An Efficient Spectral Graph Sparsification Approach to Scalable Reduction of Large Flip-Chip Power Grids Xueqian Zhao (MTU) Zhuo Feng (MTU) Cheng Zhuo (Intel) Design Automation Group Authors: Department of Electrical & Computer Engineering Michigan Technological University
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An Efficient Spectral Graph Sparsification Approach to Scalable Reduction of Large Flip-Chip Power Grids
Department of Electrical & Computer EngineeringMichigan Technological University
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Large-Scale Power Grids Reductions
Motivations– Modern power delivery networks (PDNs) integrate huge number of
components
– Direct modeling and simulation of large PDNs can be very computationally expensive and even intractable
Challenges in large-scale PDNs modeling and reductions– Reductions of large-scale power grid with massive number of ports become
challenging due to fast growing computational complexity
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Prior PDN reductions methods– 1. Krylov-subspace based model order reduction methods via moment
matching ([1], [2])
– 2. Nodal elimination methods (TICER~[3], [4])
– 3. Multigrid-like reduction methods ([5], [6])
Problem formulation– DC and transient (TR) simulations:
Background & Existing Works
TR
DC
[1].P. Feldmann, et., “Sparse and efficient reduced order modeling of linear subcircuits with large number of terminals”, ICCAD,2004[2].P. Li, et., “Model order reduction of linear networks with massive ports via frequency-dependent port packing”, DAC, 2006[3].B.N. Sheehan, “Realizable reduction of RC networks”, IEEE TCAD, 2007[4].C.S. Amin, et., “Realizable RLCK circuit crunching”, DAC, 2003[5].H. Aca, et., “Power grid reduction based on algebraic mutigrid principles”, DAC, 2003[6].Y. Su, et., “AMOR: an efficient aggregating based model order reduction method for many-terminal interconnect circuits”, DAC, 2012
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Overview of Our Method
Keys steps of our proposed method– Divide-and-conquer power grid reduction and sparsification approach:
Spectral Graph Sparsification
Grid Partition&
Block Reduction
Original Power Grid
Model Stitching
Reduced Power Grid
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Graph Laplacian
Weighted graph A and its Laplacian matrix
1 2
4
3
5
1.52
2
1.51
0.5
3.5 1.5 21.5 4 2 0.5
2 3 10.5 1 3 1.5
2 1.5 3.5
− − − − − − − − − − − − −
1
2
4
3
5
1 2 43 5
( , )
( , ) if ( , )( , ) ( , ) if
otherwise0u v E
w u v u v EA u v w u v u v
∈
− ∈
= ==
∑Grounded resistance
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Block Power Grid Reduction
For a power grid with n nodes and m current sources
If we only keep the port nodes but eliminate all non-port nodes, a much smaller equivalent system can be obtained by Schur complement method
1 111 12 12( )T T
eqG G G L L G− −= −
22TG LL=where
port:
Smaller but much
denser
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An Efficient Port-Merging Scheme
Once the number of ports remains large, the resultant reduced grid can be still costly to use
Port b
Port a
Block i
Left Right
Back
Front
Port c
Port d
Port 1
Port 2
Block i
Port a’ Port dPort 1’
Original block and ports Merge ports based on effective resistance
– Only remove less importance edges– Spectral-based sparsification
– Approximate the spectral similarity (e.g. eigenvalue distributions) of original graph by a sparser graph with updated graph edges
Reduced block grid density– Stitched block grids after reduction can also be too expensive to use– Our target is to reduce the grid dimension while maintain the grid density
Reduce Block Grid Density
Reducedimension
(billion -> million)
Sparsify(reduce edges)
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Spectral Graph Sparsification
Sub-graph à is (1+ε)–spectral approximation of A if
( , , )A V E w=
A spectral sparsifier is a sub-graph of the original whose Laplacian quadratic form is approximately the same as that of the original graph on all real vector inputs [1]
Spectral sparsifier:
0ε >
[1]. D.A. Spielman and S. Teng, “Spectral sparsification of graphs,” CoRR, vol. abs/0808.4134, 2008.
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Spectral Graph Sparsification (cont.)
In practical applications, spectral graph sparsifier can be obtained by using sampling method:
Valid samples of each edge are determined by its probability pe
( )effe e Ap w R e=
How to:
0 pe 1
1. generate M random numbers between 0 ~ 12. if K of M have values < pe, the edge is selected K times3. the new weight of edge e in spectral graph:
' ee
e
KwwMp
=
samplesunselected samplesselected samples
ee
e
ppp
=∑
[1]
[1]. I. Koutis, G. L. Miller, and R. Peng, “A fast solver for a class of linear systems,” Commun. ACM, 2012
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Weighted degree metric– Provides trade-offs between the sparsity and the accuracy of
reduced power grid.
– The weighted degree of vertex v in a graph A is defined:
– In a mesh grid, 1 (1 critical edge) ≤ wd(v) ≤ 4 (4 evenly critical edges)
• Runtime analysis between complete power grid reductions and incremental reductions.
• In the incremental analysis, 10% blocks of each benchmark are modified for updated reductions
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Conclusion
Proposed an efficient spectral graph sparsification approach to scalable reduction of large flip-chip power grids
Key Ideas:– 1. Partition the original power grids into massive number of smaller
grid blocks
– 2. Reduce each block by merging outgoing ports and Schurcomplement method
– 3. Sparsify the reduced dense grid blocks using spectral graph sparsification scheme, and at last stitch them together to build the final reduced power grid model.
Our experimental results show that the proposed method can:
– Achieve up to 20X reductions w/o loss of much accuracy in both DC and transient analysis