空間資料的查詢通常涉及到某種程度的範圍或區域，例如有關範圍交叉、區域包含和最近鄰居的查詢。最近鄰居查詢的類別中，有一種是連續最近鄰居(Continuous Nearest Neighbor)的查詢。舉例來說，人們想知道沿著高速公路，從起點到終點之間沿路最近的加油站在哪裡？由於在空間物件之間，其空間鄰近的特性沒有全序(total ordering)的性質，於是有空間填滿曲線(Space FillingCurve)這個方法被提出來保存空間鄰近的特性。基於空間填滿曲線，Chen 和Chang 提出了一些演算法來有效率地回答最近鄰居的查詢。所以，也許我們可以利用Chen 和Chang 的演算法，於集中式系統(centralized system)中，來執行一連串個別的最近鄰居查詢，來回答這種連續最近鄰居的查詢。然而，每一個最近鄰居查詢搜尋的範圍可能會發生部分重疊的現象，而且這些個別的查詢可能會存取到磁碟上數個相同的分頁，導致許多不必要的磁碟存取。在另一方面，針對無線廣播環境，Zheng 等人提出了一個基於Hilbert curve 理論的演算法，來解決連續最近鄰居的查詢。Zheng 等人的演算法分為兩個階段，在第一階段中，Zheng 等人設計一個搜索範圍來找到候選物件。在第二階段時，使用一些經驗法則去過濾候選物件來得到最後的答案。然而，Zheng 等人的演算法可能會檢查一些資料區塊兩次，或是檢查一些不需要的資料區塊，導致一些多餘的磁碟存取。在這篇論文中，針對避免Zheng 等人演算法中之缺點，我們提出了一個以Peanocurve 理論為基礎的局部擴張方法，於集中式系統中解決連續最近鄰居的查詢。在第一個階段中，我們決定出搜尋的範圍來取得所有的候選物件。基本上，我們先計算出從起點到終點之間會經過的資料區塊，然後從起點向終點方向一次移動一個區塊，並且局部地擴張搜尋範圍去尋找候選物件。在第二階段，我們使用在Zheng 等人的演算法中提出的經驗法則去過濾候選物件來得到最後的答案。根據這樣一個方法，我們提出兩個演算法：FM 和FM*演算法。這兩演算法不同之處在於其之於不同的假設而設計的。FM 演算法假設每一個物件都位於區塊的中心點，而FM*則假設物件可以在區塊中的任何一個位置。我們的局部擴張方法可以避免在Zheng 等人演算法中對於某些區域的重複檢查，並且決定出一個精確度比Zheng 等人的演算法更高的搜尋範圍。根據模擬的結果顯示，FM 或FM*演算法在精確度與處理時間方面，比起Zheng 等人的演算法有著更好的效能。

Queries on spatial data commonly concern a certain range or area, for example, queries related to intersections, containment and nearest neighbors. The Continuous Nearest Neighbor (CNN) query is one kind of the nearest neighbor queries. For example, people may want to know where those gas stations are along the super highway from the starting position to the ending position. Due to that there is no total ordering of spatial proximity among spatial objects, the space filling curve (SFC) approach has proposed to preserve the spatial locality. Chen and Chang have proposed efficient algorithms based on SFC to answer nearest neighbor queries, so we may perform a sequence of individually nearest neighbor queries to answer such a CNN query in the centralized system by one of Chen and Chang's algorithms. However, each searched range of these nearest neighbor queries could be overlapped, and these queries may access several same pages on the disk, resulting in many redundant disk accesses. On the other hand, Zheng et al. have proposed an algorithm based on the Hilbert curve for the CNN query for the wireless broadcast environment, and it contains two phases. In the first phase, Zheng et al.'s algorithm designs a searched range to find candidate objects. In the second phase, it uses some heuristics to filter the candidate objects for the final answer. However, Zheng et al.'s algorithm may check some data blocks twice or some useless data blocks, resulting in some redundant disk accesses. Therefore, in this thesis, to avoid these disadvantages in the first phase of Zheng et al.'s algorithm, we propose a local expansion approach based on the Peano curve for the CNN query in the centralized system. In the first phase, we determine the searched range to obtain all candidate objects. Basically, we first calculate the route between the starting point and the ending point. Then, we move forward one block from the starting point to the ending point, and locally spread the searched range to find the candidate objects. In the second phase, we use heuristics mentioned in Zheng et al.'s algorithm to filter the candidate objects for the final answer. Based on such an approach, we proposed two algorithms: the forward moving (FM) algorithm and the forward moving* (FM*) algorithm. The FM algorithm assumes that each object is in the center of a block, and the FM* algorithm assumes that each object could be in any place of a block. Our local expansion approach can avoid the duplicated check in Zheng et al.'s algorithm, and determine a searched range with higher accuracy than that of Zhenget al.'s algorithm. From our simulation results, we show that the performance of the FM or FM* algorithm is better than that of Zheng et al.'s algorithm, in terms of the accuracy and the processing time.

ABSTRACT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . i
LIST OF FIGURES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iv
LIST OF TABLES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ix
1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1 Spatial Databases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.2 Query Types . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.3 Continuous Nearest Neighbor . . . . . . . . . . . . . . . . . . . . . . 6
1.4 Motivations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
1.5 Organization of Thesis . . . . . . . . . . . . . . . . . . . . . . . . . . 14
2. A Survey . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
2.1 Neighbor Finding Based on the Peano Curve . . . . . . . . . . . . . . 16
2.1.1 Properties of the Peano curve . . . . . . . . . . . . . . . . . . 16
2.1.2 Nearest Neighbor-Finding . . . . . . . . . . . . . . . . . . . . 18
2.2 CNN Based on the R-tree Index . . . . . . . . . . . . . . . . . . . . . 21
2.3 CNN Based on the Hilbert Curve Index . . . . . . . . . . . . . . . . . 23
3. A Local Expansion Approach . . . . . . . . . . . . . . . . . . . . . . 27
3.1 The FM Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
3.2 The FM* Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
4. Performance Study . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
4.1 The Performance Model . . . . . . . . . . . . . . . . . . . . . . . . . 66
4.2 Simulation Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
4.2.1 The Uniform Distribution . . . . . . . . . . . . . . . . . . . . 69
4.2.2 The Diagonal Distribution . . . . . . . . . . . . . . . . . . . . 75
4.2.3 The Real Map . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
5. Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82
5.1 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82
5.2 Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83
BIBLIOGRAPHY . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84

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