本研究提出了新的處理大規模交通路網上的最短路徑規劃問題的方法。我們利用交通路網本身所具有的地理資訊，並結合quad-tree資料結構來將路網進行分群。為了減少進行路徑規劃時所需要的計算量，本研究在預處理階段即建構出適合進行最短路徑規劃的分群結構，同時，本研究進一步提出了調整結構的技巧來使得結構更適合進行路徑規劃。除此之外，我們提供了許多加速技巧來幫助本研究所提出的路徑規劃演算法，並且我們的演算法可以依據不同的資料結構設定來找出最短路徑計算時間以及所需的記憶體用量之間的好的平衡點，使得我們的演算法在實作上更具有彈性。本研究從實作中證實了所提出的路徑規劃演算法的良好成效，更具體的來說，在多達一千萬個節點以上的交通路網上，如歐洲與美國的交通路網，演算法僅需個位數毫秒以下的時間即可完成最短路徑的計算。

This study proposes a new approach for the shortest path query problem on continental-scale road networks. We exploit the inherent geographical information of road networks, and use the quad-tree data structure to partition input graphs. In order to reduce computational load of route planning, the clustering structure is constructed during the preprocessing stage; some fine-tune techniques are also applied to make the structure more effective and efficient for shortest path queries. Moreover, we provide several accelerated skills to speedup our routing algorithm based on the geographical information of the data structure, and the setting of the quad-tree data structure can be adjusted to find a good balance between query time and memory space consumption, which makes the algorithm more flexible for practical use. The empirical studies demonstrate the usefulness of our routing algorithm; in particular, each shortest path query can be answered under a few milliseconds on average in Europe and USA networks with more than ten million nodes.

[1] S. Arora. Nearly linear time approximation schemes for euclidean tsp and
other geometric problems. In Foundations of Computer Science, 1997. Pro-
ceedings., 38th Annual Symposium on, pages 554–563. IEEE, 1997.
[2] S. Arora. Polynomial time approximation schemes for euclidean traveling
salesman and other geometric problems. Journal of the ACM (JACM),
45(5):753–782, 1998.
[3] S. Arora, P. Raghavan, and S. Rao. Approximation schemes for euclidean
k-medians and related problems. In Proceedings of the thirtieth annual ACM
symposium on Theory of computing, pages 106–113. ACM, 1998.
[4] H. Bast, D. Delling, A. Goldberg, M. M¨uller-Hannemann, T. Pajor,
P. Sanders, D. Wagner, and R. F. Werneck. Route planning in transportation
networks. arXiv preprint arXiv:1504.05140, 2015.
[5] G. V. Batz, R. Geisberger, P. Sanders, and C. Vetter. Minimum time-
dependent travel times with contraction hierarchies. Journal of Experimental
Algorithmics (JEA), 18:1–4, 2013.
[6] D. Delling, A. V. Goldberg, T. Pajor, and R. F. Werneck. Customizable
route planning. In International Symposium on Experimental Algorithms,
pages 376–387. Springer, 2011.
[7] E. W. Dijkstra. A note on two problems in connexion with graphs. Nu-
merische mathematik, 1(1):269–271, 1959.
[8] R. A. Finkel and J. L. Bentley. Quad trees a data structure for retrieval on
composite keys. Acta informatica, 4(1):1–9, 1974.
[9] R. Geisberger, P. Sanders, D. Schultes, and C. Vetter. Exact routing in
large road networks using contraction hierarchies. Transportation Science,
46(3):388–404, 2012.
39[10] M. Hilger, E. K¨ohler, R. H. M¨ohring, and H. Schilling. Fast point-to-point
shortest path computations with arc-flags. The Shortest Path Problem: Ninth
DIMACS Implementation Challenge, 74:41–72, 2009.
[11] M. Holzer, F. Schulz, and D. Wagner. Engineering multilevel overlay graphs
for shortest-path queries. Journal of Experimental Algorithmics (JEA), 13:5,
2009.
[12] S. Jung and S. Pramanik. An efficient path computation model for hierarchi-
cally structured topographical road maps. Knowledge and Data Engineering,
IEEE Transactions on, 14(5):1029–1046, 2002.
[13] U. Lauther. An experimental evaluation of point-to-point shortest path cal-
culation on roadnetworks with precalculated edge-flags. The Shortest Path
Problem: Ninth DIMACS Implementation Challenge, 74:19–40, 2006.
[14] J. Maue, P. Sanders, and D. Matijevic. Goal-directed shortest-path queries
using precomputed cluster distances. Journal of Experimental Algorithmics
(JEA), 14:2, 2009.
[15] R. H. M¨ohring, H. Schilling, B. Sch¨utz, D. Wagner, and T. Willhalm. Par-
titioning graphs to speedup dijkstra’s algorithm. Journal of Experimental
Algorithmics (JEA), 11:2–8, 2007.
[16] F. Schulz, D. Wagner, and K. Weihe. Dijkstra’s algorithm on-line: an em-
pirical case study from public railroad transport. Journal of Experimental
Algorithmics (JEA), 5:12, 2000.
[17] D. Wagner, T. Willhalm, and C. Zaroliagis. Geometric containers for efficient
shortest-path computation. Journal of Experimental Algorithmics (JEA),
10:1–3, 2005.