1. New Graph Bipartizations for Double-Exposure, Bright Field Alternating Phase-Shift Mask Layout Andrew B. Kahng (UCSD) Shailesh Vaya (UCLA) Alex Zelikovsky (GSU)

2. 2 ASPDAC’2001 Outline  Subwavelength lithography  Alternating PSM  Phase assignment problem  Minimum perturbation problem  Bipartizing feature graph Fast algorithm for edge-deletion Fast algorithm for edge-deletion Approximation algorithm node-deletion Approximation algorithm node-deletion  Experimental results  Conclusions

3. 3 ASPDAC’2001 Subwavelength Gap since.35  m Subwavelength Optical Lithography Numerical Technologies, Inc.

4. 4 ASPDAC’2001 Alternating PSM for Subwavelength Technology conventional mask glass Chrome 0 Electric field at mask 0 Intensity at wafer phase shifting mask Phase shifter

5. 5 ASPDAC’2001 Double-Exposure Bright-Field PSM 0 180 +=

6. 6 ASPDAC’2001 The Phase Assignment Problem  Assign 0, 180 phase regions such that critical features with width < B are induced by adjacent phase regions with opposite phases 0180 <B shifters

7. 7 ASPDAC’2001 Phase Assignment for Bright-Field PSM  PROPER Phase Assignment: Opposite phases for opposite shifters Opposite phases for opposite shifters Same phase for overlapping shifters Same phase for overlapping shifters Overlapping shifters

8. 8 ASPDAC’2001 Key: Global 2-Colorability ? 180 0 0  Odd cycle of “phase implications”  layout cannot be manufactured layout verification becomes a global, not local, issue layout verification becomes a global, not local, issue

9. 9 ASPDAC’2001 F4 F2 F3 F1 Critical features: F1,F2,F3,F4

10. 10 ASPDAC’2001 F4 F2 F3 F1 Opposite-Phase Shifters (0,180)

11. 11 ASPDAC’2001 F4 F2 F3 F1 S1 S2 S3 S5 S4 S6 S7S8 Shifters: S1-S8 PROPER Phase Assignment: Opposite phases for opposite shifters Same phase for overlapping shifters

12. 12 ASPDAC’2001 F4 F2 F3 F1 S1 S2 S3 S5 S4 S6 S7S8 Phase Conflict Proper Phase Assignment is IMPOSSIBLE Phase Conflict

13. 13 ASPDAC’2001 F4 F2 F3 F1 S1 S2 S3 S5 S4 S6 S7S8 Phase Conflict feature shifting to remove overlap Conflict Resolution: Shifting Conflict Resolution: Shifting

14. 14 ASPDAC’2001 F4 F2 F1 S1 S2 S3S4 S7S8 Phase Conflict feature widening to turn conflict into non-conflict Conflict Resolution: Widening F3

15. 15 ASPDAC’2001 Minimum Perturbation Problem  Layout modifications feature shifting feature shifting feature widening feature widening  area increase, slowing down  area increase, slowing down  manual fixing, design cost increase  manual fixing, design cost increase  Minimum Perturbation Problem Find min # of layout modifications leading to proper phase assignment Find min # of layout modifications leading to proper phase assignment

16. 16 ASPDAC’2001 Feature Graph Feature Graph: Black - feature nodes Blue - shifter overlap Pink - extra nodes to distinguish opposite shifters

17. 17 ASPDAC’2001 Odd Cycles in Feature Graph Feature graph has ODD CYCLE Proper Phase Assignment IMPOSSIBLE

18. 18 ASPDAC’2001 Shifting in Feature Graph I feature shifting = delete EDGE of feature graph

19. 19 ASPDAC’2001 Shifting in Feature Graph II feature shifting = delete BLUE NODE of feature graph

20. 20 ASPDAC’2001 Widening in Feature Graph feature widening = delete BLACK NODE of feature graph

21. 21 ASPDAC’2001 Graph Bipartization  Proper phase assignment  Feature graph bipartite  Minimum Perturbation Problem  Graph Bipartization Problem  Graph Bipartization Problem  Layout modifications  Graph modifications feature shifting  edge deletion feature shifting  edge deletion feature widening  node deletion feature widening  node deletion both types with weights  node-weighted deletion both types with weights  node-weighted deletion

22. 22 ASPDAC’2001 Edge-Deletion Graph Bipartization  In general graphs NP-hard NP-hard Constant-factor approximation Constant-factor approximation  In planar graphs  T-join problem can be solved efficiently: reduction to min-weight matching O(n 3 ) (Hadlock) reduction to min-weight matching O(n 3 ) (Hadlock) LP-based solution (Barahona) O(n 3/2 logn) LP-based solution (Barahona) O(n 3/2 logn) t no known implementation fast reduction to matching via gadgets O(n 3/2 log n) fast reduction to matching via gadgets O(n 3/2 log n)

