在現代超大型積體電路設計中，因應不斷縮小的製成設計，多重排大小的標準元件庫變得越來越普及。需要複雜電路、較高能源需求、需要較高面積實作的電路會被設計成行高較高的標準原件，反之將成為行高較矮的原件。同時，因為不同大小的元件，我們不再能夠將每一行各自的擺置合法化問題視為單一子問題，導致擺置合法化的問題變得更加複雜且面臨新的挑戰。在這篇論文中，我們提出一個根據有向無環圖設計的演算法，此演算法可以在固定標準元件的先後順序的情況下得到減少元件位移的最大值的最佳化結果，且能在線性的時間內完成。首先，我們採用一種類似網路流的方法，藉由移動元件位置，盡量減少設計中密度較高的區域。接著藉由將元件移動到可以放置的最左方的位置確認現行的擺放狀況是否可以有合法的結果，也就是不超出設計邊界和不與設計障礙物重疊，之後再使用我們提出的有向無環圖演算法。然而，我們的演算法會因為被多重行高標準元件圍住的無效空間影響，因此必須再應用我們的演算法後重新擺放一些標準元件以達到降低平均元件位移的效果。實驗數據顯示，和 [1] 比較，我們提出的演算法可以降低約60%的最大元件位移量且同時平均元件位移量能有差不多的數值。

In modern VLSI design, standard cells of different row heights become more and more common due to the aggressive scaling of advanced technologies. Such multiple-row-height cells allow cells which are complex, need more power and areas have larger heights while simple cells be shorter. Besides, it also leads to more complicated challenges for legalization problem because we cannot consider legalization problems of each row independently. In this thesis, we propose a DAG-based algorithm which is linear-time for multiple-row-height cells placement legalization with optimal maximum displacement when row assignment and cell ordering are fixed. However, not all of the design can get feasible solution when row assignment and cell ordering are fixed. Thus, we propose a framework to deal with such situation. First, we use a flow-like spreading method to avoid region of undesirably high density and check if the layout can be legalized by pushing each cell to the left most position then examining whether there is any cell overlapped with macros, blockages or out of right boundary before applying our DAG-based algorithm. However, the dead spaces caused by multiple row height cells can limit the solution quality. So, after applying DAG-based algorithm, we use a greedy reassignment method to replace the cells which can moved into dead spaces to reduce average cell displacement. Experimental results show that our proposed algorithm can reduce maximum cell displacement by 60% while having comparable average cell displacement compared to [1].

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