由於無線通訊、定位、行動計算技術的快速進步，行動服務已經在大型空間資料庫管理上成為實際且重要的應用。行動服務使用者可能會在一定範圍內的空間移動（例：一個國家）。他們通常會透過空間資料庫來進行 K 個最近鄰居 (KNN) 查詢以取得資料物件。行動服務的挑戰是要如何有效率地回報使用者有興趣的資料物件。KNN 查詢問題的其中一種是 K 個有排序最近鄰居動態查詢 (order-sensitive MkNN) 問題。在 K 個有排序最近鄰居動態查詢問題中，查詢點是動態且不可預知的，KNN 答案必須即時回報且依物件與查詢點的距離由近到遠排序。所以在 K 個有排序最近鄰居動態查詢問題中，如何有效率地、漸進地 (incrementally) 且準確地回報 KNN 答案是一個重要議題。Nutanong 學者提出了 V*-KNN 方法來進行 K 個有排序最近鄰居動態查詢。V*-KNN 方法使用他們的 V*-diagram 方法來產生安全區域 (safe region)。V*-KNN 方法還使用他們的漸進等級更新方法 (Incremental Rank Updates) 來處理當查詢點通過中垂線或安全區域邊界時所產生的事件。然而，V*-KNN 方法使用非漸進式的最佳優先方法 (Best-First KNN) 來取得 K 個最近鄰居。這使得搜尋時間會隨著物件密度上升。更何況，他們並沒有考慮多個物件在同一排序和多個事件在一步發生這兩種情況。這些情況都可能使 KNN 答案錯誤。因此，在這篇論文中，我們提出了一個基於 Hilbert 曲線的 K 個最近鄰居查詢方法 (Hilbert curve-based KNN，HC-KNN) 來進行 K 個有排序最近鄰居動態查詢。HC-KNN 能夠處理多個事件在一步發生的情況。我們還提出了新的 KNN 答案的資料結構。接著，我們提出了垂直平分線交點方法 (Intersection of Perpendicular Bisectors algorithm，IPB) 來處理 KNN 答案更新事件。IPB 還能夠處理多個物件在同一排序的情況。最後，我們基於 Hilbert 曲線索引提出 ONHC-KNN 方法來漸進地取得 NN 並且產生安全區域。此安全區域將不會被物件的密度影響，而且比 V*-KNN 產生的安全區域還大。從我們的實驗結果可以得知，我們的 HC-KNN 方法的效能比 V*-KNN 方法還好。

　　Due to wireless communication technologies, positioning technologies, and mobile computing develop quickly, mobile services are becoming practical and important on big spatiotemporal databases management. Mobile service users move only inside a spatial space, e:g: a country. They often issue the K Nearest Neighbor (kNN) query to obtain data objects reachable through the spatial database. The challenge problem of mobile services is how to efficiently answer the data objects which users interest to the corresponding mobile users. One type of kNN query problems is the order-sensitive moving kNN (order-sensitive MkNN) query problem. In the order-sensitive MkNN query problem, the query point is dynamic and unpredictable, the kNN answers should be responded in real time and sorted by the distance in the ascending order. Therefore, how to respond the kNN answers effectively, incrementally and correctly is an important issue. Nutanong et al: have proposed the V*-kNN algorithm to process the order-sensitive MkNN query. The V*-kNN algorithm uses their the V*-diagram algorithm to generate the safe region. It also uses the Incremental Rank Updates algorithm (IRU) to handle the events while the query point passing the bisectors or the boundary of the safe region. However, the V*-kNN algorithm uses the BF-kNN algorithm to retrieve NNs, which is non-incremental. This makes the search time increase while the density of the object increases. Moreover, they do not consider the situation that there are multiple objects at the same order, and the situation that there are multiple events happen in a single step. These situations may cause that the kNN answers are incorrect. Therefore, in this thesis, we propose the Hilbert curve-based kNN algorithm (HC-kNN) algorithm to process the ordersensitive MkNN query. The HC-kNN algorithm can handle the situation that there are multiple events happen in a single step. We also propose new data structure of the kNN answers. Next, we propose the Intersection of Perpendicular Bisectors algorithm (IPB) in order to handle order update events of the kNN answers. The IPB algorithm handles the situation which there are multiple objects at the same order. Finally, based on the Hilbert curve index, we propose the ONHC-kNN algorithm to get NNs incrementally and to generate the safe region. The safe region will not be affected while the density of the object increases. The safe region of our algorithm is larger than that of the V*-kNN algorithm. From our simulation result, we show that the HC-kNN algorithm provides better performance than the V*-kNN algorithm.

