New Graph Bipartizations for New Graph Bipartizations for Double-Exposure, Double-Exposure, Bright Field Alternating Phase- Bright Field Alternating Phase- Shift Mask Layout Shift Mask Layout Andrew B. Kahng (UCSD) Andrew B. Kahng (UCSD) Shailesh Vaya (UCLA) Shailesh Vaya (UCLA) Alex Zelikovsky (GSU) Alex Zelikovsky (GSU)
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New Graph Bipartizations for Double-Exposure, Bright Field Alternating Phase-Shift Mask Layout Andrew B. Kahng (UCSD) Shailesh Vaya (UCLA) Alex Zelikovsky.
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New Graph Bipartizations for Double-Exposure,New Graph Bipartizations for Double-Exposure,
Bright Field Alternating Phase-Shift Mask LayoutBright Field Alternating Phase-Shift Mask Layout
Andrew B. Kahng (UCSD)Andrew B. Kahng (UCSD)
Shailesh Vaya (UCLA)Shailesh Vaya (UCLA)
Alex Zelikovsky (GSU)Alex Zelikovsky (GSU)
2ASPDAC’2001
OutlineOutline
Subwavelength lithographySubwavelength lithography Alternating PSMAlternating PSM Phase assignment problem Phase assignment problem Minimum perturbation problemMinimum perturbation problem Bipartizing feature graphBipartizing feature graph
Fast algorithm for edge-deletion Fast algorithm for edge-deletion Approximation algorithm node-deletionApproximation algorithm node-deletion
In general graphs In general graphs NP-hardNP-hardConstant-factor approximation Constant-factor approximation
In planar graphs In planar graphs T-join problem T-join problem can be solved efficiently:can be solved efficiently:reduction to min-weight matching O(nreduction to min-weight matching O(n33) (Hadlock) ) (Hadlock)
LP-based solution (Barahona) O(nLP-based solution (Barahona) O(n3/23/2logn)logn) no known implementation no known implementation
fast reduction to matching via gadgets O(fast reduction to matching via gadgets O(nn3/23/2 log n) log n)
23ASPDAC’2001
The T-join ProblemThe T-join Problem
How to delete How to delete minimum-costminimum-cost set of edges from set of edges from conflict graph G to eliminate odd cycles? conflict graph G to eliminate odd cycles?
Construct geometric dual graph D=dual(G)Construct geometric dual graph D=dual(G)Find odd-degree vertices T in DFind odd-degree vertices T in DSolve the Solve the T-join problemT-join problem in Din D::
find min-weight edge set J in D such thatfind min-weight edge set J in D such thatall T-vertices has all T-vertices has odd odd degreedegreeall other vertices have all other vertices have even even degreedegree
Solution J corresponds to desired min-cost edge Solution J corresponds to desired min-cost edge set in conflict graph Gset in conflict graph G
24ASPDAC’2001
T-join Problem: Reduction to MatchingT-join Problem: Reduction to Matching
Desirable properties of 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 solutionresults in a very fast solution
Solution: Solution: gadgetsgadgetsreplace 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 graphT-join in original graph
25ASPDAC’2001
T-join Problem: Reduction to MatchingT-join Problem: Reduction to Matching replace each vertex with a chain of trianglesreplace each vertex with a chain of trianglesone more edge for T-verticesone more edge for T-vertices in graph D: m = #edges, n = #vertices, t = #Tin graph D: m = #edges, n = #vertices, t = #T in gadgeted graph: 4m-2n-t vertices, 7m-5n-t edgesin gadgeted graph: 4m-2n-t vertices, 7m-5n-t edgescost of red edges = original dual edge costs cost of red edges = original dual edge costs
cost of (black) edges in triangles = 0cost of (black) edges in triangles = 0
vertex in T
vertex T
26ASPDAC’2001
Example of Gadgeted GraphExample of Gadgeted Graph
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
39ASPDAC’2001
Experiment SettingExperiment Setting
