近十幾年來，圖形是在電腦科學裡被運用最廣泛的，它能夠透過圖像化來簡化並呈現原本複雜的資訊。對於圖形繪製演算法而言，力導向演算法是比較常用的方法，因為它很容易實作，並且通常能得到好的繪製結果，如圖形都有一致的邊長，頂點也能均勻分佈等等。

本篇論文研究的問題是基於力導向演算法，並且降低其運算所花的時間。當頂點數量大於150時，傳統的力導向演算法會有著很高的運行時間。我們提出了改善方法，其概念為事先把原本的圖形分成若干個叢集，使其頂點數量便能降低。除此之外，我們為了能運用在叢集圖形上，修改了傳統力導向演算法中計算力大小的方程式。另外，對於叢集圖形我們建立了一套模組，使它能夠運行在我們所提出的修改力導向演算法。再者，我們也定義了評量圖形品質好壞的一套標準。最後，我們透過了實作與實驗方法來評測我們所提出的演算法，由實驗結果得知，我們不但有效的減少了力導向演算法的運行時間，所繪製出來的圖形也保留了傳統力導向演算法的性質。

Graph drawing can simplify complex information of graphs through visualization. One of commonly used graph drawing methods is Force-directed algorithm (FDA) because of its easy of implementation and good quality of uniform edge length, uniform vertex distribution, etc. However, classical FDA has a high running time when the number of vertices is large. In this thesis, we present a modified FDA algorithm which can reduce the running time of FDA. The major idea is to partition the graph into clusters, to run FDA of each clusters in parallel, and to merge the results in the end. We evaluated our method on several data sets, and the experimental results show the proposed method can efficiently reduce the running time without scarifying the quality of the classical FDA.

1 Introduction 1
2 Preliminary and Related Works 4
2.1 Graph . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
2.2 Clustered Graph . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
2.3 Drawing of a Graph . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
2.4 Force-directed Graph Layout . . . . . . . . . . . . . . . . . . . . . . . . . 5
2.5 Graph Drawing Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
3 The Proposed Method 8
3.1 part 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
3.2 part 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
3.2.1 Spring Force . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
3.2.2 Repulsive Force . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
3.2.3 Modified FDA . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
3.3 part 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
4 Experiments and Results 13
4.1 Test Data for Experiment . . . . . . . . . . . . . . . . . . . . . . . . . . 13
4.2 Experimental Setting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
4.3 Experimental Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
4.4 Sample Outputs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
4.4.1 Sample Layout . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
5 Conclusion and Future work 23

[1] http://math.nist.gov/matrixmarket/.
[2] http://neo4j.com/blog/social-networks-in-the-database-using-a-graph-database/.
[3] https://code.google.com/archive/p/linloglayout/.
[4] http://www.pinterest.com/pin/464011567831173796/.
[5] Giuseppe Di Battista, Peter Eades, Roberto Tamassia, and Ioannis G. Tollis. Graph
Drawing: Algorithms for the Visualization of Graphs. Prentice Hall PTR, Upper
Saddle River, NJ, USA, 1st edition, 1998.
[6] François Bertault. A force-directed algorithm that preserves edge-crossing properties.
Information Processing Letters, 74(1):7–13, 2000.
[7] P. EADES. A heuristics for graph drawing. Congressus Numerantium, 42:146–160,
1984.
[8] Peter Eades and Mao Lin Huang. Navigating clustered graphs using force-directed
methods. J. Graph Algorithms Appl., 4(3):157–181, 2000.
[9] Thomas MJ Fruchterman and Edward M Reingold. Graph drawing by force-directed
placement. Software: Practice and experience, 21(11):1129–1164, 1991.
[10] Ivan Herman, Guy Melançon, and M. Scott Marshall. Graph visualization and navigation
in information visualization: A survey. IEEE Transactions on Visualization
and Computer Graphics, 6(1):24–43, January 2000.
[11] Stephen G. Kobourov. Force-Directed Drawing Algorithms. 2004.
[12] Stephen G. Kobourov. Spring embedders and force directed graph drawing algorithms.
CoRR, abs/1201.3011, 2012.
[13] Andreas Noack. Energy models for graph clustering. J. Graph Algorithms Appl.,
11(2):453–480, 2007.
[14] Andre Suslik Spritzer and Carla Maria Dal Sasso Freitas. Navigation and interaction
in graph visualizations. Revista de informática teórica e aplicada. Porto Alegre. Vol.
15, n. 1 (2008), p. 111-136, 2008.
[15] Kozo Sugiyama and Kazuo Misue. Graph drawing by the magnetic spring model.
Journal of Visual Languages and Computing, 6(3):217¡V237, 1955.
[16] William T Tutte. How to draw a graph. Proc. London Math. Soc, 13(3):743–768,
1963.