Abstract
This paper studies the dynamic single-source shortest paths (SSSP) in Erdös-Rényi random graphs generated by G(n, p) model. In 2014, Ding and Lin (AAIM 2014, LNCS 8546, 197–207) first considered the dynamic SSSP in general digraphs with arbitrary positive weights, and devised a nontrivial local search algorithm named DSPI which takes at most \(O(n\cdot \max \{1, n\log n / m\})\) expected update time to handle a single weight increase, where n is the number of nodes and m is the number of edges in the digraph. DSPI also works on undirected graphs. This paper analyzes the expected update time of DSPI dealing with edge weight increases or edge deletions in Erdös-Rényi (a.k.a., G(n, p)) random graphs. For weighted G(n, p) random graphs with arbitrary positive edge weights, DSPI takes at most \(O(h(T_s))\) expected update time to deal with a single edge weight increase as well as \(O(pn^2 h(T_s))\) total update time, where \(h(T_s)\) is the height of input SSSP tree \(T_s\). For G(n, p) random graphs, DSPI takes \(O(\ln n)\) expected update time to handle a single edge deletion as well as \(O(pn^2 \ln n)\) total update time when \(20\ln n / n \le p < \sqrt{2\ln n / n}\), and O(1) expected update time to handle a single edge deletion as well as \(O(pn^2)\) total update time when \(p > \sqrt{2\ln n / n}\). Specifically, DSPI takes the least total update time of \(O(n\ln n h(T_s))\) for weighted G(n, p) random graphs with \(p = c\ln n / n, c > 1\) as well as \(O(n^{3/2}(\ln n)^{1/2})\) for G(n, p) random graphs with \(p = c\sqrt{\ln n / n}, c > \sqrt{2}\).
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
Ausiello, G., Italiano, G.F., Marchetti-Spaccamela, A., Nanni, U.: Incremental algorithms for minimal length paths. J. Algorithms 12, 615–638 (1991)
Bernstein, A.: Fully dynamic \((2+\epsilon )\) approximate all-pairs shortest paths with fast query and close to linear update time. In: Proceedings of the 50th FOCS, pp. 693–702 (2009)
Bernstein, A.: Maintaining shortest paths under deletions in weighted directed graphs. In: Proceedings of the 45th STOC, pp. 725–734 (2013)
Bernstein, A., Roditty, L.: Improved dynamic algorithms for maintaining approximate shortest paths under deletions. In: Proceedings of the 22th SODA, pp. 1355–1365 (2011)
Blum, A., Hopcroft, J., Kannan, R.: Foundation of Data Science, Manuscript (14 May 2015). http://www.cs.cornell.edu/jeh/bookMay2015.pdf
Bollobás, B.: Random Graphs. Cambridge University Press, Cambridge (2001)
Demetrescu, C., Italiano, G.F.: A new approach to dynamic all pairs shortest paths. J. ACM 51, 968–992 (2004)
Dijkstra, E.W.: A note on two problems in connection with graphs. Numer. Math. 1, 269–271 (1959)
Ding, W., Lin, G.: Partially dynamic single-source shortest paths on digraphs with positive weights. In: Gu, Q., Hell, P., Yang, B. (eds.) AAIM 2014. LNCS, vol. 8546, pp. 197–207. Springer, Heidelberg (2014)
Erdös, P., Rényi, A.: On random graphs-I. Publicationes Mathematicae (Debrecen) 6, 290–297 (1959)
Erdös, P., Rényi, A.: On the Evolution of Random Graphs. Akad. Kiado, Budapest (1960)
Even, S., Shiloach, Y.: An on-line edge-deletion problem. J. ACM 28, 1–4 (1981)
Fakcharoemphol, J., Rao, S.: Planar graphs, negative weight edges, shortest paths, and near linear time. In: Proceedings of the 42nd FOCS, pp. 232–241 (2001)
Feller, W.: An Introduction to Probability Theory and Its Applications. Wiley, New York (1968)
Fredman, M.L., Tarjan, R.E.: Fibonacci heaps and their uses in improved network optimization algorithms. J. ACM 34(3), 596–615 (1987)
Gilbert, E.N.: Random graphs. Ann. Math. Stat. 30(4), 1141–1144 (1959)
Henzinger, M., King, V.: Fully dynamic biconnectivity and transitive closure. In: Proceedings of the 36th FOCS, pp, 664–672 (1995)
Henzinger, M., Krinninger, S., Nanongkai, D.: A subquadratic-time algorithm for dynamic single-source shortest paths. In: Proceedings of the 25th SODA, pp, 1053–1072 (2014)
Henzinger, M., Krinninger, S., Nanongkai, D.: Sublinear-time decremental algorithms for single-source reachability and shortest paths on directed graphs. In: Proceedings of the 46th STOC, pp. 674–683 (2014)
Henzinger, M., Krinninger, S., Nanongkai, D.: Decremental single-source shortest paths on undirected graphs in near-linear total update time. In: Proceedings of the 55th FOCS, pp. 146–155 (2014)
King, V.: Fully dynamic algorithms for maintaining all-pairs shortest paths and transitive closure in digraphs. In: Proceedings of the 40th FOCS, pp. 81–99 (1999)
Madry, A.: Faster approximation schemes for fractional multicommodity flow problems via dynamic graph algorithms. In: Proceedings of the 42th STOC, pp. 121–130 (2010)
Peres, Y., Sotnikov, D., Sudakov, B., Zwick, U.: All-pairs shortest paths in \(O(n^2)\) time with high probability. In: Proceedings of the 51th FOCS, pp. 663–672 (2010)
Roditty, L., Zwick, U.: Dynamic approximate all-pairs shortest paths in undirected graphs. In: Proceedings of the 45th FOCS, pp. 499–508 (2004)
Roditty, L., Zwick, U.: On dynamic shortest paths problems. In: Albers, S., Radzik, T. (eds.) ESA 2004. LNCS, vol. 3221, pp. 580–591. Springer, Heidelberg (2004)
Solomonoff, R., Rapoport, A.: Connectivity of random nets. Bull. Math. Biol. 13(2), 107–117 (1951)
Thorup, M.: Fully-dynamic all-pairs shortest paths: faster and allowing negative cycles. In: Hagerup, T., Katajainen, J. (eds.) SWAT 2004. LNCS, vol. 3111, pp. 384–396. Springer, Heidelberg (2004)
Thorup, M.: Worst-case update times for fully-dynamic all-pairs shortest paths. In: Proceedings of the 37th STOC, pp. 112–119 (2005)
Acknowledgement
We thank the reviewers for their valuable comments and suggestions.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2015 Springer International Publishing Switzerland
About this paper
Cite this paper
Ding, W., Qiu, K. (2015). Dynamic Single-Source Shortest Paths in Erdös-Rényi Random Graphs. In: Lu, Z., Kim, D., Wu, W., Li, W., Du, DZ. (eds) Combinatorial Optimization and Applications. Lecture Notes in Computer Science(), vol 9486. Springer, Cham. https://doi.org/10.1007/978-3-319-26626-8_39
Download citation
DOI: https://doi.org/10.1007/978-3-319-26626-8_39
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-26625-1
Online ISBN: 978-3-319-26626-8
eBook Packages: Computer ScienceComputer Science (R0)