Abstract
We obtain the following results related to dynamic versions of the shortest-paths problem:
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(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.
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(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).
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(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
Althöfer, I., Das, G., Dobkin, D., Joseph, D., Soares, J.: On sparse spanners of weighted graphs. Discrete Comput. Geom. 9, 81–100 (1993)
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)
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)
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)
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)
Chan, T.M.: Dynamic subgraph connectivity with geometric applications. SIAM J. Comput. 36(3), 681–694 (2006)
Chandra, B., Das, G., Narasimhan, G., Soares, J.: New sparseness results on graph spanners. Int. J. Comput. Geom. Appl. 5, 125–144 (1995)
Coppersmith, D., Winograd, S.: Matrix multiplication via arithmetic progressions. J. Symb. Comput. 9, 251–280 (1990)
Demetrescu, C., Italiano, G.F.: Experimental analysis of dynamic all pairs shortest path algorithms. In: Proc. of 15th SODA, pp. 362–371 (2004)
Even, S., Shiloach, Y.: An on-line edge-deletion problem. J. ACM 28(1), 1–4 (1981)
Fredman, M.L., Tarjan, R.E.: Fibonacci heaps and their uses in improved network optimization algorithms. J. ACM 34, 596–615 (1987)
Galil, Z., Margalit, O.: All pairs shortest distances for graphs with small integer length edges. Inf. Comput. 134, 103–139 (1997)
Gudmundsson, J., Levcopoulos, C., Narasimhan, G.: Fast greedy algorithm for constructing sparse geometric spanners. SIAM J. Comput. 31, 1479–1500 (2002)
Hagerup, T.: Improved shortest paths on the word RAM. In: Proc. of 27th ICALP, pp. 61–72 (2000)
Henzinger, M., King, V.: Fully dynamic biconnectivity and transitive closure. In: Proc. of 36th FOCS, pp. 664–672 (1995)
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)
Peleg, D., Schäffer, A.A.: Graph spanners. J. Graph Theory 13, 99–116 (1989)
Pettie, S.: A new approach to all-pairs shortest paths on real-weighted graphs. Theor. Comput. Sci. 312(1), 47–74 (2004)
Pettie, S., Ramachandran, V.: A shortest path algorithm for real-weighted undirected graphs. SIAM J. Comput. 34(6), 1398–1431 (2005)
Ramalingam, G., Reps, T.W.: An incremental algorithm for a generalization of the shortest-path problem. J. Algorithms 21(2), 267–305 (1996)
Roditty, L., Zwick, U.: Improved dynamic reachability algorithms for directed graphs. In: Proc. of 43rd FOCS, pp. 679–688 (2002)
Roditty, L., Zwick, U.: Dynamic approximate all-pairs shortest paths in undirected graphs. In: Proc. of 45th FOCS, pp. 499–508 (2004)
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)
Roditty, L., Thorup, M., Zwick, U.: Deterministic constructions of approximate distance oracles and spanners. In: ICALP, pp. 261–272 (2005)
Seidel, R.: On the all-pairs-shortest-path problem in unweighted undirected graphs. J. Comput. Syst. Sci. 51, 400–403 (1995)
Shoshan, A., Zwick, U.: All pairs shortest paths in undirected graphs with integer weights. In: Proc. of 40th FOCS, pp. 605–614 (1999)
Thorup, M.: Undirected single-source shortest paths with positive integer weights in linear time. J. ACM 46, 362–394 (1999)
Thorup, M.: Fully-dynamic all-pairs shortest paths: Faster and allowing negative cycles. In: Proc. of 9th SWAT, pp. 384–396 (2004)
Thorup, M., Zwick, U.: Approximate distance oracles. J. ACM 52(1), 1–24 (2005)
Ullman, J.D., Yannakakis, M.: High-probability parallel transitive-closure algorithms. SIAM J. Comput. 20, 100–125 (1991)
Yuster, R., Zwick, U.: Fast sparse matrix multiplication. ACM Trans. Algorithms 1(1), 2–13 (2005)
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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