Title: New Graph Bipartizations for Double-Exposure, Bright Field Alternating Phase-Shift Mask Layout
1New Graph Bipartizations for Double-Exposure,
Bright Field Alternating Phase-Shift Mask Layout
- Andrew B. Kahng (UCSD)
- Shailesh Vaya (UCLA)
- Alex Zelikovsky (GSU)
2Outline
- Subwavelength lithography
- Alternating PSM
- Phase assignment problem
- Minimum perturbation problem
- Bipartizing feature graph
- Fast algorithm for edge-deletion
- Approximation algorithm node-deletion
- Experimental results
- Conclusions
3Subwavelength Optical Lithography
Subwavelength Gap since .35 ?m
Numerical Technologies, Inc.
4Alternating PSM for Subwavelength Technology
5Double-Exposure Bright-Field PSM
6The Phase Assignment Problem
- Assign 0, 180 phase regions such that critical
features with width lt B are induced by adjacent
phase regions with opposite phases
0
180
ltB
7Phase Assignment for Bright-Field PSM
- PROPER Phase Assignment
- Opposite phases for opposite shifters
- Same phase for overlapping shifters
Overlapping shifters
8Key Global 2-Colorability
- Odd cycle of phase implications layout
cannot be manufactured - layout verification becomes a global, not local,
issue
?
180
0
180
0
180
180
9Critical features F1,F2,F3,F4
F2
F4
F1
F3
10F2
F4
F1
F3
Opposite-Phase Shifters (0,180)
11F2
S3
S4
F4
S7
S8
S1
F1
S2
F3
S5
S6
Shifters S1-S8
- PROPER Phase
Assignment - Opposite phases for opposite shifters
- Same phase for overlapping shifters
12Phase Conflict
F2
S3
S4
F4
S7
S8
S1
F1
S2
F3
S5
S6
Phase Conflict
Proper Phase Assignment is IMPOSSIBLE
13 Conflict Resolution Shifting
F2
S3
S4
F4
S7
S8
S1
F1
S2
F3
S5
S6
Phase Conflict
feature shifting to remove overlap
14Conflict Resolution Widening
F2
S3
S4
F4
S7
S8
S1
F1
S2
F3
Phase Conflict
feature widening to turn conflict into
non-conflict
15Minimum Perturbation Problem
- Layout modifications
- feature shifting
- feature widening
- ? area increase, slowing down
- ? manual fixing, design cost increase
- Minimum Perturbation Problem
- Find min of layout modifications leading to
proper phase assignment
16Feature Graph
17Odd Cycles in Feature Graph
Feature graph has ODD CYCLE
Proper Phase Assignment IMPOSSIBLE
18Shifting in Feature Graph I
19Shifting in Feature Graph II
20Widening in Feature Graph
21Graph Bipartization
- Proper phase assignment ? Feature graph bipartite
- Minimum Perturbation Problem
- ? Graph Bipartization Problem
- Layout modifications ? Graph modifications
- feature shifting ? edge deletion
- feature widening ? node deletion
- both types with weights ? node-weighted deletion
22Edge-Deletion Graph Bipartization
- In general graphs
- NP-hard
- Constant-factor approximation
- In planar graphs ? T-join problem
can be solved efficiently - reduction to min-weight matching O(n3) (Hadlock)
- LP-based solution (Barahona) O(n3/2logn)
- no known implementation
- fast reduction to matching via gadgets O(n3/2 log
n)
23The T-join Problem
- How to delete minimum-cost set of edges from
conflict graph G to eliminate odd cycles? - Construct geometric dual graph Ddual(G)
- Find odd-degree vertices T in D
- Solve the T-join problem in D
- find min-weight edge set J in D such that
- all T-vertices has odd degree
- all other vertices have even degree
- Solution J corresponds to desired min-cost edge
set in conflict graph G
24T-join Problem Reduction to Matching
- Desirable properties of reduction to matching
- exact (i.e., optimal)
- not much memory (say 2-3Xmore)
- results in a very fast solution
- Solution gadgets
- replace each edge/vertex with gadgets s.t.
