在許多網路上，找出兩點間一條短的路徑往往是很重要的。在沒有任何預處理的情況下，要找出最短路徑至少需要 O(n + m) 的時間，而在現實世界的網路中，頂點數 n 和邊數 m 的數量級通常是相當大的，這樣的話，即使是 O(n + m) 的方法也可能需要數分鐘或數小時才能找出給定兩點間的一條最短路徑，但在現實中，能夠快速的計算出一條路徑的重要性往往遠大於計算出一條最短路徑(卻要花許多時間)的。在本篇論文中，我們研究如何能利用預處理來找出任意給定兩點間的一條路徑。實驗證明了我們提出的方法不但在時間效率上是相當高的，同時找到的路徑其長度也很接近最短路的長度。

In many networks, it is important to find a short path between two vertices. Without any preprocessing, it requires at least O(n+m) time to find a shortest path, consider the magnitude of n (vertices) and m (edges) in a real world graph are often so large, it could require minutes or even up to hours to find a shortest path. However, in reality, a quick computation to find a short-enough path is often more important than finding a shortest path. In this thesis, we propose two versions of algorithm that find a near-shortest path between any two given vertices based on a preprocessing stage. Experiments on two real world networks shown that the proposed algorithms each not only have good performances in terms of processing time, but also can find paths that the lengths are close to the lengths of shortest paths.

中文摘要i
Abstract ii
Contents iii
List of Figures v
List of Tables vi
1 Introduction 1
2 Network Model and Problem Definition 4
3 Preprocessing 6
3.1 Reachability Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
3.2 Overall ideas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
3.3 Preprocessing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
4 Algorithm 13
iii
4.1 Distance Estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
4.2 Finding a Short Path . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
4.3 Differences Between Our Algorithm and the A* Algorithm . . . . . . 22
5 Experiments 27
5.1 Experiments on the Network of ego-Twitter . . . . . . . . . . . . . . 28
5.1.1 Experiment 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
5.2 Experiments on the Network of ego-Gplus . . . . . . . . . . . . . . . 32
5.2.1 Experiment 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
6 Discussion 36
6.1 Unweighted Undirected Graphs . . . . . . . . . . . . . . . . . . . . . 36
6.2 Weighted Graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
7 Conclusion 40
Bibliography 41

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