Adaptive Path-Delay Testing [ShintaniUT09] Test patterns are specific to process condition Select test pattern based on measured process condition  reduced test cost! Critical paths for process condition V j Critical path sets for various process conditions Test patterns for process condition V j Test pattern sets for various process conditions ATPG Measure process condition of a chip Select a test pattern set based on the measured process condition Path delay testing Test pattern generationAdaptive testing

3. 3 Clustering Example Process conditions {V 1, V 2,V 3 } Critical path sets {S 1, S 2, S 3 } 10 S1S1 S2S2 S3S3 5 5 Test 35 paths if process condition = V 1 or V 2 Test 25 paths if process condition = V 3 20 10 5 25 20 S3S3 S2S2 S1S1 C1C1 C2C2 Clustering Solution B 10 15 5 20 S3S3 S2S2 S1S1 C1C1 C2C2 Venn diagrams of critical path sets Clustering Solution A No clustering: Test 40 paths per chip Test 15 paths if process condition = V 1 Test 35 paths if process condition = V 2 or V 3

4. 4 Clustering for Min Expected Cost Objective : minimize f(C) Input : V, Q and k Output : k disjoint clusters, C = {C 1, C 2, …, C k } V j = the j th process condition, j = 1,...,M P = {P 1,...,P N } = set of all critical paths S j  P = set of critical paths for process condition V j Q j = occurrence probability of process condition V j k = maximum number of clusters 10 S1S1 S2S2 S3S3 5 5 20 10 5 25 20 S3S3 S2S2 S1S1 C1C1 C2C2 C 1 : (0.2 + 0.5) x (5 + 10 + 20) = 17.50 C 2 : (0.3) x (25) = 0.75 f(C) = 17.5 + 0.75 = 18.25 Q 1 = 0.2 Q 2 = 0.5 Q 3 = 0.3 Expected testing cost:

5. 5 Previous Work: Greedy Algorithm [Uezono10] Calculate cost of merging any two clusters Perform the cluster merge with minimum cost Repeat until number of clusters = k S1S2S3S4 N/2-2 C2C2 C1C1 Optimal solution Greedy method 1 1 N/2-2 1 1 C2C2 C1C1 C3C3 1 1 C2C2 C1C1 1 1 11 11 1 1 11 When Q 1 = Q 4 = 0.5-  and Q 2 = Q 3 = ,  ≈ 0

6. 6 Proposed Method I: KL-FM Analog Model clustering problem as a hypergraph Goal: partition the graph with minimum cost Recursively partition a hypergraph into two subgraphs Random bipartition Calculate gain of moving a node Move node with highest gain to other partition Lock the moved node All nodes are moved? Select partition with minimum cost KL-FM approach V2V2 V3V3 P3P3 V4V4 P2P2 P1P1 cut V1V1

7. 7 General Testcase Represent clustering problem with a hypergraph e h,j : Process condition j needs to test critical path h b j,d : Process condition j belongs to cluster d Goal: find the connections b j,d that minimizes test cost e h,j are generated using random graph model G(n,P) Probability of process conditions are generated randomly (uniform, gaussian, power law …) V1V1 V2V2 P2P2 … P1P1 PNPN c1c1 c2c2 ckck … VMVM … P3P3 ClustersProcess conditionsCritical paths e h,j b j,d QMQM Q3Q3 Q2Q2 Q1Q1

8. 8 Experiment Results (1) When k = M, only one feasible solution  Performance ratio = 1.0 For k < M, performance ratio < 1.0  Proposed method has a lower test cost Greedy method prone to generating suboptimal solution in merging operation Total number of merging operations = Total number of process conditions – number of clusters = M-k

9. 9 Industrial Testcase Critical/test paths have strong correlations, and “containment” property

10. 10 Experiment Results (2) Greedy+ only merges adjacent clusters to avoid suboptimal merging solutions FM method does not take advantage of correlation among process conditions Test cost : Greedy+ < FM < Greedy

11. 11 Proposed Method II: Greedy+ DP-RP Greedy + Dynamic programming Greedy method provides a good initial solution Still prone to suboptimal merging operation Refine merging with dynamic programming S1 S2 S3 S4 S3 S4 S1 S2 S3 S4 S1 S2 S3S4S1S2 Step 1: Run Greedy+ and order process conditions accordingly Step 2: Optimally partition 1D array into k clusters with “DP-RP”: DAC 1994, Alpert et al. For j = 1,2, …, M For partition = 1, 2, …, M-1 calc min cost end end S3S4S1S2

12. 12 Experiment Results (3) Test cost is reduced by 0 to 5% Similar runtime complexity, O(M 2 N) DP-RP takes 10% more time than Greedy+

13. 13 Summary Formulation of the clustering problem in adaptive path-delay testing Proposed a hypergraph representation and clustering algorithm based on FM partitioning Improve simple Greedy method for random testcases Greedy+ works well for highly correlated testcases Further improvement on Greedy+ with DP-RP Future/ongoing work: DP-RP + Greedy ordering is suboptimal: better ordering? Critical path extraction for multi-dimensional process variations

14. 14 Acknowledgment Professor Takashi Sato, Graduate School of Informatics, Kyoto University. Dr. Takumi Uezono, Integrated Research Institute, Tokyo Institute of Technology.

15. Thank You

16. 16 References [Alpert94] C. J. Alpert and A. B. Kahng, “Multi-Way Partitioning Via Spacefilling Curves and Dynamic Programming”, Proc. Design Automation Conference, 1994, pp. 652-657. [Shintani09] M. Shintani, T. Uezono, T. Takahashi, H. Ueyama, T. Sato, K. Hatayama, T. Aikyo and K. Masau, “An Adaptive Test for Parametric Faults Based on Statistical Timing Information,” Proc. IEEE Asian Test Symposium, 2009, pp. 151-156. [Uezono10] T. Uezono, T. Takahashi, M. Shintani, K. Hatayama, K. Masu, H. Ochi and T. Sato, “Path Clustering for Adaptive Test,” Proc. IEEE VLSI Test Symposium, 2010, pp. 15-20.