在先進製程節點中，為了滿足各種設計需求，標準元件庫會由多重排高的標準元件所組成。在標準元件庫中，擁有多重排高的複雜標準元件通常能降低其面積。此外，在混合不同高度的列高下，也能達到更好的佈局密度與效能。考量混合列高下的多重排高元件之擺置大大提高了擺置合法化的難度。為了有效地使如此複雜的擺置合法化，我們提出了一個三階段的方法來解決混合列高佈局下的多重排高元件合法化之問題。首先，我們提出了一個基於戴克斯特拉演算法的方法來分散標準元件到擺置網格來緩和高密度區域的局部擁堵。接著，我們提出一個以窗口為基礎的方法來有效地插入標準元件至擺置區域。實驗結果顯示我們的方法能有效地使全域擺置的結果合法化且相較於[1]只在平均位移量和線長增加率擁有微小的增幅。

Standard cell libraries consisting of multiple-row-height cells have become popular in advanced process nodes to achieve various design demands. Multi-height cells are typically used to reduce the area of complex cells in a cell library. In addition, mixing different row heights may also be used for better design density
and performance. Mixed-cell-height placement with mixed row height awareness greatly increase the difficulty of placement legalization. To legalize such placement efficiently, we propose a three-stage approach to address the mixed-cell height legalization problem of mixed-row-height designs. First, we propose a method based on Dijkstra’s algorithm to spread the cells to placement bins which alleviate local congestion in high density regions. Then we propose a window-based method to insert cells into the placement region efficiently. Experimental results show that our approach can efficiently legalize global placement results with marginal quality loss compared to the work [1].

1 Introduction 1
1.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Previous Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.3 Our Contribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.4 Organization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
2 Preliminaries 5
2.1 Legalization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
2.2 Problem Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . 5
3 Proposed Approach 7
3.1 Overall Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
3.2 Preprocessing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
3.3 Cell Spreading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
3.3.1 Flow-based Cell Spreading . . . . . . . . . . . . . . . . . . . 9
3.3.1.1 Budgeting . . . . . . . . . . . . . . . . . . . . . . 9
3.3.1.2 Cell Moving . . . . . . . . . . . . . . . . . . . . . 11
3.3.1.3 Multi-Round Flow-based Cell Spreading . . . . . . 12
3.3.2 Dijkstra-based Cell Spreading . . . . . . . . . . . . . . . . . 12
3.3.2.1 Framework Overview . . . . . . . . . . . . . . . . 12
ii
3.3.2.2 Minimum Cost Path . . . . . . . . . . . . . . . . . 13
3.4 Window-based Legalization . . . . . . . . . . . . . . . . . . . . . . 16
3.4.1 Enumerating Insertion Range . . . . . . . . . . . . . . . . . 16
3.4.2 Evaluating Insertion Range . . . . . . . . . . . . . . . . . . 17
3.4.3 Time Complexity Analysis . . . . . . . . . . . . . . . . . . . 20
4 Experimental Results 21
4.1 Experiment Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
4.2 Comparisons of Different Cell Spreading Method . . . . . . . . . . 22
4.3 Effectiveness of Our Approach . . . . . . . . . . . . . . . . . . . . . 23
5 Conclusion 26
Bibliography 27

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