考慮顏色平衡的雙重曝光

Abstract

在Double Patterning Lithography(DPL) Layout Decomposition的問題裡，任兩個圖案的距離若小於最小著色間距(Minimum Coloring Spacing)，則此兩個圖案必須被分配成不同的顏色，表示它們是屬於不同次的曝光。以往的DPL演算法大多著重於最小化衝突(Conflict)與縫合(Stitch)的數量，然而本篇論文更進一步考慮到兩次曝光之間的面積平衡。針對此三個議題，本篇論文使用整數線性規劃(Integer Linear Programming)技巧解決之:最小化衝突與縫合的數量，也同時考慮顏色平衡(Color Balancing)。由於整數線性規劃的時間複雜度為指數成長(exponential growth)，我們提出不同的演算法來拆解衝突圖(conflict graph)，除了降低求解整數線性規劃的運算時間外，同時確保好的解能被保留下來。我們提出有效率的後續最佳化(post-optimization)演算法，能更進一步篩選出好的解。

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Previous works of DPL focus on the Previous works of DPL focus on the Previous works of DPL focus on the Previous works of DPL focus on the Previous works of DPL focus on the Previous works of DPL focus on the Previous works of DPL focus on the Previous works of DPL focus on the Previous works of DPL focus on the Previous works of DPL focus on the Previous works of DPL focus on the Previous works of DPL focus on the Previous works of DPL focus on the Previous works of DPL focus on the Previous works of DPL focus on the Previous works of DPL focus on the Previous works of DPL focus on the Previous works of DPL focus on the Previous works of DPL focus on the Previous works of DPL focus on the Previous works of DPL focus on the Previous works of DPL focus on the Previous works of DPL focus on the Previous works of DPL focus on the Previous works of DPL focus on the Previous works of DPL focus on the Previous works of DPL focus on the Previous works of DPL focus on the Previous works of DPL focus on the Previous works of DPL focus on the Previous works of DPL focus on the following two issues: following two issues: following two issues: following two issues: following two issues: following two issues: following two issues: following two issues: following two issues: following two issues: following two issues: following two issues: following two issues: following two issues: following two issues: following two issues: following two issues: following two issues: following two issues: following two issues: following two issues: one is conflict minimization and the other one is conflict minimization and the other one is conflict minimization and the other one is conflict minimization and the other one is conflict minimization and the other one is conflict minimization and the other one is conflict minimization and the other one is conflict minimization and the other one is conflict minimization and the other one is conflict minimization and the other one is conflict minimization and the other one is conflict minimization and the other one is conflict minimization and the other one is conflict minimization and the other one is conflict minimization and the other one is conflict minimization and the other one is conflict minimization and the other one is conflict minimization and the other one is conflict minimization and the other one is conflict minimization and the other one is conflict minimization and the other one is conflict minimization and the other one is conflict minimization and the other one is conflict minimization and the other one is conflict minimization and the other one is conflict minimization and the other one is conflict minimization and the other one is conflict minimization and the other one is conflict minimization and the other one is conflict minimization and the other one is conflict minimization and the other one is conflict minimization and the other one is conflict minimization and the other one is conflict minimization and the other one is conflict minimization and the other one one one is stitch minimizationstitch minimizationstitch minimizationstitch minimizationstitch minimization stitch minimizationstitch minimizationstitch minimizationstitch minimizationstitch minimizationstitch minimizationstitch minimizationstitch minimization stitch minimizationstitch minimizationstitch minimization. In this thesis, In this thesis, In this thesis, In this thesis, In this thesis, In this thesis, In this thesis, In this thesis, In this thesis, In this thesis, In this thesis, In this thesis, In this thesis, we model DPL layout decomposition e model DPL layout decomposition e model DPL layout decomposition e model DPL layout decomposition e model DPL layout decomposition e model DPL layout decomposition e model DPL layout decomposition e model DPL layout decomposition e model DPL layout decomposition e model DPL layout decomposition e model DPL layout decomposition e model DPL layout decomposition e model DPL layout decomposition e model DPL layout decomposition e model DPL layout decomposition e model DPL layout decomposition e model DPL layout decomposition e model DPL layout decomposition e model DPL layout decomposition e model DPL layout decomposition e model DPL layout decomposition e model DPL layout decomposition e model DPL layout decomposition e model DPL layout decomposition e model DPL layout decomposition e model DPL layout decomposition e model DPL layout decomposition e model DPL layout decomposition as Integer Linear Programming (ILP) problem. as Integer Linear Programming (ILP) problem.as Integer Linear Programming (ILP) problem. as Integer Linear Programming (ILP) problem.as Integer Linear Programming (ILP) problem.as Integer Linear Programming (ILP) problem.as Integer Linear Programming (ILP) problem. as Integer Linear Programming (ILP) problem.as Integer Linear Programming (ILP) problem.as Integer Linear Programming (ILP) problem.as Integer Linear Programming (ILP) problem. as Integer Linear