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Abstract

Shortest path problems are fundamental network optimization problems arising in many contexts and having a wide range of applications, including dynamic programming, project management, knapsack problems, routing in data networks, and transportation problems. The scope of this chapter is to provide an extensive treatment of shortest path algorithms covering both classical and recently proposed approaches.

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Bibliography

  • R.K. Ahuja, T.L. Magnanti, and J.B. Orlin. Network Flows: Theory, Algorithms and Applications. Prentice-Hall Englewood Cliffs, 1993.

    Google Scholar 

  • R.K. Ahuja, K. Mehlhorn, J.B. Orlin, and R.E. Tarjan. Faster algorithms for the shortest path problem. Journal of ACM, 37:213–223, 1990.

    Article  MATH  MathSciNet  Google Scholar 

  • R.E. Bellman. Dynamic Programming. Princeton Univ. Press, 1957.

    Google Scholar 

  • R.E. Bellman. On a routing problem. Quart. Appl. Math., 16:87–90, 1958.

    MATH  MathSciNet  Google Scholar 

  • D.P. Bertsekas. A distributed algorithm for the assignment problem. Technical report, Lab. for Information and Decision Systems Working Paper, MIT, Boston, USA, 1979.

    Google Scholar 

  • D.P. Bertsekas. A distributed asynchronous relaxation algorithm for the assignment problem. In Proceedings of the 24th IEEE Conference on Decision and Control, pages 1703–1704, Ft. Lauderdale, Fla., USA, 1985.

    Google Scholar 

  • D.P. Bertsekas. The auction algorithm: A distributed relaxation method for the assignment problems. Annals of Operation Research, 14:105–123, 1988.

    Article  MATH  MathSciNet  Google Scholar 

  • D.P. Bertsekas. An auction algorithm for shortest paths. SIAM Journal on Optimization, 1:425–447, 1991a.

    Article  MATH  MathSciNet  Google Scholar 

  • D.P. Bertsekas. Linear networks optimization: Algorithms and Codes. MIT Press, 1991b.

    Google Scholar 

  • D.P. Bertsekas. Network Optimization: Continuous and Discrete Models. Athena Scientific, 1998.

    Google Scholar 

  • D.P. Bertsekas and D.A. Castanon. The auction algorithm for minimum cost network flow problem. Technical Report Report LIDS-P-1925, Lab. for Information and Decision Systems, MIT, Boston, USA, 1989a.

    Google Scholar 

  • D.P. Bertsekas and D.A. Castanon. The auction algorithm for transportation problems. Annals of Operation Research, 20:67–96, 1989b.

    Article  MATH  MathSciNet  Google Scholar 

  • D.P. Bertsekas and D.A. Castanon. A generic auction algorithm for the minimum cost network flow problem. Technical Report Report LIDS-P-2084, Lab. for Information and Decision Systems, MIT, Boston, USA, 1991.

    Google Scholar 

  • D.P. Bertsekas and D.A. Costanon. Asynchronous hungarian methods for the assignment problem. ORSA Journal on Computing, 5:261–274, 1993.

    MATH  Google Scholar 

  • D.P. Bertsekas, F. Guerriero, and R. Musmanno. Parallel asynchronous label correcting methods for shortest paths. Journal of Optimization: Theory and Applications, 88:297–320, 1996.

    Article  MATH  MathSciNet  Google Scholar 

  • D.P. Bertsekas, S. Pallottino, and M.G. Scutellá. Polynomial auction algorithms for shortest paths. Computational Optimization and Application, 4:99–125, 1995.

    Article  MATH  Google Scholar 

  • R. Cerulli, P. Festa, and G. Raiconi. Graph collapsing in shortest path auction algorithms. Computational Optimization and Applications, 18:199–220, 2001.

    Article  MATH  MathSciNet  Google Scholar 

  • R. Cerulli, P. Festa, and G. Raiconi. Shortest path auction algorithm without contractions using virtual source concept. Computational Optimization and Applications, 26(2): 191–208, 2003.

    Article  MATH  MathSciNet  Google Scholar 

  • Z.L. Chen and W.B. Powell. A note on bertsekas’s small-label-first strategy. Networks, 29:111–116, 1997.

    Article  MATH  MathSciNet  Google Scholar 

  • B.V. Cherkassky and A.V. Goldberg. Negative-cycle detection algorithms. Mathematical Programming, 85:277–311, 1999.

    Article  MATH  MathSciNet  Google Scholar 

  • B.V. Cherkassky, A.V. Goldberg, and T. Radzik. Shortest path algorithms: Theory and experimental evaluation. Mathematical Programming, 73:129–174, 1996.

    MathSciNet  Google Scholar 

  • B.V. Cherkassky, A.V. Goldberg, and C. Silverstein. Buckets, heaps, lists, and monotone priority queues. SIAM Journal of Computing, 28:1326–1346, 1999.

