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  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 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 S3S3 S2S2 S1S1 C1C1 C2C2 Clustering Solution B 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 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 S3S S3S3 S2S2 S1S1 C1C1 C2C2 C 1 : ( ) x ( ) = C 2 : (0.3) x (25) = 0.75 f(C) = = Q 1 = 0.2 Q 2 = 0.5 Q 3 = 0.3 Expected testing cost:

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/ C2C2 C1C1 C3C3 1 1 C2C2 C1C When Q 1 = Q 4 = 0.5-  and Q 2 = Q 3 = ,  ≈ 0

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 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 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 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 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 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.

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 [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 [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