an efficient linear time triple patterning solver haitong tian hongbo zhang zigang xiao martin d.f....
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An Efficient Linear Time Triple Patterning Solver
Haitong TianHongbo Zhang
Zigang XiaoMartin D.F. Wong
ASP-DAC’15
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Outline• Introduction• Preliminaries• An Optimal Algorithm• Hierarchical Approach• Experimental result• Conclusions
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Introduction• Triple patterning lithography (TPL) has
been recognized as one of the most promising techniques for 14/10nm technology node.
• The general TPL problem is a three coloring problem,which is a well-known NP-Complete problem.
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Introduction• The contributions of this paper are summarized
as follows:
• A TPL algorithm is proposed which essentially explores all solution space incorporating all legal stitch candidates, and compute a TPL decomposition with optimal number of stitches if one exists.
• A novel graph model is proposed to minimize the number of vertices in the solution graph.
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Preliminaries• A TPL algorithm targeting on standard cell based
designs is proposed in [10], which guarantees to find a solution if one exists for stitch-free designs.
• Problem Definition:• Given a standard cell based row structure layout
and a minimum coloring distance dmin, our objective is to find a legal triple patterning decomposition while minimizing the number of stitches.
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An Optimal Algorithm• Although the algorithm in [10] is able to find a
stitch-free decomposition if one exists, it may uses an excessive amount of runtime and memory than necessary.
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A Novel Graph Model
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Fast computation of cutting line
sets• Constructing such a graph model is expensive.
For each original cutting line set, we need to compute all its subsets and enumerate the number of legal TPL solutions for all the subsets.
• The key observation here is that to reduce the number of TPL solutions for a cutting line set.
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• Any two adjacent cutting line sets are merged together if one of them is a subset of the other.
• Lemma 1. For any feature pi which first appears in the cutting line set sj , all conflicting features of pi with smaller left boundaries are included in the cutting line set sj−1.
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Stitch candidatesThe approach of finding all legal TPL stitch candidates in [11] is embedded into our algorithm.
When stitches exist, the solution graph becomes a weighted graph, with the weight of an edge computed as the number of stitches needed for the connected vertices.
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Hierarchical Approach• To further speed up the algorithm, the solution
graphs of different types of cells can be computed and stored in a look up table.
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Experimental result• The algorithm is implemented in C++ and run on
a Linux server with 8GB RAM and a 3.0 GHZ CPU. The algorithm in [10] is also implemented to compare with our approach
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Conclusion• In this paper, we propose a linear time triple
patterning solver that guarantees to compute a TPL decomposition with optimal number of stitches if one exists.
• A fast approach is also proposed to achieve simultaneous memory and runtime reductions compared with state-of-the-art TPL algorithm.