    Article  MATH  MathSciNet  Google Scholar 

  • T.H. Cormen, C.E. Leiserson, and R.L. Rivest. Introduction to Algorithms. MIT Press, 1990.

    Google Scholar 

  • G.B. Dantzig. On the shortest route through a network. Management Science, 6: 187–190, 1960.

    Article  MATH  MathSciNet  Google Scholar 

  • E.V. Denardo and B.L. Fox. Shortest route methods: 2. group knapsacks, expanded networks, and branch-and-bound. Operational Research, 27:548–566, 1979a.

    MATH  MathSciNet  Google Scholar 

  • E.V. Denardo and B.L. Fox. Shortest route methods: reaching pruning, and buckets. Operational Research, 27:161–186, 1979b.

    MATH  MathSciNet  Google Scholar 

  • N. Deo and C. Pang. Shortest path algorithms: taxonomy and annotation. Networks, 14:275–323, 1984.

    Article  MATH  MathSciNet  Google Scholar 

  • R.B. Dial. Algorithm 360: Shortest path forest with topological ordering. Comm. ACM, 12:632–633, 1969.

    Article  Google Scholar 

  • E. Dijkstra. A note on two problems in connection with graphs. Numer. Math., 1: 269–271, 1959.

    Article  MATH  MathSciNet  Google Scholar 

  • P. Festa. New Auction Algorithms for Shortest Path Problems. PhD thesis, University of Naples FEDERICO II, Naples, Italy, February 2000.

    Google Scholar 

  • L.R. Ford and D.R. Fulkerson. Flows in Networks. Princeton Univ. Press, 1962.

    Google Scholar 

  • L.R. Jr. Ford. Network flow theory. Technical Report Report P-923, The Rand Corporation, Santa Monica, California, USA, 1956.

    Google Scholar 

  • M.L. Fredman and R.E. Tarjan. Fibonacci heaps and their uses in improved network optimization algorithms. Journal of ACM, 34:596–615, 1987.

    Article  MathSciNet  Google Scholar 

  • G. Gallo and S. Pallottino. Shortest path methods: A unified approach. Math. Programming Study, 26:38–64, 1986.

    MATH  MathSciNet  Google Scholar 

  • G. Gallo and S. Pallottino. Shortest path methods. Ann. Oper. Res., 7:3–79, 1988.

    MathSciNet  Google Scholar 

  • F. Glover, D. Klingman, and N. Phillips. New polynomially bounded on finding shortest path trees. Operations Research, 33:65–73, 1985a.

    Article  MATH  MathSciNet  Google Scholar 

  • F. Glover, D. Klingman, N. Phillips, and R.F. Schneider. Network Models in Optimization and Their Applications in Practice. Wiley, 1985b.

    Google Scholar 

  • A.V. Goldberg. A simple shortest path algorithm with linear average time. Technical Report STAR-TR-01-03, InterTrust Technologies Corp., 2001.

    Google Scholar 

  • A.V. Goldberg and C. Silverstein. Implementations of dijkstra’s algorithm based on multi-level buckets. In P.M. Pardalos, D.W. Hearn, and W.W. Hager, editors, Network Optimization, pages 292–327. Springer Lecture Notes in Economics and Mathematical Systems, 1997.

    Google Scholar 

  • J.A. Hillier and P.D. Whiting. A method for finding the shortest route through a road network. Operation Research Quarterly, 11:37–40, 1960.

    Article  Google Scholar 

  • A. Kershenbaum. A note on finding shortest path trees. Networks, 11:399–400, 1981.

    Article  MathSciNet  Google Scholar 

  • M. Klein. A primal method for minimal cost flows with applications to the assignment and transportation problems. Management Science, 14:205–220, 1967.

    Article  MATH  Google Scholar 

  • E.F. Moore. The shortest path through a maze. In Proceedings of the International Symposium on the Theory of Switching, pages 285–292, 1959.

    Google Scholar 

  • S. Pallottino. Shortest path methods: Complexity, interrelations and new propositions. Networks, 14:257–267, 1984.

    Article  MATH  Google Scholar 

  • S. Pallottino and M.G. Scutellá. Strongly polynomial auction algorithms for shortest path. Ricerca Operativa, 21:60–72, 1991.

    Google Scholar 

  • D.R. Shier and C. Witzgall. Properties of labeling methods for determining shortest path trees. Journals of Research of the National Bureau of Standards, 86:317–330, 1981.

    MATH  MathSciNet  Google Scholar 

  • R.E. Tarjan. Shortest paths. In Y. Alavi et al., editor, Graph Theory with Applications to Algorithms and Computer Science, pages 753–759. John Wiley, 1985.

    Google Scholar 

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Festa, P. (2006). Shortest Path Algorithms. In: Resende, M.G.C., Pardalos, P.M. (eds) Handbook of Optimization in Telecommunications. Springer, Boston, MA. https://doi.org/10.1007/978-0-387-30165-5_8

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