在傳統的製程當中，通常以單一列高之標準元件庫設計一個電路。為了滿足時序限制，通常關鍵時序之元件會使用列高較高之標準元件庫，這是因為列高較高之元件有比較強的驅動能力，但其面積較大且功率耗損更多。然而，並非所有的元件皆為關鍵時序之元件，若是所有元件皆使用列高較高之標準元件，將會導致擺放晶片整體的面積增加，且增加功率耗損量。將較高與較矮之標準元件庫結合可以達到更好的效益及面積的優化；除此之外，若密度較高的區域之高或矮列資源被過度使用，則寬度較窄的兩倍列高之標準元件可提供幫助，兩倍列高之元件的高度為一個高列與一個矮列高之總和。雖然混合列高的設計有諸多優勢，但這也造成擺放問題之難度的提升。
在本論文中，我們提出一個方法將混合列高之設計進行元件擺放合法化。受到Abacus[1]的啟發，我們提出一個二次規劃方程式來定義我們的混合列高之元件擺放合法化問題，並且展示如何處理元件重疊的問題。為了要有效利用兩倍列高之元件所造成之空白區間，我們使單一列高之元件擺放至最近之空白區間以達到位移量減少之目的。最後，我們提出一個概念：元件可以換成其它版本，其列高及功能必須與之前的版本相同以降低位移量，且其驅動能力不可降低。實驗結果顯示：我們的程式可以在合理時間內執行完畢，而當我們導入最後提出之概念時，平均位移量與最大位移量皆有進一步地降低。此外，我們也展示導入混合列高之標準元件的效益。

In tradition, a single-row-height cell library is used to design a circuit. In order to meet the timing constraints, a cell library with taller row height is chosen for timing critical cells. That is, cells with taller row height have larger driving strength but larger area and power consumption. However, not all cells are timing critical, using taller cell-height versions for all cells will cause the waste in area and power consumption. The combination of taller cell-height and shorter cell-height cell libraries can provide a better performance and area co-optimization compared to designs with single-row-height cells. In addition, thinner double-row-height cells, whose heights are the sum of the heights of a short row and a tall row, can be helpful for dense local regions, where one kind of the resources, which is required by tall or short cells, is over used. Though there are some advantages with mixed-row-height designs, it turns out that the placement problem becomes much more complex.
In this thesis, we propose a method to legalize cells in mixed-row-height designs with the mixed-row-height feature. Inspired by Abacus [1], we first propose a quadratic program to model our mixed-row-height legalization problem, and illustrate how to handle overlapping cells. To well utilize the whitespaces caused by double-row-height cells, we place single-row-height cells into the nearest whitespaces if better displacements can be achieved. Finally, we propose a concept that cells can change to other versions with different cell heights but with the same function so that the cell displacements can be further reduced under the constraint that driving strengths cannot be weakened. The experimental results show that the running time of our legalization method is reasonable, and the average and maximal displacements are further reduced if the version change is performed. We also show the effectiveness of introducing double-row-height cells in mixed-row-height design with our legalization method.

Contents
誌謝 i
摘要 ii
Abstract iii
1 Introduction 1
1.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Previous Works . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.3 Our Contributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
2 Preliminaries 7
2.1 Problem Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
2.2 Algorithm Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
3 Details of Our Methodology 11
3.1 Review of Abacus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
3.2 Legalization Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
3.2.1 Quadratic Program . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
3.2.2 Cell Placement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
3.2.3 Version Change . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
3.2.4 Cell Insertion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
4 Experimental Results 25
4.1 Experiment Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
4.2 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
5 Conclusions 39
Bibliography 41

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