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Kombinatorische Optimierung pp 167–184Cite as

KürzesteWege

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Zusammenfassung

Eines der bekanntesten kombinatorischen Optimierungsprobleme ist, einen kürzesten Weg zwischen zwei bestimmten Knoten eines gerichteten oder ungerichteten Graphen zu finden. Wir behandeln Algorithmen für verschiedene Varianten des Problems.

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Literatur

Allgemeine Literatur:

  • Ahuja, R.K., Magnanti, T.L., und Orlin, J.B. [1993]: Network Flows. Prentice-Hall, Englewood Cliffs 1993, Kapitel 4 und 5

    Google Scholar 

  • Cormen, T.H., Leiserson, C.E., Rivest, R.L., und Stein, C. [2009]: Introduction to Algorithms. 3. Aufl. MIT Press, Cambridge 2009, Kapitel 24 und 25

    Google Scholar 

  • Dreyfus, S.E. [1969]: An appraisal of some shortest path algorithms. Operations Research 17 (1969), 395–412

    Article  Google Scholar 

  • Gallo, G., und Pallottino, S. [1988]: Shortest paths algorithms. Annals of Operations Research 13 (1988), 3–79

    Article  MathSciNet  Google Scholar 

  • Gondran, M., und Minoux, M. [1984]: Graphs and Algorithms. Wiley, Chichester 1984, Kapitel 2

    Google Scholar 

  • Lawler, E.L. [1976]: Combinatorial Optimization: Networks and Matroids. Holt, Rinehart and Winston, New York 1976, Kapitel 3

    Google Scholar 

  • Schrijver, A. [2003]: Combinatorial Optimization: Polyhedra and Efficiency. Springer, Berlin 2003, Kapitel 6–8

    Google Scholar 

  • Tarjan, R.E. [1983]: Data Structures and Network Algorithms. SIAM, Philadelphia 1983, Kapitel 7

    Google Scholar 

Zitierte Literatur:

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

    Article  MathSciNet  Google Scholar 

  • Albrecht, C., Korte, B., Schietke, J., und Vygen, J. [2002]: Maximum mean weight cycle in a digraph and minimizing cycle time of a logic chip. Discrete Applied Mathematics 123 (2002), 103–127

    Article  MathSciNet  Google Scholar 

  • Bazaraa, M.S., und Langley, R.W. [1974]: A dual shortest path algorithm. SIAM Journal on Applied Mathematics 26 (1974), 496–501

    Article  MathSciNet  Google Scholar 

  • Bellman, R.E. [1958]: On a routing problem. Quarterly of Applied Mathematics 16 (1958), 87–90

    Article  MathSciNet  Google Scholar 

  • Cherkassky, B.V., und Goldberg, A.V. [1999]: Negative-cycle detection algorithms. Mathematical Programming A 85 (1999), 277–311

    Article  MathSciNet  Google Scholar 

  • Dial, R.B. [1969]: Algorithm 360: shortest path forest with topological order. Communications of the ACM 12 (1969), 632–633

    Article  Google Scholar 

  • Dijkstra, E.W. [1959]: A note on two problems in connexion with graphs. Numerische Mathematik 1 (1959), 269–271

    Article  MathSciNet  Google Scholar 

  • Edmonds, J., und Karp, R.M. [1972]: Theoretical improvements in algorithmic efficiency for network flow problems. Journal of the ACM 19 (1972), 248–264

    Article  Google Scholar 

  • Floyd, R.W. [1962]: Algorithm 97 – shortest path. Communications of the ACM 5 (1962), 345

    Article  Google Scholar 

  • Ford, L.R. [1956]: Network flow theory. Paper P-923, The Rand Corporation, Santa Monica 1956

    Google Scholar 

  • Fredman, M.L., und Tarjan, R.E. [1987]: Fibonacci heaps and their uses in improved network optimization problems. Journal of the ACM 34 (1987), 596–615

    Article  MathSciNet  Google Scholar 

  • Fredman, M.L., und Willard, D.E. [1994]: Trans-dichotomous algorithms for minimum spanning trees and shortest paths. Journal of Computer and System Sciences 48 (1994), 533–551

    Article  MathSciNet  Google Scholar 

  • Goldberg, A.V. [1995]: Scaling algorithms for the shortest paths problem. SIAM Journal on Computing 24 (1995), 494–504

    Article  MathSciNet  Google Scholar 

  • Goldberg, A.V., und Harrelson, C. [2005]: Computing the shortest path: A* search meets graph theory. Proceedings of the 16th Annual ACM-SIAM Symposium on Discrete Algorithms (2005), 156–165

    Google Scholar 

  • Han, Y., und Takaoka, T. [2016]: An \( O\left( {n^{3} \,\text{log}\,\text{log }n/\,\,\text{log}^{2} \,n} \right) \) time algorithm for all pairs shortest paths. Journal of Discrete Algorithms 38–41 (2016), 9–19

    Article  Google Scholar 

  • Henzinger, M.R., Klein, P., Rao, S., und Subramanian, S. [1997]: Faster shortest-path algorithms for planar graphs. Journal of Computer and System Sciences 55 (1997), 3–23

