Transcript (.ppt)

Improved Path Clustering for
Adaptive Path-Delay Testing
Tuck-Boon Chan* and Prof. Andrew B. Kahng*#
UC San Diego
ECE* & CSE# Departments
Adaptive Path-Delay Testing [ShintaniUT09]
• Test patterns are specific to process condition
• Select test pattern based on measured process
condition  reduced test cost!
Critical path sets
for various
process conditions
Critical paths
for process
condition Vj
ATPG
Test pattern
sets for various
process conditions
Test patterns
for process
condition Vj
Test pattern generation
Measure process
condition of a chip
Select a test pattern set
based on the measured
process condition
Path delay testing
Adaptive testing
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Clustering Example
• Process conditions {V1, V2 ,V3}
• Critical path sets {S1, S2, S3}
No clustering:
Test 40 paths per chip
S2
10
Clustering Solution A
C1
S1
S3
20
5
5
5
S3
20
10
C2
25
Test 35 paths if process condition = V1 or V2
Test 25 paths if process condition = V3
Clustering Solution B
S1
Venn diagrams of
critical path sets
S2
C1
S1
15
S2
10
S3
20
C2
5
Test 15 paths if process condition = V1
Test 35 paths if process condition = V2 or V3
3
Clustering for Min Expected Cost
Expected
testing cost:
S1
5
k
f (C)  
i 1
S2
10
S3
20
5

S j Ci
 
Qj 
S1
5
PhCi
| Ph |
S2
10
20
C1
Q1 = 0.2
Q2 = 0.5
Q3 = 0.3

S3
25
C2
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
• Objective : minimize f(C)
• Input
: V, Q and k
• Output : k disjoint clusters, C = {C1, C2, …, Ck}
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
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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
Greedy method
S1
S2 S3 S4
C1
Optimal
solution
1
N/2-2
1
1
N/2-2
1
C1
C2
1
N/2-2
1
1
1
N/2-2
1
When Q1= Q4 = 0.5- and Q2= Q3 = ,  ≈ 0
C1
1
N/2-2
1
C2
1
N/2-2
C3
1
N/2-2
1
N/2-2
1
1
C2
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Proposed Method I: KL-FM Analog
cut
V1
V2
P1
Random bipartition
V4
P3
V3
P2
• Model clustering problem as a
hypergraph
• Goal: partition the graph with
minimum cost
• Recursively partition a hypergraph
into two subgraphs
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
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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 …)
Critical paths
Process conditions
P1
V1
P2
V2
P3
PN
eh,j
Q2
Q3
QM
bj,d
c1
c2
…
…
…
VM
Q1
Clusters
ck
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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
9
Experiment Results (2)
• Greedy+ only merges adjacent clusters to avoid suboptimal
merging solutions
• FM method does not take advantage of correlation among
process conditions
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• Test cost : Greedy+ < FM < Greedy
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
S3
S3
S4
S4
S1
S3
S1
S3
S1
S3
S2
S4
S2
S4
S2
S4
S1
S1
S2
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
end
end
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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.
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Thank You
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.
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