由於近年來可攜式電子產品與網際網路的普及，許多書籍等紙本刊物逐漸地數位化或電子化。不同於舊有的紙本地圖，電子地圖應用了地理視覺化系統(Geographical visualization systems)提供了使用者互動性操作(User-interactive operations)的介面，因而發展出動態地圖標記問題(Dynamic map labeling problem)。而此問題為組合最佳化領域中著名的「最大化獨立矩形集合」(Maximum independent set of rectangles)的一種延伸問題。
本研究以理論演算法角度探討動態地圖標記問題：給定一組矩形標籤，在所有標籤需具有連續性以及與其他標籤彼此不可重疊的條件之下，求得所有標籤合適的顯示範圍(active range)，而此研究的目標為最大化所有標籤的顯示範圍之總和(sum of total active ranges)，故動態地圖標記問題又可稱為ARO問題(Active Range Optimization Problem, ARO problem)。在此，本研究探討兩種ARO問題模型，即簡化ARO問題(simple ARO problem)和一般化ARO問題(general ARO problem)。
依據上述模型，本研究提出了多個演算法以分別求解不同模型的ARO問題。在簡化ARO問題方面，本研究提出了3c log n倍近似比率的近似演算法以求解可隨著階層成比例縮放的統一寬度矩形金字塔標籤，其中，c為求解在平面時的標籤放置問題之近似演算法的近似比率；另提出O(log n log log n) 倍近似比率的近似演算法以求解可隨著階層成比例縮放的任意矩形金字塔標籤。在一般化ARO問題方面，本研究探討所有標籤皆具有相同距離顯示高度的限制下，提出了9倍近似比率的近似演算法求解常數縮放的任意正方形柱體標籤以及統一寬度的矩形柱體標籤；此外，本研究亦提出5倍近似比率的近似演算法，求解常數縮放的全等正方形柱體標籤。
關鍵詞：動態地圖標記；電子地圖；線上地圖；標籤放置問題

Electronic maps have been widely used in recent years, especially on portable devices. Such geographical visualization systems provide user-interactive operations such as continuous zooming. Thus, the interface provides to a new model in map labeling problems. Been et al. initiated the consistent dynamic map labeling problem whose objective is to maximize the sum of total visible ranges, each of which corresponds to the consistent interval of scales at which the label is visible; in other words, the aim is to maximize the number of consistent labels at every scale. This is a generalization of the well-known maximum independent set of rectangles problem in combinatorial optimization.
In contrast with the two-dimensional static map labeling problem, this three-dimensional dynamic map labeling problem can be considered a traditional map labeling by incorporating ‘scale’ as an additional dimension. During map zooming in and out, the labeling is a function of scale and area. In this study, we consider the dynamic map labeling problem. The goal of this problem is, given a set of rectangular labels on the map, to select visible ranges for all the labels such that no two consistent labels overlap at every scale. The objective is to maximize the sum of total visible ranges, i.e., so-called active ranges. The dynamic map labeling problem is also called the active range optimization (ARO) problem.
We consider both simple and general ARO problem, and several algorithms for approximating this problem are provided. For the simple ARO problem, we consider the case in which labels are proportional dilation, that is, labels increase and decrease proportionally when continuous zooming. Next, we provide a 3c log n-approximation algorithm for unit-width rectangular labels, where c is an approximation ratio for the label placement problem in the plane. Furthermore, we present an O(log n log log n)-approximation algorithm for arbitrary rectangular labels in a similar manner. For the general ARO problem, we consider the model in which labels are constant dilation and uniform active range height; that is, labels do not change their size when continuous zooming and labels have equal visible ranges. We contribute a 9-approximation algorithm for both arbitrary square labels and unit-width rectangular labels, and a 5-approximation algorithm for congruent square labels. We also show that the approximation analysis and proofs are tight.

摘要
Abstract
目錄
圖目錄
表目錄
第一章 緒論
1.1 研究背景與動機
1.2 各種地圖回顧
1.3 文獻回顧
第二章 問題定義與貢獻
2.1 問題定義
2.2 貢獻
第三章 簡化可顯示範圍問題之近似演算法
3.1 統一高度的矩形金字塔標籤之近似演算法與極端範例
3.1.1 統一高度的矩形金字塔標籤之近似演算法
3.1.2 統一高度的矩形金字塔標籤之極端範例
3.2 矩形金字塔標籤之近似演算法
第四章 一般化可顯示範圍問題之近似演算法
4.1 正方形柱體標籤之近似演算法與極端範例
4.1.1 正方形柱體標籤之近似演算法
4.1.2 正方形柱體標籤之極端範例
4.2 統一高度的矩形柱體標籤之近似演算法
4.3 全等正方形柱體標籤之近似演算法與極端範例
4.3.1 全等正方形柱體標籤之近似演算法
4.3.2 全等正方形柱體標籤之極端範例
第五章 結論與未來展望
5.1 結論
5.2 未來展望
參考文獻
附錄A
A.1 一維度之最多標籤數目集合
A.2 二維度之統一寬度矩形標籤放置問題
A.3二維度之任意矩形標籤放置問題
附錄B
B.1 3.1.1節近似演算法推論1之證明
B.1 3.2節近似演算法理論2之證明