    Article  MathSciNet  Google Scholar 

  • Hougardy, S. [2010]: The Floyd-Warshall algorithm on graphs with negative cycles. Information Processing Letters 110 (2010), 279–281

    Article  MathSciNet  Google Scholar 

  • Johnson, D.B. [1977]: Efficient algorithms for shortest paths in sparse networks. Journal of the ACM 24 (1977), 1–13

    Article  MathSciNet  Google Scholar 

  • Johnson, D.B. [1982]: A priority queue in which initialization and queue operations take \( O\left( {\text{log}\,\text{log}\,D} \right) \) time. Mathematical Systems Theory 15 (1982), 295–309

    Google Scholar 

  • Karp, R.M. [1978]: A characterization of the minimum cycle mean in a digraph. Discrete Mathematics 23 (1978), 309–311

    Article  MathSciNet  Google Scholar 

  • Khuller, S., Raghavachari, B., und Young, N. [1995]: Balancing minimum spanning trees and shortest-path trees. Algorithmica 14 (1995), 305–321

    Article  MathSciNet  Google Scholar 

  • Megiddo, N. [1979]: Combinatorial optimization with rational objective functions. Mathematics of Operations Research 4 (1979), 414–424

    Article  MathSciNet  Google Scholar 

  • Megiddo, N. [1983]: Applying parallel computation algorithms in the design of serial algorithms. Journal of the ACM 30 (1983), 852–865

    Article  MathSciNet  Google Scholar 

  • Moore, E.F. [1959]: The shortest path through a maze. Proceedings of the International Symposium on the Theory of Switching, Part II, Harvard University Press, 1959, 285–292

    Google Scholar 

  • Mozes, S., und Wulff-Nilsen, C. [2010]: Shortest paths in planar graphs with real lengths in \( O\left( {n\,\text{log}^{\text{2}} \,n/\,\,\text{log}\,\text{log}\,n} \right) \) time. In: Algorithms – Proceedings of the 18th European Symposium on Algorithms (ESA), Part II; LNCS 6347 (M. de Berg, U. Meyer, Hrsg.), Springer, Berlin 2010, pp. 206–217

    Google Scholar 

  • Orlin, J.B. [1993]: A faster strongly polynomial minimum cost flow algorithm. Operations Research 41 (1993), 338–350

    Article  MathSciNet  Google Scholar 

  • Orlin, J.B., Madduri, K., Subramani, K., und Williamson, M. [2010]: A faster algorithm for the single source shortest path problem with few distinct positive lengths. Journal of Discrete Algorithms 8 (2010), 189–198

    Article  MathSciNet  Google Scholar 

  • Orlin, J.B., und Sedeño-Noda, A. [2017]: An \( O\left( {nm} \right) \) time algorithm for finding the min length directed cycle in a graph. Proceedings of the 28th Annual ACM-SIAM Symposium on Discrete Algorithms (2017), 1866–1879

    Google Scholar 

  • Pettie, S. [2004]: A new approach to all-pairs shortest paths on real-weighted graphs. Theoretical Computer Science 312 (2004), 47–74

    Article  MathSciNet  Google Scholar 

  • Pettie, S., und Ramachandran, V. [2005]: Computing shortest paths with comparisons and additions. SIAM Journal on Computing 34 (2005), 1398–1431

    Article  MathSciNet  Google Scholar 

  • Radzik, T. [1993]: Parametric flows, weighted means of cuts, and fractional combinatorial optimization. In: Complexity in Numerical Optimization (P.M. Pardalos, Hrsg.), World Scientific, Singapore 1993

    Chapter  Google Scholar 

  • Raman, R. [1997]: Recent results on the single-source shortest paths problem. ACM SIGACT News 28 (1997), 81–87

    Article  Google Scholar 

  • Thorup, M. [1999]: Undirected single-source shortest paths with positive integer weights in linear time. Journal of the ACM 46 (1999), 362–394

    Article  MathSciNet  Google Scholar 

  • Thorup, M. [2000]: On RAM priority queues. SIAM Journal on Computing 30 (2000), 86–109

    Article  MathSciNet  Google Scholar 

  • Thorup, M. [2004]: Integer priority queues with decrease key in constant time and the single source shortest paths problem. Journal of Computer and System Sciences 69 (2004), 330–353

    Article  MathSciNet  Google Scholar 

  • Warshall, S. [1962]: A theorem on boolean matrices. Journal of the ACM 9 (1962), 11–12

    Article  MathSciNet  Google Scholar 

  • Zwick, U. [2002]: All pairs shortest paths using bridging sets and rectangular matrix multiplication. Journal of the ACM 49 (2002), 289–317

    Article  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Forschungsinstitut für Diskrete Mathematik, Universität Bonn, Bonn, Deutschland

    Bernhard Korte & Jens Vygen

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  1. Bernhard Korte
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  2. Jens Vygen
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Correspondence to Bernhard Korte .

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Korte, B., Vygen, J. (2018). KürzesteWege. In: Kombinatorische Optimierung. Masterclass. Springer Spektrum, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-57691-5_7

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