ABSTRACT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . i
LIST OF FIGURES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iv
LIST OF TABLES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viii
1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1 The Difficulty of Spatial Queries . . . . . . . . . . . . . . . . . . . . . 1
1.2 The Basic Concept of the KNN Query . . . . . . . . . . . . . . . . . 3
1.2.1 The Static KNN Query Problem . . . . . . . . . . . . . . . . . 3
1.2.2 The Dynamic KNN Query Problem . . . . . . . . . . . . . . . 5
1.2.3 The Order-Sensitive Moving KNN Query Problem . . . . . . . 7
1.2.4 The Related Work of the Order-Sensitive Moving KNN Query
Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
1.3 Motivations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
1.4 Organizations of the Thesis . . . . . . . . . . . . . . . . . . . . . . . 13
2. A Survey of Order-Sensitive K Nearest Neighbors Query Processing
Algorithms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
2.1 The Kth-Order Vorinoi Diagram . . . . . . . . . . . . . . . . . . . . . 14
2.2 The RIS-kNN Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . 16
2.3 The IRU Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
2.4 The V*-kNN Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . 19
2.4.1 The V*-Diagram Algorithm . . . . . . . . . . . . . . . . . . . 20
2.4.2 The V*-kNN algorithm . . . . . . . . . . . . . . . . . . . . . . 22
3. The HC-KNN Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . 27
ii
Page
3.1 Processing of the HC-KNN Algorithm . . . . . . . . . . . . . . . . . . 27
3.2 Modeling the Order-Sensitive KNN Answer . . . . . . . . . . . . . . . 31
3.2.1 The KnnList Node Data Structure . . . . . . . . . . . . . . . 31
3.2.2 The KnnList Data Structure and Rules to Generate the Perpendicular
Bisector Set . . . . . . . . . . . . . . . . . . . . . . 32
3.3 The IPB Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
3.3.1 The Problem of Updating KNN Answers of the V*-KNN Algorithm
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
3.3.2 Processing of the IPB Algorithm . . . . . . . . . . . . . . . . 33
3.4 The ONHC-KNN Algorithm and the Hilbert-Curve-Based Safe Region
Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
3.4.1 Introduction to the Hilbert Curve . . . . . . . . . . . . . . . . 43
3.4.2 The ONHC-KNN Algorithm . . . . . . . . . . . . . . . . . . . 44
3.4.3 The Safe Region of the HC-KNN Algorithm . . . . . . . . . . 60
4. Performance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
4.1 The Performance Model . . . . . . . . . . . . . . . . . . . . . . . . . 64
4.2 Simulation Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
4.2.1 The Effect of n . . . . . . . . . . . . . . . . . . . . . . . . . . 66
4.2.2 The Effect of k . . . . . . . . . . . . . . . . . . . . . . . . . . 68
4.2.3 The Effect of t . . . . . . . . . . . . . . . . . . . . . . . . . . 68
5. Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72
5.1 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72
5.2 Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
BIBLIOGRAPHY . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75

[1] H. L. Chen and Y. I. Chang, “All-Nearest-Neighbors Finding Based on the Hilbert Curve,” Expert Systems With Applications, Vol. 38, No. 6, pp. 7462–7475, Jun 2011.
[2] Y. Chen and J. M. Patel, “Efficient Evaluation of All-Nearest-Neighbor Queries,” Proc. of the 23rd Int. Conf. on Data Eng., pp. 1056–1065, 2007.
[3] M. de Berg, M. van Kreveld, and M. Overmars, “Computational Geometry: Algorithms and Applications,” Springer-Verlag, second edition, 2000.
[4] M. Hasan, M. A. Cheema, X. Lin, and Y. Zhang, “Efficient Construction of Safe Regions for Moving KNN Queries over Dynamic Datasets,” Proc. of the 11th Int. Conf. on Advances in Spatial and Temporal Databases, Vol. 5644, pp. 373–379, 2009.
[5] G. R. Hjaltason and H. Samet, “Ranking in Spatial Databases,” Proc. of Int. Symp. on Large Spatial Databases, Vol. 951, pp. 83–95, 1995.