23. 23 ASPDAC’2001 The T-join Problem  How to delete minimum-cost set of edges from conflict graph G to eliminate odd cycles?  Construct geometric dual graph D=dual(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 find min-weight edge set J in D such that t all T-vertices has odd degree t all other vertices have even degree  Solution J corresponds to desired min-cost edge set in conflict graph G

24. 24 ASPDAC’2001 T-join Problem: Reduction to Matching  Desirable properties of reduction to matching: exact (i.e., optimal) exact (i.e., optimal) not much memory (say 2-3Xmore) not much memory (say 2-3Xmore) results in a very fast solution results in a very fast solution  Solution: gadgets replace each edge/vertex with gadgets s.t. replace each edge/vertex with gadgets s.t. matching all vertices in gadgeted graph matching all vertices in gadgeted graph  T-join in original graph  T-join in original graph

25. 25 ASPDAC’2001 T-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

26. 26 ASPDAC’2001 Example of Gadgeted Graph Dual Graph Gadgeted graph black + red edges == min-cost perfect matching

27. 27 ASPDAC’2001 Node-Deletion Graph Bipartization  Difficult for general graphs MAX SNP-hard  no very good approximation MAX SNP-hard  no very good approximation  For planar graphs Primal-dual algorithm (GW98) Primal-dual algorithm (GW98) t takes in account weights (to distinguish two modification types) t simple for implementation t t provably good: 9/4 approximation t quadratic runtime Greedy Vertex Cover Algorithm Greedy Vertex Cover Algorithm

28. 28 ASPDAC’2001 Primal-Dual Approximation Algorithm (GW) Input: Planar graph (with node weights) Input: Planar graph (with node weights) Output: Bipartite subgraph 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

29. 29 ASPDAC’2001 Greedy Vertex Cover Algorithm (GVC) Input: Planar graph (with node weights) Input: Planar graph (with node weights) Output: Bipartite subgraph 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) Wile there are violating edges do Delete node incident to maximum # of violating edges

30. 30 ASPDAC’2001 Experimental Results Benchmark Algorithm Cost Ratio L1 L2 GVC 1.5 430 371 GW 1.5 267 225 GVC 2.0 461 408 GW 2.0 306 263 GVC 3.0 475 438 GW 3.0 344 307 Edge-deletion 314 234 Cost Ratio = cost of feature widening cost of feature shifting

31. 31 ASPDAC’2001 Experimental Graph Bipartization  GW algorithm is 2-2.5 times better than Greedy Vertex Cover Algorithm 2-2.5 times better than Greedy Vertex Cover Algorithm quite good for small cost ratio quite good for small cost ratio may be very bad (>50% worse) for large cost ratio may be very bad (>50% worse) for large cost ratio  Exact edge deletion algorithm is better than GW for cost ratio > 2 better than GW for cost ratio > 2 faster than GW faster than GW

32. 32 ASPDAC’2001 Conclusions/Future Work  Contributions first formulation of the minimum perturbation problem for bright-field Alternating PSM technology first formulation of the minimum perturbation problem for bright-field Alternating PSM technology unified approach for feature widening and shifting unified approach for feature widening and shifting optimal solution for feature shifting and approximate solution when feature when both modifications are allowed 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

33. EXTRA SLIDES

34. 34 ASPDAC’2001 Standard-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

35. 35 ASPDAC’2001 Taxonomy 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 G=guaranteed composability, NG=not guaranteed Adj-G/Same-G = free composability Adj-G/Same-G = free composability Adj-G/Same-NG Adj-G/Same-NG Adj-NG/Same-G Adj-NG/Same-G Adj-NG/Same-NG Adj-NG/Same-NG

36. 36 ASPDAC’2001 Taxonomy of Composability VDD GND VDD GND VDD GND Adj-G/Same-NG Adj-NG/Same-G Adj-NG/Same-NG

37. 37 ASPDAC’2001 Adj-G/Same-NG GIVEN: order of cells in a row version compatibility matrix FIND: version assignment such that versions of adjacent cells are compatible such that versions of adjacent cells are compatible  (BFS) traversal of DAG nodes = versions nodes = versions arcs = compatibility arcs = compatibility

38. 38 ASPDAC’2001 Adj-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 either total # of extra sites or total/max displacement from optimal placement or total/max displacement from optimal placement  Dynamic Programming O(kV), k=max displacement Restricted DP Restricted DP

39. 39 ASPDAC’2001 T-join Problem in Sparse Graphs  Reduction to matching construct a complete graph T(G) construct a complete graph T(G) t vertices = T-vertices t edge costs = shortest-path cost find minimum-cost perfect matching find minimum-cost perfect matching  Typical example = sparse (not always planar) graph note that conflict graphs are sparse note that conflict graphs are sparse #vertices = 1,000,000 #vertices = 1,000,000 #edges  5  #vertices #edges  5  #vertices # T-vertices  10% of #vertices = 100,000 # T-vertices  10% of #vertices = 100,000  Drawback: finding all shortest paths too slow and memory-consuming #vertices = 100,000  #edges = 5,000,000,000 #vertices = 100,000  #edges = 5,000,000,000