An Efficient Linear Time Triple Patterning Solver Haitong Tian Hongbo Zhang Zigang Xiao Martin D.F. Wong ASP-DAC’15

Outline Introduction Preliminaries An Optimal Algorithm Hierarchical Approach Experimental result Conclusions

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.

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.

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 d min, our objective is to find a legal triple patterning decomposition while minimizing the number of stitches.

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.

A Novel Graph Model

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.

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 s j, all conflicting features of pi with smaller left boundaries are included in the cutting line set s j−1.

Stitch candidates The 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.

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.

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

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.