Improved Path Clustering for Adaptive Path-Delay Testing

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

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

3. Clustering Example • Process conditions {V1, V2 ,V3} • Critical path sets {S1, S2, S3} Clustering Solution A No clustering: Test 40 paths per chip C1 S1 S2 S3 C2 S2 20 S3 10 5 25 10 20 5 Test 35 paths if process condition = V1 or V2 Test 25 paths if process condition = V3 5 Clustering Solution B S1 C1 S2 S3 C2 S1 Venn diagrams of critical path sets 15 20 10 5 Test 15 paths if process condition = V1 Test 35 paths if process condition = V2 or V3

4. Clustering for Min Expected Cost • Objective : minimize f(C) • Input : V, Q and k • Output : k disjoint clusters, C = {C1, C2, …, Ck} Expected testing cost: S1 S2 S3 S1 S2 S3 10 20 20 10 5 5 25 5 C1 C2 Q1 = 0.2 Q2 = 0.5 Q3 = 0.3 C1: (0.2 + 0.5) x (5 + 10 + 20) = 17.50 C2: (0.3) x (25) = 0.75 f(C) = 17.5 + 0.75 = 18.25 Vj = the jth process condition, j = 1, ...,M P = {P1, ...,PN} = set of all critical paths Sj P = set of critical paths for process condition Vj Qj = occurrence probability of process condition Vj k = maximum number of clusters

5. Previous Work: Greedy Algorithm [Uezono10] Greedy method • Calculate cost of merging any two clusters • Perform the cluster merge with minimum cost • Repeat until number of clusters = k S1 S2 S3 S4 1 1 N/2-2 N/2-2 1 1 C1 C3 C2 1 1 N/2-2 N/2-2 1 1 When Q1= Q4 = 0.5- and Q2=Q3 = ,  ≈0 C1 C1 C2 Optimal solution 1 1 1 1 C2 N/2-2 N/2-2 N/2-2 N/2-2 1 1 1 1

6. Proposed Method I: KL-FM Analog cut • Model clustering problem as a hypergraph • Goal: partition the graph with minimum cost • Recursively partition a hypergraph into two subgraphs Random bipartition P1 V1 V4 Calculate gain of moving a node P3 V2 V3 P2 Move node with highest gain to otherpartition Lock the moved node All nodes are moved? Select partition with minimum cost KL-FM approach

7. General Testcase Critical paths Process conditions Clusters • Represent clustering problem with a hypergraph • eh,j : Process condition j needs to test critical path h • bj,d : Process condition j belongs to cluster d • Goal: find the connections bj,d that minimizes test cost • eh,j are generated using random graph model G(n,P) • Probability of process conditions are generated randomly (uniform, gaussian, power law …) Q1 V1 P1 c1 Q2 P2 V2 c2 Q3 P3 … … … QM VM ck PN eh,j bj,d

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

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. 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 S3 S1 S3 S1 S3 S2 S4 S2 S4 S2 S4 Step 1: Run Greedy+ and order process conditions accordingly S3 S4 S1 S2 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 S3 S4 S1 S2

12. Experiment Results (3) • Test cost is reduced by 0 to 5% • Similar runtime complexity, O(M2N) • DP-RP takes 10% more time than Greedy+

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

15. Thank You

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.