[6] Y. L. Hsueh, R. Zimmermann, H. Wang, and W. S. Ku, “Partition-Based Lazy Updates for Continuous Queries over Moving Objects,” Proc. of the 15th Int. Symp. on Advances in Geographic Information Systems, pp. 1–8, 2007.
[7] X. Huang, C. S. Jensen, H. Lu, and S. Saltenis, “S-GRID: A Versatile Approach to Efficient Query Processing in Spatial Networks,” Proc. of the 10th Int. Conf. on Advances in Spatial and Temporal Databases, pp. 93–111, 2007.
[8] D. P. J. Zhang, N. Mamoulis and Y. Tao, “All-Nearest-Neighbors Queries in Spatial Databases,” Proc.of the 16th Int. Conf. on Scientific and Statistical Database Management, pp. 297–306, 2004.
[9] D. V. Kalashnikov, S. Prabhakar, W. G. Aref, and S. E. Hambrusch, “Efficient Evaluation of Continuous Range Queries on Moving Objects,” Proc. of the 13th Int. Conf. on Database and Expert Systems Applications, pp. 731–740, 2002.
[10] D. V. Kalashnikov, S. Prabhakar, and S. E. Hambrusch, “Main Memory Evaluation of Monitoring Queries on Moving Objects,” Distributed and Parallel Databases, Vol. 15, No. 2, pp. 117–135, Oct 2004.
[11] L. Kulik and E. Tanin, “Incremental Rank Updates for Moving Query Points,” Proc. of the 4th Int. Conf. on Geographic Information Science, pp. 251–268, 2006.
[12] C. Li, Y. Gu, F. L. Li, and M. Chen, “Moving K-Nearest Neighbor Query over Obstructed Regions,” Proc. of 12th Int. Asia-Pacific Web Conf., pp. 29–35, 2010.
[13] S. Nutanong, R. Zhang, E. Tanin, and L. Kulik, “The V*-Diagram: A Query Dependent Approach to Moving KNN Queries,” Proc. of the VLDB Endowment, Vol. 1, No. 1, pp. 1095–1106, 2008.
[14] S. Nutanong, R. Zhang, E. Tanin, and L. Kulik, “Analysis and Evaluation of V*-KNN: An Efficient Algorithm,” The Int. Journal on Very Large Data Bases, Vol. 19, No. 3, pp. 307–332, Jun. 2010.
[15] A. Okabe, B. Boots, K. Sugihara, and D. S. N. Chiu, “Spatial Tessellations: Concepts and Applications of Voronoi Diagrams,” John Wiley and Sons, second edition, pp. 291–410, 1992.
[16] K. Patroumpas, T. Minogiannis, and T. Sellis, “Approximate Order-K Voronoi Cells over Positional Streams,” Proc. of The 15th Annual ACM Int. Symp. on Advances in Geographic Information Systems, pp. 1–8, 2007.
[17] N. Roussopoulos, S. Kelley, and F. Vincent, “Nearest Neighbor Queries,” Proc. of ACM SIGMOD Int. Conf. on Management of Data, pp. 71–79, 1995.
[18] Z. Song and N. Roussopoulos, “K-Nearest Neighbor Search for Moving Query Point,” Proc. of the 7th Int. Symp. on Advances in Spatial and Temporal Databases, pp. 79–96, 2001.
[19] Y. Tao and D. Papadias, “Time-Parameterized Queries in Spatiotemporal Databases,” Proc. of ACM SIGMOD Int. Conf. on Management of Data, pp. 334–345, 2002.
[20] Y. Tao, D. Papadias, and Q. Shen, “Continuous Nearest Neighbor Search,” Proc. of the 28th Int. Conf. on Very Large Data Bases, pp. 287–298, 2002.
[21] H.Wang, R. Zimmermann, and W. S. Ku, “Distributed Continuous Range Query Processing on Moving Objects,” Proc. of the 17th Int. Conf. on Database and Expert Applications, pp. 655–665, 2006.
[22] J. Zhang, M. Zhu, D. Papadias, Y. Tao, and D. L. Lee, “Location-Based Spatial Queries,” Proc. of ACM SIGMOD Int. Conf. on Management of Data, pp. 443–454, 2003.
[23] B. Zheng, W. C. Lee, and D. L. Lee, “Search Continuous Nearest Neighbors on the Air,” Proc. of the 1st Annual Int. Conf. on Mobile and Ubiquitous Systems: Networking and Services, pp. 236–245, 2004.