研究

最大流算法统计

对最大流算法历史文献的一个调研

Table: Polynomial algorithms for the max flow problem

  No   Duo to     Year     Running Time;
1 Ford & Fulkerson [1] 1956 $O(nmU)$
2 Edmonds and Karp [2] 1972 $O(nm^2)$
3 Dinic [3] 1970 $O(n^2m)$
4 Karzanov [4] 1974 $O(n^3)$
5 Cherkasky [5] 1977 $O(n^2\sqrt{m})$
6 Malhotra, Kumar & Maheshwari [6] 1977 $O(n^3)$
7 Galil [7] 1980 $O(n^(5/3)m^(2/3))$
8 Galil & Naaman [8] 1980 $O(nmlog^2n)$
9 Sleator & Tarjan [9] 1983 $O(nmlogn)$
10 Gabow [10] 1985 $O(nmlogU)$
11 Goldberg & Tarjan [11] 1988 $O(nmlog(n^2/m))$
12 Ahuja & Orlin [12] 1989 $O(nm + n^2logU)$
13 Ahuja, Orlin & Tarjan [13] 1989 $O(nmlog(n\sqrt{U}/(m + 2))$
14 King, Rao & Tarjan [14] 1992 $O(nm+n^{2+e})$
15 King, Rao & Tarjan [15] 1994 $O(nmlog_{m/nlogn}n)$
16 Cheriyan, Hagerup & Mehlhorn [16] 1996 $O(n^3/logn)$
17 Goldberg & Rao [17] 1998 $O(min{n^(2/3),m^{1/2}}mlog{n2/m}logU)$
18 Orlin [18] 2012 $O(nm)$
19 Orlin [18] 2012 $O(n^2/logn) if m = O(n)$
  • [1] L. R. Ford and D. R. Fulkerson. Maximal flow through a network. Canadian Journal of Mathematics, 8:399-404, 1956.
  • [2] J. Edmonds and R. M. Karp. Theoretical improvements in algorithmic eciency for network flow problems. Journal of the ACM, 19:248-264, 1972.
  • [3] E. A. Dinic. Algorithm for solution of a problem of maximum flow in networks with power estimation. Soviet Mathematics Doklady, 11:1277{1280, 1970
  • [4] A. V. Karzanov. Determining the maximal flow in a network by the method of preflows. Soviet Mathematics Doklady, 15:434-437, 1974.
  • [5] B. V. Cherkasky. Algorithm for construction of maximal flow in networks with complexity of $O(V^2\sqrt{E})$ operations. Mathematical Methods of Solution of Economical Problems, 17:112-125, 1977. (In Russian).
  • [6] V. M. Malhotra, P. Kumar, and S. N. Maheshwari. An $O(V^3)$ algorithm for fi})nding the maximum flows in networks. Information Processing Letters, 7:277-278, 1978.
  • [7] Z. Galil. An $O(V^{5/3}E^{2/3})$ algorithm for the maximal flow problem. Acta Informatica, 14(3):221-242, 1980.
  • [8] Z. Galil and A. Naaman. An $O(VElog^2E)$ algorithm for the maximal flow problem. J.Computer and System Sciences, 21:203-217., 1980.
  • [9] D. D. Sleator and R. E. Tarjan. A data structure for dynamic trees. J. Computer and System Sciences, 24:362-391, 1983.
  • [10] H. N. Gabow. A data structure for dynamic trees. J. Computer and System Sciences, 31:148-168, 1985.
  • [11] A. V. Goldberg and R. E. Tarjan. A new approach to the maximum flow problem. Journal of the ACM, 35:921-940, 1988.
  • [12] R. K. Ahuja and J. B. Orlin. A fast and simple algorithm for the maximum flow problem. Operations Research, 37:748-759, 1989.
  • [13] R. K. Ahuja, J. B. Orlin, and R. E. Tarjan. Improved time bounds for the maximum flow problem. SIAM Journal on Computing, 18:939-954, 1989.
  • [14] V. King, S. Rao, and R. Tarjan. A faster deterministic maximum flow algorithm. In Proceedings of the 8th Annual ACM-SIAM Symposium on Discrete Algorithms, pages 157{164, 1992.
  • [15] V. King, S. Rao, and R. Tarjan. A faster deterministic maximum flow algorithm. J. Algorithms, 23:447-474, 1994.
  • [16] J. Cheriyan, T. Hagerup, and K. Mehlhorn. An $O(n^3)$ time maximum-flow algorithm. SIAM Journal on Computing, 45:1144-1170, 1996.
  • [17] A. V. Goldberg and S. Rao. Beyond the flow decomposition barrier. Journal of the ACM, 45:783-797, 1998.
  • [18] J. B. Orlin, “Max flows in $o(nm)$ time, or better,” in Proceedings of the 45th annual ACM symposium on Symposium on theory of computing, ser. STOC ’13. New York, NY,USA: ACM, 2013, pp. 765–774. [Online]. Available: http://doi.acm.org/10.1145/2488608.2488705}