Principal lattice of partitions of submodular functions on graphs: Fast algorithms for principal partition and generic rigidity

  • Sachin Patkar
  • H. Narayanan
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 650)


In this paper we use a single unifying approach (which we call the Principal Lattice of Partitions approach) to construct simple and fast algorithms for problems including and related to the “Principal Partition” and the “Generic Rigidity” of graphs. Most of our algorithms are at least as fast as presently known algorithms for these problems, while our algorithm for Principal Partition problem (complete partition and the partial orders for all critical values) runs in O(¦E∥V¦2log2¦V¦) time and is the fastest known so far.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [1]
    Gabow, H.N. and Westermann, H.H.: Forests, Frames and Games: Algorithms for Matroid sums and Applications, in Proc. 20th STOC, 1988, pp. 407–421.Google Scholar
  2. [2]
    Imai, H.: Network flow algorithms for lower truncated transversal polymatroids, J. of the Op. Research Society of Japan, vol. 26, 1983, pp. 186–210.Google Scholar
  3. [3]
    Iri, M. and Fujishige, S.: Use of Matroid Theory in Operations Research, Circuits and Systems Theory, Int. J. Systems Sci.,vol. 12, no. 1, 1981, pp. 27–54.Google Scholar
  4. [4]
    Nakamura, M.: On the Representation of the Rigid Sub-systems of a Plane Link System, J. Op. Res. Soc. of Japan, vol. 29, No. 4, 1986, pp. 305–318.Google Scholar
  5. [5]
    Narayanan, H.: Theory of Matroids and Network Analysis, Ph.D. thesis, Department of Electrical Engineering, IIT Bombay, INDIA, 1974.Google Scholar
  6. [6]
    Narayanan, H.: The Principal Lattice of Partitions of a Submodular function, Linear Algebra and its Applications, 144, 1991, pp. 179–216.CrossRefGoogle Scholar
  7. [7]
    Patkar, S. and Narayanan, H.: Principal Lattice of Partitions of the Rank Function of a Graph, Technical Report VLSI-89-3, IIT Bombay, INDIA, 1989.Google Scholar
  8. [8]
    Patkar, S. and Narayanan, H.: Fast algorithm for the Principal Partition of a graph, in Proc. 11th ann. symp. on Foundations of Software Technology and Theoretical Computer Science (FST & TCS-11), LNCS-560, 1991, pp. 288–306.Google Scholar
  9. [9]
    Patkar, S.:Investigations into the structure of graphs through the Principal Lattice of Partitions approach, Ph.D. thesis, Dept. of Computer Sci. and Engg., IIT Bombay, INDIA, 1992.Google Scholar
  10. [10]
    Sleator, D.D. and Tarjan, R.E.: A data structure for dynamic trees, J. Comp. and System Sci., vol. 26, 1983, pp. 362–391.CrossRefGoogle Scholar
  11. [11]
    Sugihara, K.: On Redundant Bracing in Plane Skeletal Structures, Bulletin of the Electrotechnical Laboratory, vol. 44, 1980, pp. 376–386.Google Scholar
  12. [12]
    Tomizawa, N.: Strongly Irreducible Matroids and Principal Partition of a Matroid into Strongly Irreducible Minors (in Japanese), Transactions of the Institute of Electronics and Communication Engineers of Japan, vol. J59A, 1976, pp. 83–91.Google Scholar
  13. [13]
    Welsh, D. J. A.: Matroid Theory, Academic Press, New York, 1976.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1992

Authors and Affiliations

  • Sachin Patkar
    • 1
  • H. Narayanan
    • 2
  1. 1.Dept. of Computer Science and Engg.IIT BombayBombayIndia
  2. 2.Dept. of Electrical Engg.IIT BombayBombayIndia

Personalised recommendations