對最大流算法歷史文獻的一個調研算法
Table: Polynomial algorithms for the max flow problemapp
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