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On Dynamic Shortest Paths Problems

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Abstract

We obtain the following results related to dynamic versions of the shortest-paths problem:

  1. (i)

    Reductions that show that the incremental and decremental single-source shortest-paths problems, for weighted directed or undirected graphs, are, in a strong sense, at least as hard as the static all-pairs shortest-paths problem. We also obtain slightly weaker results for the corresponding unweighted problems.

  2. (ii)

    A randomized fully-dynamic algorithm for the all-pairs shortest-paths problem in directed unweighted graphs with an amortized update time of \(\tilde {O}(m\sqrt{n})\) (we use \(\tilde {O}\) to hide small poly-logarithmic factors) and a worst case query time is O(n 3/4).

  3. (iii)

    A deterministic O(n 2log n) time algorithm for constructing an O(log n)-spanner with O(n) edges for any weighted undirected graph on n vertices. The algorithm uses a simple algorithm for incrementally maintaining single-source shortest-paths tree up to a given distance.

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References

  1. Althöfer, I., Das, G., Dobkin, D., Joseph, D., Soares, J.: On sparse spanners of weighted graphs. Discrete Comput. Geom. 9, 81–100 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  2. Baswana, S., Sen, S.: A simple and linear time randomized algorithm for computing sparse spanners in weighted graphs. Random Struct. Algorithms 30(4), 532–563 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  3. Baswana, S., Hariharan, R., Sen, S.: Maintaining all-pairs approximate shortest paths under deletion of edges. In: Proc. of 14th SODA, pp. 394–403 (2003)

  4. Baswana, S., Hariharan, R., Sen, S.: Improved decremental algorithms for maintaining transitive closure and all-pairs shortest paths. J. Algorithms 62(2), 74–92 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  5. Bernstein, A.: Fully dynamic approximate all-pairs shortest paths with query and close to linear update time. In: Proc. of the 50th FOCS, Atlanta, GA, pp. 50–60 (2009)

  6. Chan, T.M.: Dynamic subgraph connectivity with geometric applications. SIAM J. Comput. 36(3), 681–694 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  7. Chandra, B., Das, G., Narasimhan, G., Soares, J.: New sparseness results on graph spanners. Int. J. Comput. Geom. Appl. 5, 125–144 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  8. Coppersmith, D., Winograd, S.: Matrix multiplication via arithmetic progressions. J. Symb. Comput. 9, 251–280 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  9. Demetrescu, C., Italiano, G.F.: Experimental analysis of dynamic all pairs shortest path algorithms. In: Proc. of 15th SODA, pp. 362–371 (2004)

  10. Even, S., Shiloach, Y.: An on-line edge-deletion problem. J. ACM 28(1), 1–4 (1981)

    Article  MathSciNet  MATH  Google Scholar 

  11. Fredman, M.L., Tarjan, R.E.: Fibonacci heaps and their uses in improved network optimization algorithms. J. ACM 34, 596–615 (1987)

    Article  MathSciNet  Google Scholar 

  12. Galil, Z., Margalit, O.: All pairs shortest distances for graphs with small integer length edges. Inf. Comput. 134, 103–139 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  13. Gudmundsson, J., Levcopoulos, C., Narasimhan, G.: Fast greedy algorithm for constructing sparse geometric spanners. SIAM J. Comput. 31, 1479–1500 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  14. Hagerup, T.: Improved shortest paths on the word RAM. In: Proc. of 27th ICALP, pp. 61–72 (2000)

  15. Henzinger, M., King, V.: Fully dynamic biconnectivity and transitive closure. In: Proc. of 36th FOCS, pp. 664–672 (1995)

  16. Karger, D.R., Koller, D., Phillips, S.J.: Finding the hidden path: time bounds for all-pairs shortest paths. SIAM J. Comput. 22, 1199–1217 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  17. Peleg, D., Schäffer, A.A.: Graph spanners. J. Graph Theory 13, 99–116 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  18. Pettie, S.: A new approach to all-pairs shortest paths on real-weighted graphs. Theor. Comput. Sci. 312(1), 47–74 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  19. Pettie, S., Ramachandran, V.: A shortest path algorithm for real-weighted undirected graphs. SIAM J. Comput. 34(6), 1398–1431 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  20. Ramalingam, G., Reps, T.W.: An incremental algorithm for a generalization of the shortest-path problem. J. Algorithms 21(2), 267–305 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  21. Roditty, L., Zwick, U.: Improved dynamic reachability algorithms for directed graphs. In: Proc. of 43rd FOCS, pp. 679–688 (2002)

  22. Roditty, L., Zwick, U.: Dynamic approximate all-pairs shortest paths in undirected graphs. In: Proc. of 45th FOCS, pp. 499–508 (2004)

  23. Roditty, L., Zwick, U.: A fully dynamic reachability algorithm for directed graphs with an almost linear update time. In: Proc. of 36th STOC, pp. 184–191 (2004)

  24. Roditty, L., Thorup, M., Zwick, U.: Deterministic constructions of approximate distance oracles and spanners. In: ICALP, pp. 261–272 (2005)

  25. Seidel, R.: On the all-pairs-shortest-path problem in unweighted undirected graphs. J. Comput. Syst. Sci. 51, 400–403 (1995)

    Article  MathSciNet  Google Scholar 

  26. Shoshan, A., Zwick, U.: All pairs shortest paths in undirected graphs with integer weights. In: Proc. of 40th FOCS, pp. 605–614 (1999)

  27. Thorup, M.: Undirected single-source shortest paths with positive integer weights in linear time. J. ACM 46, 362–394 (1999)

    MathSciNet  MATH  Google Scholar 

  28. Thorup, M.: Fully-dynamic all-pairs shortest paths: Faster and allowing negative cycles. In: Proc. of 9th SWAT, pp. 384–396 (2004)

  29. Thorup, M., Zwick, U.: Approximate distance oracles. J. ACM 52(1), 1–24 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  30. Ullman, J.D., Yannakakis, M.: High-probability parallel transitive-closure algorithms. SIAM J. Comput. 20, 100–125 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  31. Yuster, R., Zwick, U.: Fast sparse matrix multiplication. ACM Trans. Algorithms 1(1), 2–13 (2005)

    Article  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Dept. of Computer Science, Bar-Ilan University, Ramat-Gan, 52900, Israel

    Liam Roditty

  2. School of Computer Science, Tel-Aviv University, Tel-Aviv, 69978, Israel

    Uri Zwick

Authors
  1. Liam Roditty
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  2. Uri Zwick
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Correspondence to Liam Roditty.

Additional information

A preliminary version of this article appeared in Proceedings of 12th Annual European Symposium on Algorithms (ESA 2004), pp. 568–579.

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Roditty, L., Zwick, U. On Dynamic Shortest Paths Problems. Algorithmica 61, 389–401 (2011). https://doi.org/10.1007/s00453-010-9401-5

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  • DOI: https://doi.org/10.1007/s00453-010-9401-5

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