圖形處理器(GPUs)由於其龐大的計算內核數量而被廣泛用於計算密集型應用。然而只有少數的演算法才能充分利用GPU的計算能力，因為它不僅需要足夠的並行度，而且還需要最細粒度的計算與數據存取之間的協同控制。
在此篇論文中，我們針對VLSI佈局密度計算提出了在GPU上的多掃描線（MSL）演算法，以此來計算佈局中組件的聯集面積。大多數組件的形狀都是矩形，MSL演算法將輸入佈局分層劃分為窗口(windows)，x軸範圍(slabs)和掃描線區域(sweep line regions)，以探索大程度的並行性。此外，為了克服GPU的記憶體限制，根據記憶體使用情況，通過抽樣算法進行評估將任務劃分為合適的大小。並且使用快速分段排序(fast segmented sort)，減少原子操作指令(atomic instruction)，負載平衡(load balance)等技術來進一步提高性能。透過實驗結果可以得知，我們在GPU GTX 1080 Ti上的實作可以比在CPU版本上加速大約75到160倍。

Graphics Processing Units (GPUs) have been widely used for computational intensive applications owing to its massive number of computing cores. However, only few algorithms can fully utilize the computational power of GPUs, because it requires not only enough degree of parallelism, but also the synergetic control between computation and data access in the finest granularity.
In this thesis, we proposed the Multi-Sweep-Line (MSL) algorithm on GPU for the VLSI layout density calculation, which computes the union area of components in a layout. The shapes of most components are rectilinear rectangles. The MSL algorithm divides the input layout hierarchically into windows, slabs, and sweep line regions to explore the large degree of parallelism. In addition, to overcome the memory limitation of GPU, tasks are partitioned into batches based on memory usage, estimated by sampling algorithms. Optimization techniques, fast segmented sort, reducing atomic instruction, load balance, are also applied to further improve the performance. The experimental results show that our MLS implementation can achieve 75 to nearly 160 times speedup over the CPU version on GTX 1080 ti.

Chinese Abstract......................i
Abstract.............................ii
Contents............................iii
List of Figures.......................v
List of Tables.....................viii
List of Algorithms....................x
1. Introduction.......................1
2. Background.........................3
3. Multi-Sweep-Line Algorithm........11
4. Experiments.......................26
5. Conclusion........................45

[1] Area of union of rectangles. https://www.hackerearth.com/zh/practice/basic-programming/implementation/basics-of-implementation/practice-problems/algorithm/area-of-union-of-rectangles/.
[2] Programming guide::Cuda toolkit documentation. https://docs.nvidia.com/cuda/cuda-c-programming-guide/index.html.
[3] Jon Louis Bentley and Thomas Ottmann. Algorithms for reporting and counting geometric intersections. IEEE Transactions on Computers, C-28:643–647,1979.
[4] Mark Harris, Shubhabrata Sengupta, and John Owens. Parallel prefix sum(scan) with cuda. In Hubert Nguyen, editor, GPU Gems 3, pages 851–876.Addison-Wesley, 2008.
[5] Kaixi Hou, Weifeng Liu, Hao Wang, and Wu chun Feng. Fast segmented sort on gpus. InICS, 2017.
[6] Andrei B. Khlopotine, Vikram Jandhyala, and Desmond Kirkpatrick. A variant of parallel plane sweep algorithm for multicore systems. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems, 32:966–970, 2013.[7] Shih-Hsiang Lo, Che-Rung Lee, I-Hsin Chung, and Yeh-Ching Chung. Optimizing pairwise box intersection checking on gpus for large-scale simulations. ACM Trans. Model. Comput. Simul., 23:19:1–19:22, 2013.
[8] Mark McKenney, Gabriel De Luna, Schiller Hill, and Logan Lowell. Geospatial overlay computation on the gpu. In Proceedings of the 19th ACM SIGSPATIAL International Conference on Advances in Geographic Information Systems, GIS’11, pages 473–476, New York, NY, USA, 2011. ACM.
[9] Mark McKenney and Tynan McGuire. A parallel plane sweep algorithm formulti-core systems. InGIS, 2009.
[10] Sushil K. Prasad, Michael McDermott, Xi He, and Satish Puri. Gpu-based parallel r-tree construction and querying. InProceedings of the 2015 IEEE Inter-national Parallel and Distributed Processing Symposium Workshop, IPDPSW’15, pages 618–627, Washington, DC, USA, 2015. IEEE Computer Society.
[11] Franco P. Preparata and Michael I. Shamos. Computational Geometry: AnIntroduction. Springer-Verlag New York, Inc., New York, NY, USA, 1985.
[12] Qiang Qiu, Xiaomin Zhu, Xiangzhen Yao, and Jinyun Fang. A parallel strategy for plane sweep algorithm in multi-core system.2013 IEEE International Geoscience and Remote Sensing Symposium - IGARSS, pages 3642–3645, 2013.
[13] Boris Schling. The Boost C++ Libraries. XML Press, 2011.
[14] Michael Ian Shamos and Dan Hoey. Geometric intersection problems.17thAnnual Symposium on Foundations of Computer Science (sfcs 1976), pages208–215, 1976.
[15] Lucanus J. Simonson and Gyuszi Suto. Geometry template library for stl-like2d operations. 2009.
[16] Peter van Oosterom. An r-tree based map-overlay algorithm. pages 318–327.1994.