Post on 02-Jan-2016
description
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}
10
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.