[1] P. K. Agarwal, M. van Kreveld, S. Suri. Label placement by maximum independent set in rectangIes. Computational Geometry: Theory and Application, 11, 209-218, 1998.
[2] K. Been, E. Daiches, and C. Yap. Dynamic map labeling. IEEE Transactions on Visualization and Computer Graphics, 12(5):773-780, 2006.
[3] K. Been, M. No ̈llenburg, S.-H. Poon, and A. Wolff. Optimizing active ranges for consistent dynamic map labeling, Computational Geometry: Theory and Application, 43 (3): 312–328, 2010.
[4] M. A. Bekos, M. Kaufmann, E. Potika, and A. Symvonis. On multi-stack boundary labeling problems. In: Proc. 〖10〗^th WSEAS International Conference on COMPUTERS, Vouliagmeni (ICCOMP'06), 1209-1214, 2006.
[5] M. de Berg, O. Cheong, M. van Kreveld, M. Overmars, Computational Geometry: Algorithms and Applications, third ed., Springer-Verlag, Berlin, 2008.
[6] P. Berman, B. DasGupta, S. Muthukrishnan, S. Ramaswami, Efficient approximation algorithms for tiling and packing problems with rectangles, Journal of Algorithms, 41, 443–470. 2001.
[7] P. Chalermsook, J. Chuzhoy, Maximum independent set of rectangles, In: Proc. 〖20〗^th Annual ACM-SIAM Symposium on Discrete Algorithms (SODA’09), 892-901, 2009.
[8] T. M. Chan, Polynomial-time approximation schemes for packing and piercing fat objects, Journal of Algorithms, 46, 178–189, 2003.
[9] T. M. Chan, A note on maximum independent sets in rectangle intersection graphs, Information Processing Letters. 89(1): 19–23, 2004.
[10] B. Chazelle and 36 co-authors, The computational geometry impact task force report, Advances in Discrete and Computational Geometry, 223, American Mathematical Society, Providence, 223, 407–463, 1999
[11] S. Doddi, M.V. Marathe, A. Mirzaian, B.M.E. Moret, and B. Zhu, Map labeling and generalizations, in: Proc.8^thAnnual ACM-SIAM Symposium on Discrete Algorithms (SODA’97), 148–157, 1997.
[12] D. Ebner, G. W. Klau, and R. Weiskircher. Label number maximization in the slider model. In: Proc. 〖12〗^th International Symposium on Graph Drawing (GD 2004), Springer-Verlag, 3383, 144-154, 2005.
[13] M. Formann, F. Wagner, A packing problem with applications to lettering of maps, In: Proc. 7^thannual symposium on Computational geometry (SoCG’91), 281–288, 1991.
[14] M.A. Garrido, C. Iturriaga, A.Márquez, J.R. Portillo, P. Reyes and A. Wolff, “Labeling subway lines,” in: Proc. 〖12〗^th Annual International Symposium on Algorithms and Computation (ISAAC’01), Springer-Verlag, 2223, 649–659, 2001.
[15] A. Gemsa, M. No ̈llenburg, and I. Rutter. Sliding labels for dynamic point labeling, in: Proc. 〖23〗^th Canadian Conference on Computational Geometry. (CCCG’11), 205–210, 2011.
[16] Google Map,
(https://maps.google.com.tw/)
[17] D.S. Hochbaum, W. Maas, Approximation schemes for covering and packing problems in image processing and VLSI, Journal of the ACM (JACM), 32(1): 130–136, 1985.
[18] JR-East Japan Railway Company
(http://www.jreast.co.jp/tc/routemaps/index.html)
[19] G. W. Klau and P. Mutzel. Optimal labelling of point features in the slider model. in: Proc. 6^th Annual International Computing and Combinatorics Conference (COCOON 2000), Springer-Verlag, 1858, 340-350, 2000.
[20] D. T. Lee , Yu-Shin Chen and C.-S. Liao, “Labeling points on a single line,” International Journal of Computational Geometry and Applications (IJCGA), 15(3):261–277, 2005.
[21] S.-H. Poon, C.-S. Shin, Adaptive zooming in point set labeling, In: Proc. 〖15〗^th International Symposium on Fundamentals of Computation Theory (FCT 2005), Springer-Verlag, 3623, 233–244, 2005.
[22] T. Strijk and A. Wolff. Labeling points with circles. International Journal of Computational Geometry & Applications. 11(2): 181-195, 2001.
[23] T. Strijk and M. van Kreveld. Practical extensions of point labeling in the slider model. GeoInformatica, 6(2):181-197, 2002.
[24] Taiwan Administrative Region Map (http://taiwanarmap.moi.gov.tw/moi/run.htm)
[25] Taipei Metro Map
(http://www.trtc.com.tw/)
[26] F. Wagner, A. Wolff, V. Kapoor, and T. Strijk. Three rules suffice for good label placement. Algorithmica, 30(2): 334-349, 2001.
[27] A. Wolff, T. Strijk, The map-labeling bibliography, (http://illwww.ira.uka.de/maplabeling/bibliography).
[28] C.K. Yap, Open problem in dynamic map labeling, in: Proc. International Workshop On Combinatoral Algorithms (IWOCA’09), 2009.