Improved Path Clustering for Adaptive Path-Delay Testing

Tuck-Boon Chan* and Prof. Andrew B. Kahng*#

UC San Diego

ECE* & CSE# Departments

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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 Vj

Critical path sets

for various process conditions

Test patterns for

process condition Vj

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 generation

Adaptive testing

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Clustering Example• Process conditions {V1, V2 ,V3}

• Critical path sets {S1, S2, S3}

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S1

S2 S3

5 5 Test 35 paths if process condition = V1 or V2

Test 25 paths if process condition = V3

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105 2520

S3S2S1C1 C2

Clustering Solution B

1015

5 20

S3S2S1C1 C2Venn diagrams of

critical path sets

Clustering Solution ANo clustering:Test 40 paths per chip

Test 15 paths if process condition = V1

Test 35 paths if process condition = V2 or V3

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Clustering for Min Expected Cost

• Objective : minimize f(C)• Input : V, Q and k • Output : k disjoint clusters, C = {C1, C2, …, Ck}

Vj = the jth process condition, j = 1, ...,MP = {P1, ...,PN} = set of all critical pathsSj P = set of critical paths for process condition Vj

Qj = occurrence probability of process condition Vj

k = maximum number of clusters

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S1 S2 S3

5 5 20 105 2520

S3S2S1

C1 C2

ihij CP h

k

iCS j PQf ||)(

1

C

C1: (0.2 + 0.5) x (5 + 10 + 20) = 17.50C2: (0.3) x (25) = 0.75f(C) = 17.5 + 0.75 = 18.25

Q1 = 0.2Q2 = 0.5Q3 = 0.3

Expected testing cost:

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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

S1 S2 S3 S4

N/2-2 N/2-2

C2

C1Optimal solution

Greedy method

1 1

N/2-2 N/2-2

1 1

C2C1 C3

N/2-2 N/2-2

1 1

C2C1

N/2-2 N/2-2

1 1

1 1

1 1

1 11 1

When Q1= Q4 = 0.5- and Q2= Q3 = , ≈ 0

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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

V2 V3

P3

V4

P2

P1cut

V1

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General Testcase• 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 …)

V1

V2P2

…

P1

PN

c1

c2

ck

…

VM

…

P3

ClustersProcess conditions

Critical paths

eh,j bj,d

QM

Q3

Q2

Q1

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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

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Industrial Testcase

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

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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

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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

S3 S4 S1 S2 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 endend

S3 S4 S1 S2

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Experiment Results (3)• Test cost is reduced by 0 to 5%• Similar runtime complexity, O(M2N)

• DP-RP takes 10% more time than Greedy+

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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

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Acknowledgment• Professor Takashi Sato, Graduate School of Informatics,

Kyoto University.• Dr. Takumi Uezono, Integrated Research Institute, Tokyo

Institute of Technology.

Thank You

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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.