在P2P (peer to peer) 系統中，空間資料 (spatial data) 發生在許多重要且不同的應用上，例如: P2P 虛擬城市 (virtual city)，地理資訊系統 (GIS)，土地發展計畫等。在P2P環境中，對於回答空間區域資料完全符合搜尋的問題，以 R-tree 為基礎的架構，也許是個不錯的選擇。然而，因為peer系統是動態的，因此，在 R-tree中，對於資料增減時會全域 (global) 更改的特色，使得無法在P2P系統中有效的作業。再者，R-tree中overlap的問題，造成大量的磁碟存取次數(在P2P系統中，則是被考慮成大量的訊息 (message) 傳遞次數)。因此，對於P2P系統來說，一個以P2PR-tree為索引基礎的方法被提出，當資料增減時，它只有區域 (local) 的索引架構被更改。對於資料增減時，雖然，P2PR-tree能達成區域性的更改目標，但是 overlapping 的現象仍然難以解決。近年來，對於空間資料的存取，一個勝過以R-tree為基礎的方法，NA-tree，已經被提出。再者，它沒有R-tree可能會發生overlap的問題。基本上，一個NA-tree並不對空間區域做分割，但是，它只是藉由某些規則對空間資料做分類。另一方面，Chord系統是一個著名的結構化 (structured) P2P系統。在Chord中，是以雜湊函數來對資料做搜尋，來取代在大部分非結構化 (unstructured) P2P系統中flooding的方式。因為，Chord是一個雜湊的方法，它能夠容易地處理資料增減時只做區域性的更改。然而，目前的Chord系統無法有效的處理區域資料，因為，它只處理單一key值。因此，在此篇論文中，我們提出在Chord系統中，使用在雜湊函數上，資料key值的部份，利用NA-tree對空間區域資料做編碼的方式做資料的搜尋。也就是說，我們仍然使用Chord中的一個雜湊函數來把節點配置給 peer，並且使用另一個雜湊函數，藉由 NA-tree對空間區域資料編碼的資料key值來配置資料。首先，我們用三個 bit來表示NA-tree中八種情況。接著，我們提出兩種方法來產生key值中剩餘的bit數。我們第一個提出的方法，是藉由加0的方式來產生剩餘的 bit數。這個方法簡單，是適用在P2P系統裡物件數量很少時。為了避免一個 peer 擁有太多物件，第二個方法把區域的中心點考慮進來。這個方法適用在P2P系統中有很多物件存在的情況。最後，我們把前三個 bit 和剩餘的 bit連接在一起，得到物件的key值。因此，我們結合了NA-tree和Chord系統，並解決P2PR-tree無法解決的overlapping問題。在我們模擬的研究中，我們使用六種不同的資料分佈來比較我們的方法以及P2PR-tree。從我們模擬的結果顯示出，在資料搜尋上，在我們的方法裡比P2PR-tree需要走訪較少的peer次數。

Spatial data occurs in several important and diverse applications in P2P systems, for example, P2P virtual cities, GIS, development planning, etc. For the problem of answering exact queries for spatial region data in the P2P environment, an R–tree based structure probably is a good choice. However, since a peer system is dynamic, the global update characteristics of data insertion/deletion in an R–tree can not work well in a P2P system. Moreover, the problem of overlaps in an R–tree results in large number of the disk accesses (which will be considered as large number of messages in P2P systems). Therefore, a P2PR–tree based indexing method for P2P systems has been proposed which has only local update to the proposed index structure when data insertion/deletion occurs. Although the P2PR–tree can achieve the goal of the local update for data insertion/deletion, the overlapping phenomenon is still hard to solve. Recently, for region data access, an NA–tree has been proposed which outperforms R–tree–like data structures. Moreover, it does not have the problem of overlaps which may occur in an R–tree. Basically, an NA–tree does not split the spatial space, but it just classifies the spatial data objects by some rules. On the other hand, the Chord system is a well–known structured P2P system in which the data search is performed by a hash function, instead of flooding used in most of the unstructured P2P system. Since the Chord system is a hash approach, it is easy to deal with data insertion/deletion with only local update. However, the current Chord system can not work well with the region data, since it only works well with a single key value. Therefore, in this thesis, we propose to apply an NA-tree in the Chord system to encode spatial region data in the data key part used in the hash function to data search. That is, we still use one hash function of Chord to assign nodes to peers, and use another hash function to do data assignment by applying an NA–tree to encode the spatial region data to data keys. First, we use three bits to present the first eight children in the NA–tree. Next, we propose two methods to generate the key value of the remaining bits. For our first proposed method, it generates the remaining bits by adding 0’s. This method is simple and applicable to the case that there are few objects in P2P systems. To avoid the case that a peer may own too many objects, the second method takes the central points of regions into consideration. This method is applicable to the case that there are too many objects in the P2P system. Finally, we concatenate the first three and the remaining bits to get the key values of objects. Thus, we combine the NA–tree with the Chord system to solve the overlapping problem which the P2PR–tree can not deal with. In our simulation study, we use six different data distributions to compare our method with the P2PR–tree. From our simulation results, we show that the number of visited peers in our approach is less than that in the P2PR–tree.

ABSTRACT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . i
LIST OF FIGURES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iv
LIST OF TABLES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vii
1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1 Peer–to–Peer Systems . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Searching Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.3 Multi–Dimensional Queries . . . . . . . . . . . . . . . . . . . . . . . . 7
1.4 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
1.5 Organization of the Thesis . . . . . . . . . . . . . . . . . . . . . . . . 13
2. A Survey . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
2.1 Unstructured P2P Systems . . . . . . . . . . . . . . . . . . . . . . . . 15
2.1.1 Napster . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
2.1.2 Gnutella . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
2.2 Structured P2P Systems . . . . . . . . . . . . . . . . . . . . . . . . . 17
2.3 The Feature–Based Signature . . . . . . . . . . . . . . . . . . . . . . 19
2.4 The Multi–Dimensional Query . . . . . . . . . . . . . . . . . . . . . . 21
2.5 Queries Based on the Space Filling Curve . . . . . . . . . . . . . . . . 25
3. An NA–Tree Approach . . . . . . . . . . . . . . . . . . . . . . . . . . 28
3.1 Data Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
3.2 The Insertion Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . 31
3.3 Answering the Spatial Exact Match Query . . . . . . . . . . . . . . . 44
4. Performance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
4.1 Simulation Study . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
4.2 Simulation Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
5. Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
5.1 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
5.2 Future Works . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
BIBLIOGRAPHY . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

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