經典的「前k短路徑問題」在導航系統和其他應用領域(如社群網路的推薦好友機制)中，一直扮演著重要的角色。在實際使用情境下，回傳的前k短路徑可能會十分相似。在這篇論文中，我們研究近年來被提出的前k短路徑問題的一個變形，稱為「前k短多樣路徑問題」。這個變形和原本的前k短路徑問題不同之處，在於我們將路徑的多樣性也納入考量。由於人們經常使用地標去表示自己的位置或指引方向，我們提出使用地標去區分不同路徑的方法。除此之外，這篇論文也展示在我們提出的模型下，如何為圖(Graph)設計一個省空間的索引，以降低尋找前k短多樣路徑的時間複雜度。實驗結果展示我們的模型相較傳統模型在效率上有所提升。

Classical top-k shortest paths (kSP) problem has played an important role in vehicle routing services as well as other application domains such as friend recommendation in social networks. Considering the real-world scenarios that the returning paths may be quite similar, in this thesis, we study a recent variant of the problem, top-k shortest paths with diversity (kSPD), which takes into account of the diversity of the $k$ shortest paths to be reported. As people usually use landmarks to locate their positions or to give directions of how to walk a path, we propose to use landmarks as a way to differentiate one path from the other. Furthermore, based on our landmark-based model, we show how to design a space-efficient index for the input graph, so that the time complexity of the $k$-SPD problem can be improved. Experimental results demonstrate
how this framework improves the efficiency over the traditional architecture.

1 Introduction 1
2 Related Work 5
2.1 Top-k shortest path problem...................... 5
2.2 Usage of Landmarks .......................... 6
3 Problem Definition 8
4 Algorithm 10
4.1 BaselineAlgorithm........................... 10
4.2 OurFramework:Indexing ....................... 11
4.3 OurFramework:QueryAlgorithm .................. 11
4.4 Performance Analysis.......................... 13
5 Evaluation 16
5.1 Experimental Setup........................... 16
5.2 Experimental Results.......................... 17
6 Conclusion 20

[1]Jin Y. Yen.Finding thekShortest Loopless Paths in a Network.ManagementScience, Volume 17, Issue 11, pages 712–716, 1971.

[2]David Eppstein.Finding thekShortest Paths.SIAM Journal on Computing(SICOMP), Volume 28, Issue 2, pages 652–673, 1997.

[3]Huiping Liu, Cheqing Jin, Bin Yang, and Aoying Zhou.Finding Top-kShortestPaths with Diversity.IEEE Transactions on Knowledge and Data Engineering(TKDE), Volume 30, Number 3, pages 488–502, 2017.

[4]Samir Khuller, Balaji Raghavachari, and Azriel Rosenfeld.Landmarks inGraphs.Discrete Applied Mathematics (DAM), Volume 70, Issue 3, pages 217–229, October 1996.

[5]Andrew V. Goldberg and Chris Harrelson.Computing the Shortest Path: A∗Search Meets Graph Theory.Proceedings of ACM-SIAM Symposium on Dis-crete Algorithms (SODA), pages 156–165, 2005.

[6]Daniel Delling and Dorothea Wagner.Landmark-Based Routing in DynamicGraphs.Proceedings of Workshop on Experimental Algorithms (WEA), pages52–65, 2007.

[7]Kevin Grant and David Mould.LPI: Approximating Shortest Paths using Land-marks.European Conference on Artificial Intelligence Workshop on AI andGames, 2008