Green, Smart and Connected Transportation Systems pp 141-160 | Cite as

# Dynamic Programming Approaches for Solving Shortest Path Problem in Transportation: Comparison and Application

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## Abstract

This paper seeks to investigate the performance of two different dynamic programming approaches for shortest path problem of transportation road network in different context, including the Bellman’s dynamic programming approach and the Dijkstra’s algorithm. The procedures to implement the two algorithms are discussed in detail in this study. The application of the Bellman’s approach shows that it is computationally expensive due to a lot of repetitive calculations. In comparison, the Dijkstra’s algorithm can effectively improve the computational efficiency of the backward dynamic programming approach. According to whether the shortest path from the node to the original node has been found, the Dijkstra’s algorithm marked the node with permanent label and temporal label. In each step, it simultaneously updates both the permanent label and temporal label to avoid the repetitive calculations in the backward dynamic programming approach. In addition, we also presented an algorithm using dynamic programming theory to solve the K shortest path problem. The K shortest path algorithm is particular useful to find the possible paths for travelers in real-world. The computational performance of the three approaches in large network is explored. This study will be useful for transportation engineers to choose the approaches to solve the shortest path problem for different needs.

## Notes

### Acknowledgements and Conflict of Interest Statement

This research is supported by the Natural Science Foundation of Zhejiang province (LQ17E080007), National Natural Science Foundation of China (71501009) and (51408323) and Sponsored by K. C. Wong Magna Fund in Ningbo University. The authors are very grateful to the anonymous reviewers for their valuable suggestions and comments. The author(s) declare(s) that there is no conflict of interest regarding the publication of this paper.

## References

- 1.Aljazzar H, Leue SK (2011) A heuristic search algorithm for finding the k shortest paths. Artif Intell 175(18):2129–2154MathSciNetCrossRefGoogle Scholar
- 2.Bauer R, Delling D (2009) SHARC: fast and robust unidirectional routing. ACM J Exp Algorithmics 14(2–4):1–29MathSciNetzbMATHGoogle Scholar
- 3.Bellman R (1958) On a routing problem. Q Appl Math 16:87–90CrossRefGoogle Scholar
- 4.Byers TH, Waterman MS (1984) Technical note—determining all optimal and near-optimal solutions when solving shortest path problems by dynamic programming. Oper Res 32(6):1381–1384Google Scholar
- 5.Delling D, Goldberg AV, Pajor T, Werneck RF (2013) Customizable route planning in road networks. Announced at SEA 2011 and SEA 2013, full version available online at http://research.microsoft.com/pubs/198358/crp_web_130724.pdf
- 6.Denardo EV, Fox BL (1979) Shortest-route methods: 1. Reaching, pruning, and buckets. Oper Res 27:161–186MathSciNetCrossRefGoogle Scholar
- 7.Dijkstra EW (1959) A note on two problems in connexion with graphs. Numer Math 1(1):269–271MathSciNetCrossRefGoogle Scholar
- 8.Eppstein D (1998) Finding the k shortest paths. In: 35th IEEE symposium. SIAM J Comput 28(2):652–673Google Scholar
- 9.Feillet D, Dejax P, Gendreau M, Gueguen C (2004) An exact algorithm for the elementary shortest path problem with resource constraints: application to some vehicle routing problems. Networks 44:216–229MathSciNetCrossRefGoogle Scholar
- 10.Floyd Robert W (1962) Algorithm 97: shortest path. Commun ACM 5(6):345CrossRefGoogle Scholar
- 11.Goldberg AV (2001) Shortest path algorithms: engineering aspects. In: ISAAC 2001, pp 502–513Google Scholar
- 12.Güner AR, Murat A, Chinnam RB (2009) Dynamic routing using real-time ITS information. Intelligent Vehicle Controls and Intelligent Transportation Systems—Proceedings of the 3rd International WorkshopGoogle Scholar
- 13.Güner AR, Murat A, Chinnam RB (2012) Dynamic routing under recurrent and non-recurrent congestion using real-time ITS information. Comput Oper Res 39(2):358–373CrossRefGoogle Scholar
- 14.Hart PE, Nilsson NJ, Raphael B (1968) A formal basis for the heuristic determination of minimum cost paths. IEEE Trans Syst Sci Cybern SSC4 4(2):100–107Google Scholar
- 15.Holzer M, Schulz F, Wagner D (2008) Engineering multi-level overlay graphs for shortest-path queries. ACM JEA 13(2.5):1–26Google Scholar
- 16.Maue J, Sanders P, Matijevic D (2006) Goal-directed shortest-path queries using precomputed cluster distances. ACM J Exp Algorithmics 4007:316–327CrossRefGoogle Scholar
- 17.Maue J, Sanders P, Matijevic D (2010) Goal-directed shortest-path queries using precomputed cluster distances. J Exp Algorithmics 14:2.3.2–2.3.27Google Scholar
- 18.Pettie S (2004) A new approach to all-pairs shortest paths on real-weighted graphs. Theoret Comput Sci 312:47–74MathSciNetCrossRefGoogle Scholar
- 19.Schulz F, Wagner D, Weihe K (2000) Dijkstra’s algorithm on-line: an empirical case study from public railroad transport. ACM JEA 5(12):1–23MathSciNetzbMATHGoogle Scholar
- 20.Seidel R (1995) On the all-pairs-shortest-path problem in unweighted undirected graphs. J Comput Syst Sci 51(3):400–403Google Scholar
- 21.Thorup M (1999) Undirected single-source shortest paths with positive integer weights in linear time. J ACM 46(3):362–394MathSciNetCrossRefGoogle Scholar
- 22.Warshall S (1962) A theorem on Boolean matrices. J ACM 9(1):11–12MathSciNetCrossRefGoogle Scholar
- 23.Wilson DB, Zwick U (2013) A forward-backward single-source shortest paths algorithm. In: IEEE 54th annual symposium on foundations of computer science, pp 707–716Google Scholar
- 24.Yen JY (1971) Finding the K-shortest loopless paths in a network. Manage Sci 17(1):712–716MathSciNetCrossRefGoogle Scholar
- 25.Zhan F (1998) Three fastest shortest path algorithms on real road networks: data structures and procedures. J Geogr Inf Decis Anal 1(1):69–82Google Scholar