Programming (ILP) problem. as Integer Linear Programming (ILP) problem.as Integer Linear Programming (ILP) problem. as Integer Linear Programming (ILP) problem.as Integer Linear Programming (ILP) problem.as Integer Linear Programming (ILP) problem.as Integer Linear Programming (ILP) problem.as Integer Linear Programming (ILP) problem.as Integer Linear Programming (ILP) problem. as Integer Linear Programming (ILP) problem. as Integer Linear Programming (ILP) problem.as Integer Linear Programming (ILP) problem. as Integer Linear Programming (ILP) problem.as Integer Linear Programming (ILP) problem.as Integer Linear Programming (ILP) problem. as Integer Linear Programming (ILP) problem. Our ILP formulations not Our ILP formulations not Our ILP formulations not Our ILP formulations not Our ILP formulations not Our ILP formulations not Our ILP formulations not Our ILP formulations not Our ILP formulations not Our ILP formulations not Our ILP formulations not Our ILP formulations not Our ILP formulations not Our ILP formulations not Our ILP formulations not Our ILP formulations not Our ILP formulations not Our ILP formulations not Our ILP formulations not Our ILP formulations not onlonlonly optimize an objective of number conflicts and stitches but also y optimize an objective of number conflicts and stitches but also y optimize an objective of number conflicts and stitches but also y optimize an objective of number conflicts and stitches but also y optimize an objective of number conflicts and stitches but also y optimize an objective of number conflicts and stitches but also y optimize an objective of number conflicts and stitches but also y optimize an objective of number conflicts and stitches but also y optimize an objective of number conflicts and stitches but also y optimize an objective of number conflicts and stitches but also y optimize an objective of number conflicts and stitches but also y optimize an objective of number conflicts and stitches but also y optimize an objective of number conflicts and stitches but also y optimize an objective of number conflicts and stitches but also y optimize an objective of number conflicts and stitches but also y optimize an objective of number conflicts and stitches but also y optimize an objective of number conflicts and stitches but also y optimize an objective of number conflicts and stitches but also y optimize an objective of number conflicts and stitches but also y optimize an objective of number conflicts and stitches but also y optimize an objective of number conflicts and stitches but also y optimize an objective of number conflicts and stitches but also y optimize an objective of number conflicts and stitches but also y optimize an objective of number conflicts and stitches but also y optimize an objective of number conflicts and stitches but also y optimize an objective of number conflicts and stitches but also y optimize an objective of number conflicts and stitches but also y optimize an objective of number conflicts and stitches but also y optimize an objective of number conflicts and stitches but also y optimize an objective of number conflicts and stitches but also y optimize an objective of number conflicts and stitches but also y optimize an objective of number conflicts and stitches but also y optimize an objective of number conflicts and stitches but also y optimize an objective of number conflicts and stitches but also y optimize an objective of number conflicts and stitches but also y optimize an objective of number conflicts and stitches but also y optimize an objective of number conflicts and stitches but also y optimize an objective of number conflicts and stitches but also y optimize an objective of number conflicts and stitches but also y optimize an objective of number conflicts and stitches but also y optimize an objective of number conflicts and stitches but also y optimize an objective of number conflicts and stitches but also y optimize an objective of number conflicts and stitches but also y optimize an objective of number conflicts and stitches but also y optimize an objective of number conflicts and stitches but also y optimize an objective of number conflicts and stitches but also y optimize an objective of number conflicts and stitches but also y optimize an objective of number conflicts and stitches but also y optimize an objective of number conflicts and stitches but also y optimize an objective of number conflicts and stitches but also y optimize an objective of number conflicts and stitches but also y optimize an objective of number conflicts and stitches but also y optimize an objective of number conflicts and stitches but also y optimize an objective of number conflicts and stitches but also y optimize an objective of number conflicts and stitches but also y optimize an objective of number conflicts and stitches but also y optimize an objective of number conflicts and stitches but also consider color balancing consider color balancingconsider color balancingconsider color balancingconsider color balancingconsider color balancing consider color balancing consider color balancingconsider color balancingconsider color balancing consider color balancingconsider color balancing consider color balancingconsider color balancingconsider color balancing consider color balancingconsider color balancing. 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solution. keeping good layout decompositions in the final solution. 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