- matching all vertices in gadgeted graph
- Û T-join in original graph
25T-join Problem Reduction to Matching
- replace each vertex with a chain of triangles
- one more edge for T-vertices
- in graph D m edges, n vertices, t T
- in gadgeted graph 4m-2n-t vertices, 7m-5n-t
edges - cost of red edges original dual edge costs
cost of (black) edges in
triangles 0
vertex in T
vertex ? T
26Example of Gadgeted Graph
Gadgeted graph
Dual Graph
black red edges min-cost perfect matching
27Node-Deletion Graph Bipartization
- Difficult for general graphs
- MAX SNP-hard ? no very good approximation
- For planar graphs
- Primal-dual algorithm (GW98)
- takes in account weights
(to distinguish two modification types) - simple for implementation
- provably good 9/4 approximation
- quadratic runtime
- Greedy Vertex Cover Algorithm
28Primal-Dual Approximation Algorithm (GW)
Input Planar graph (with node weights)
Output Bipartite subgraph
For each face F age(F)? 0 While there are odd
faces do for each odd face F age(F) ?
age(F)1 delete v with max weight (v) sum of
ages of faces with v for new face F age(F)?
0 In reverse order of node deletions do bring
node v back if an odd face appears, then
delete v permanently
29Example of GW Algorithm
30Example of GW Algorithm
31Example of GW Algorithm
1
1
3
2
3
1
2
2
2
0
1
0
1
32Example of GW Algorithm
2
2
5
4
5
2
4
4
4
1
2
0
2
33Example of GW Algorithm
2
2
5
4
5
2
4
4
4
1
2
0
2
34Example of GW Algorithm
2
2
5
4
2
4
4
4
1
2
0
2
35Example of GW Algorithm
2
2
5
5
2
6
6
5
1
3
0
3
36Example of GW Algorithm
2
2
5
5
2
6
6
5
1
3
0
3
37Example of GW Algorithm
2
2
5
5
2
6
5
1
3
0
3
38Greedy Vertex Cover Algorithm (GVC)
Input Planar graph (with node weights)
Output Bipartite subgraph
Color all nodes into 2 colors using BFS node
traversal Find the set T of all violating edges
(endpoints of the same color) Greedily cover
with vertices violating edges Wile there
are violating edges do Delete node incident to
maximum of violating edges
39Experiment Setting
- Compact layouts aggressively
- design rule between features should be single
shifter - Determine shifter overlaps
- Find minimum of modifications
to resolve all phase conflicts
- Two industrial benchmarks Metal Layers
- wires 8622 (L1) and 4539 (L2)
- overlaps 7805 (L1) and 5439 (L2)
40Experimental Results
Benchmark Algorithm Cost Ratio L1
L2
Edge-deletion 314
234
- GW algorithm is 2 times better than Greedy
Vertex Cover Algorithm
- Exact edge-deletion algorithm is better than GW
for cost ratio gt 2
- Runtime
- GVC is linear and very fast
- Exact edge-deletion is 2x faster than GW for
benchmarks
41Conclusions/Future Work
- Contributions
- first formulation of the minimum perturbation
problem for bright-field Alternating PSM
technology - unified approach for feature widening and
shifting - optimal solution for feature shifting and
approximate solution when feature when both
modifications are allowed
- Future work develop a model for PSM in
hierarchical designs - standard cell overlapping
- composability of standard cells
- multiple PSM-aware versions of master cells
42 43Standard-Cell PSM
- Hierarchical layout vs flat layout
- Free composability of standard cells
- Cells may overlap unique master cell causes
area loss - Multiple PSM-aware versions of master cell
- Version-composability matrix
44Taxonomy of Composability
- (Same) Same row composability any cell can be
placed immediately adjacent to any other - (Adj) Adjacent row composability any two
cells from adjacent rows are freely combined - Four cases of cell libraries
Gguaranteed composability, NGnot
guaranteed - Adj-G/Same-G free composability
- Adj-G/Same-NG
- Adj-NG/Same-G
- Adj-NG/Same-NG
45Taxonomy of Composability
46Adj-G/Same-NG
- GIVEN
- order of cells in a row
- version compatibility matrix
- FIND version assignment
- such that versions of adjacent cells are
compatible - (BFS) traversal of DAG
- nodes versions
- arcs compatibility
47Adj-G/Same-NG
- GIVEN
- order of cells in a row (or optimal placement)
- version compatibility weighted matrix (weight
extra sites) - FIND version assignment minimizing
- either total of extra sites
- or total/max displacement from optimal
placement - Dynamic Programming O(kV), kmax displacement
- Restricted DP