I/O Efficient Algorithms for the Minimum Cut Problem on Unweighted Undirected Graphs

  • Alka Bhushan
  • G. Sajith
Part of the Lecture Notes in Computer Science book series (LNCS, volume 8344)


The problem of finding the minimum cut of an undirected unweighted graph is studied on the external memory model. First, a lower bound of Ω((E/V) Sort(V)) on the number of I/Os is shown for the problem, where V is the number of vertices and E is the number of edges. Then the following are presented, for M = Ω(B 2), (1) a minimum cut algorithm that uses \(O(c \log E ({\rm MSF}{(V,E)} + \frac{V}{B} {\rm Sort}({V})))\) I/Os; here MSF(V,E) is the number of I/Os needed to compute a minimum spanning tree of the graph, and c is the value of the minimum cut. The algorithm performs better on dense graphs than the algorithm of [7], which requires O(E + c 2 V log(V/c)) I/Os, when executed on the external memory model. For a δ-fat graph (for δ > 0, the maximum tree packing of the graph is at least (1 + δ)c/2), our algorithm computes a minimum cut in O(c logE (MSF(V,E) + Sort(E))) I/Os. (2) a randomized algorithm that computes minimum cut with high probability in \(O(c \log E \cdot{\rm MSF}{(V,E)} + {\rm Sort}{(E)} \log^2 V + \frac{V}{B} {\rm Sort}{(V)} \log V)\) I/Os. (3) a (2 + ε)-minimum cut algorithm that requires O((E/V) MSF(V,E)) I/Os and performs better on sparse graphs than our exact minimum cut algorithm.


Span Tree Minimum Span Tree Tree Packing Cluster Versus Tree Edge 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


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  1. 1.
    Aggarwal, A., Vitter, J.S.: The input/output complexity of sorting and related problems. Commun. ACM 31(9), 1116–1127 (1988)CrossRefMathSciNetGoogle Scholar
  2. 2.
    Aggarwal, G., Datar, M., Rajagopalan, S., Ruhl, M.: On the streaming model augmented with a sorting primitive. In: Proc. IEEE Symposium on Foundations of Computer Science, pp. 540–549 (2004)Google Scholar
  3. 3.
    Alka: Efficient algorithms and data structures for massive data sets, Ph.D. thesis, Indian Institute of Technology Guwahati, India (2009),
  4. 4.
    Brinkmeier, M.: A simple and fast min-cut algorithm. Theor. Comp. Sys. 41(2), 369–380 (2007)CrossRefzbMATHMathSciNetGoogle Scholar
  5. 5.
    Chiang, Y.J., Goodrich, M.T., Grove, E.F., Tamassia, R., Vengroff, D.E., Vitter, J.S.: External memory graph algorithms. In: Proc. ACM-SIAM Symposium on Discrete Algorithms, pp. 139–149 (1995)Google Scholar
  6. 6.
    Ford, L.R., Fulkerson, D.R.: Maximal flow through a network. Can. J. Math. 8, 399–404 (1956)CrossRefzbMATHMathSciNetGoogle Scholar
  7. 7.
    Gabow, H.N.: A matroid approach to finding edge connectivity and packing arborescences. J. Comput. Syst. Sci. 50(2), 259–273 (1995)CrossRefzbMATHMathSciNetGoogle Scholar
  8. 8.
    Goldschlager, L.M., Shaw, R.A., Staples, J.: The maximum flow problem is LOGSPACE complete for P. Theoret. Comput. Sci. 21(1), 105–111 (1982)CrossRefzbMATHMathSciNetGoogle Scholar
  9. 9.
    Hao, J., Orlin, J.: A faster algorithm for finding the minimum cut in a graph. J. Algorithms 17(3), 424–446 (1994)CrossRefzbMATHMathSciNetGoogle Scholar
  10. 10.
    Karger, D.R.: Minimum cuts in near linear time. J. ACM 47(1), 46–76 (2000)CrossRefzbMATHMathSciNetGoogle Scholar
  11. 11.
    Karger, D.R., Motwani, R.: An NC algorithm for minimum cuts. SIAM J. Comput. 26(1), 255–272 (1997)CrossRefzbMATHMathSciNetGoogle Scholar
  12. 12.
    Matula, D.W.: A linear time 2 + ε approximation algorithm for edge connectivity. In: Proc. ACM-SIAM Symposium on Discrete Algorithms, pp. 500–504 (1993)Google Scholar
  13. 13.
    Mungala, K., Ranade, A.: I/O-complexity of graph algorithms. In: Proc. ACM-SIAM Symposium on Discrete Algorithms, pp. 687–694 (1999)Google Scholar
  14. 14.
    Nagamochi, H., Ibaraki, T.: A linear-time algorithm for finding a sparse k-connected spanning subgraph of a k-connected graph. Algorithmica 7(5-6), 583–596 (1992)CrossRefzbMATHMathSciNetGoogle Scholar
  15. 15.
    Nash-Williams, C.S.J.A.: Edge-disjoint spanning tree of finite graphs. J. London Math. Soc. 36, 445–450 (1961)CrossRefzbMATHMathSciNetGoogle Scholar
  16. 16.
    Plotkin, S.A., Shmoys, D.B., Tardos, É.: Fast approximation algorithms for fractional packing and covering problems. Math. Oper. Res. 20(2), 257–301 (1995)Google Scholar
  17. 17.
    Stoer, M., Wagner, F.: A simple min-cut algorithm. J. ACM 44(4), 585–591 (1997)CrossRefzbMATHMathSciNetGoogle Scholar
  18. 18.
    Thorup, M., Karger, D.R.: Dynamic graph algorithms with applications. In: Halldórsson, M.M. (ed.) SWAT 2000. LNCS, vol. 1851, pp. 1–9. Springer, Heidelberg (2000)CrossRefGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2014

Authors and Affiliations

  • Alka Bhushan
    • 1
  • G. Sajith
    • 1
  1. 1.Department of Computer Science and EngineeringIndian Institute of Technology GuwahatiIndia

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