Compact layouts Compact layouts aggressivelyaggressively::design rule = between features should be design rule = between features should be single single shiftershifter
Find minimum # of modifications Find minimum # of modifications
to resolve all phase conflictsto resolve all phase conflicts
Two industrial benchmarks: Metal LayersTwo industrial benchmarks: Metal Layers# wires = # wires = 8622 (L1) and 8622 (L1) and 4539 (L2)4539 (L2)
# overlaps = # overlaps = 7805 (L1) and 7805 (L1) and 5439 (L2)5439 (L2)
40ASPDAC’2001
Experimental ResultsExperimental Results
BenchmarkAlgorithm Cost Ratio L1 L2
GVC 1.5 430 371 GW 1.5 267 225GVC 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 wideningcost of feature shifting
GW algorithm is 2 times better than Greedy Vertex Cover Algorithm Exact edge-deletion algorithm is better than GW for cost ratio > 2Exact edge-deletion algorithm is better than GW for cost ratio > 2
Runtime:Runtime: GVC is linear and very fastGVC is linear and very fast Exact edge-deletion is 2x faster than GW for benchmarksExact edge-deletion is 2x faster than GW for benchmarks
first formulation of the minimum perturbation problem for first formulation of the minimum perturbation problem for bright-field Alternating PSM technologybright-field Alternating PSM technology
unified approach for feature widening and shiftingunified approach for feature widening and shiftingoptimal solution for feature shifting and approximate optimal solution for feature shifting and approximate
solution when feature when both modifications are allowedsolution when feature when both modifications are allowed
Future work: develop a model for PSM in hierarchical Future work: develop a model for PSM in hierarchical designs:designs:standard cell overlappingstandard cell overlappingcomposability of standard cellscomposability of standard cells
multiple PSM-aware versions of master cellsmultiple PSM-aware versions of master cells
EXTRA SLIDESEXTRA SLIDES
43ASPDAC’2001
Standard-Cell PSM Standard-Cell PSM
Hierarchical layout vs flat layoutHierarchical layout vs flat layout
Free composability of standard cellsFree composability of standard cells
Cells may overlap: unique master cell causes area loss Cells may overlap: unique master cell causes area loss
Multiple PSM-aware versions of master cellMultiple PSM-aware versions of master cell
Taxonomy of ComposabilityTaxonomy of Composability
(Same)(Same) Same rowSame row composability: any cell can be placed composability: any cell can be placed immediately adjacent to any otherimmediately adjacent to any other
(Adj)(Adj) Adjacent rowAdjacent row composability: any two cells from composability: any two cells from adjacent rows are freely combined adjacent rows are freely combined
Four cases of cell libraries Four cases of cell libraries G=guaranteed composability, NG=not guaranteedG=guaranteed composability, NG=not guaranteedAdj-G/Same-G = free composabilityAdj-G/Same-G = free composabilityAdj-G/Same-NG Adj-G/Same-NG Adj-NG/Same-G Adj-NG/Same-G Adj-NG/Same-NGAdj-NG/Same-NG
45ASPDAC’2001
Taxonomy of ComposabilityTaxonomy of Composability
VDD
VDD
GND
VDD
VDD
GND
VDD
VDD
GND
Adj-G/Same-NG
Adj-NG/Same-G
Adj-NG/Same-NG
46ASPDAC’2001
Adj-G/Same-NGAdj-G/Same-NG
GIVEN: GIVEN:
order of cells in a row order of cells in a row
version compatibility matrix version compatibility matrix
FIND: FIND: version assignment version assignment
such that versions of adjacent cells are compatiblesuch that versions of adjacent cells are compatible
(BFS) traversal of DAG(BFS) traversal of DAGnodes = versionsnodes = versionsarcs = compatibilityarcs = compatibility
47ASPDAC’2001
Adj-G/Same-NGAdj-G/Same-NG
GIVEN: GIVEN:
order of cells in a row (or “optimal” placement)order of cells in a row (or “optimal” placement)