傳統的標準元件庫包含了有著相同高度但具備不同功能的元件，統一的高度讓設計師可以將元件對準在排中的擺置點上，進而加速超大型積體電路的設計流程。但是，在先進製程中，元件被設計成具有不同的高度。有了不同的高度，我們可以將一些簡單的元件（例如反向器）設計成單排高，將複雜的元件（例如正反器）設計成多排高。然而，使用多重排高元件庫會使得擺置合法化變得更加困難。在這篇論文中，我們提出了一個針對多重排高標準元件庫的平行合法化方法來最小化總位移。實驗結果顯示我們的方法相較於目前最新的研究而言，在平均的位移上有18%的進步。

A traditional standard cell library consists of various functional cells with the same height, which could speed up VLSI design flow since designers could align cells to placement sites in rows. However, in advanced nodes, cells are designed with different heights. With mixed cell heights, we can design simple cells (e.g. inverters) as single-row-height cells, and complex cells (e.g. flip-flops) as multiple-row-height cells. Nevertheless, placement legalization becomes more difficult with mixed-cell-height libraries. In this thesis, we propose a parallel legalization method for mixed-cell-height standard cell libraries to minimize total displacement. Experimental results show that our method has a 18% improvement on average in displacement compared with the state-of-the-art work.

誌謝 v
Acknowledgements vii
摘要 ix
Abstract xi
1 Introduction 1
1.1 Mixed-Cell-Height Standard Cell Library . . . . . . . . . . . . . . . . 1
1.2 Standard Cell Placement . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.3 Previous Works . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.4 Organization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
2 Preliminaries 5
2.1 Legalization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
2.2 Power-Rail Alignment Constraint . . . . . . . . . . . . . . . . . . . . 5
2.3 Problem Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
3 Proposed Algorithm 7
3.1 Overall Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
3.2 Cell Spreading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
3.3 Parallel Legalization . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
3.4 LP-based Refinement . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
4 Experimental Results 17
5 Conclusion 21
References 23

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