A graph G = (V,E) can be described by the characteristic function of the edge set \(\mathcal{X}_E\) which maps a pair of binary encoded nodes to 1 iff the nodes are adjacent. Using Ordered Binary Decision Diagrams (OBDDs) to store \(\mathcal{X}_E\) can lead to a (hopefully) compact representation. Given the OBDD as an input, symbolic/implicit OBDD-based graph algorithms can solve optimization problems by mainly using functional operations, e.g., quantification or binary synthesis. While the OBDD representation size can not be small in general, it can be provable small for special graph classes and then also lead to fast algorithms. In this paper, we show that the OBDD size of unit interval graphs is O( ∣ V ∣ /log ∣ V ∣ ) and the OBDD size of interval graphs is O( ∣ V ∣ log ∣ V ∣ ) which both improve a known result from Nunkesser and Woelfel (2009). Furthermore, we can show that using our variable order and node labeling for interval graphs the worst-case OBDD size is Ω( ∣ V ∣ log ∣ V ∣ ). We use the structure of the adjacency matrices to prove these bounds. This method may be of independent interest and can be applied to other graph classes. We also develop a maximum matching algorithm on unit interval graphs using O(log ∣ V ∣ ) operations and evaluate the algorithm empirically.


Boolean Function Adjacency Matrix Variable Order Interval Graph Maximum Match 
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.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Arnold, D.B., Sleep, M.R.: Uniform random generation of balanced parenthesis strings. ACM Trans. Program. Lang. Syst. 2(1), 122–128 (1980)CrossRefGoogle Scholar
  2. 2.
    Bloem, R., Gabow, H.N., Somenzi, F.: An algorithm for strongly connected component analysis in n log n symbolic steps. Formal Methods in System Design 28(1), 37–56 (2006)CrossRefzbMATHGoogle Scholar
  3. 3.
    Bollig, B.: On symbolic OBDD-based algorithms for the minimum spanning tree problem. TCS 447, 2–12 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Bollig, B., Gillé, M., Pröger, T.: Implicit computation of maximum bipartite matchings by sublinear functional operations. In: Agrawal, M., Cooper, S.B., Li, A. (eds.) TAMC 2012. LNCS, vol. 7287, pp. 473–486. Springer, Heidelberg (2012)CrossRefGoogle Scholar
  5. 5.
    Bollig, B., Löbbing, M., Wegener, I.: On the effect of local changes in the variable ordering of ordered decision diagrams. IPL 59(5), 233–239 (1996)CrossRefzbMATHGoogle Scholar
  6. 6.
    Bollig, B., Pröger, T.: An efficient implicit OBDD-based algorithm for maximal matchings. In: Dediu, A.-H., Martín-Vide, C. (eds.) LATA 2012. LNCS, vol. 7183, pp. 143–154. Springer, Heidelberg (2012)CrossRefGoogle Scholar
  7. 7.
    Bollig, B., Wegener, I.: Asymptotically optimal bounds for OBDDs and the solution of some basic OBDD problems. Journal of Computer and System Sciences 61(3), 558–579 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Bryant, R.E.: Graph-based algorithms for boolean function manipulation. IEEE Transactions on Computers 35(8), 677–691 (1986)CrossRefzbMATHGoogle Scholar
  9. 9.
    Burch, J.R., Clarke, E.M., McMillan, K.L., Dill, D.L., Hwang, L.J.: Symbolic model checking: 1020 states and beyond. Information and Computation 98(2), 142–170 (1992)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Chung, Y., Park, K., Cho, Y.: Parallel maximum matching algorithms in interval graphs. In: Proc. of the 4th ICPADS, pp. 602–609 (1997)Google Scholar
  11. 11.
    Coudert, O.: Doing two-level logic minimization 100 times faster. In: Proc. of the 6th SODA, pp. 112–121 (1995)Google Scholar
  12. 12.
    Gentilini, R., Piazza, C., Policriti, A.: Computing strongly connected components in a linear number of symbolic steps. In: Proc. of the 14th SODA, pp. 573–582 (2003)Google Scholar
  13. 13.
    Gentilini, R., Piazza, C., Policriti, A.: Symbolic graphs: Linear solutions to connectivity related problems. Algorithmica 50(1), 120–158 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Gillé, M.: OBDD-Based Representation of Interval Graphs. ArXiv e-prints (2013),
  15. 15.
    Golumbic, M.C.: Algorithmic Graph Theory and Perfect Graphs. Annals of Discrete Mathematics, vol. 57. North-Holland Publishing Co. (2004)Google Scholar
  16. 16.
    Hachtel, G.D., Somenzi, F.: A symbolic algorithms for maximum flow in 0-1 networks. Formal Methods in System Design 10(2/3), 207–219 (1997)CrossRefGoogle Scholar
  17. 17.
    Hosaka, K., Takenaga, Y., Kaneda, T., Yajima, S.: Size of ordered binary decision diagrams representing threshold functions. Theor. Comput. Sci. 180(1-2), 47–60 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Lai, Y.-T., Pedram, M., Vrudhula, S.B.K.: EVBDD-based algorithms for integer linear programming, spectral transformation, and function decomposition. IEEE Transactions on CAD of Integrated Circuits and Systems 13(8), 959–975 (1994)CrossRefGoogle Scholar
  19. 19.
    Meer, K., Rautenbach, D.: On the OBDD size for graphs of bounded tree- and clique-width. Discrete Mathematics 309(4), 843–851 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Meinel, C., Theobald, T.: On the influence of the state encoding on OBDD-representations of finite state machines. ITA 33(1), 21–32 (1999)MathSciNetzbMATHGoogle Scholar
  21. 21.
    Mertzios, G.B.: A matrix characterization of interval and proper interval graphs. Applied Mathematics Letters 21(4), 332–337 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Nunkesser, R., Woelfel, P.: Representation of graphs by OBDDs. Discrete Applied Mathematics 157(2), 247–261 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Roberts, F.S.: Indifference graphs. In: Harary, F. (ed.) Proof Techniques in Graph Theory, pp. 139–146 (1969)Google Scholar
  24. 24.
    Saitoh, T., Yamanaka, K., Kiyomi, M., Uehara, R.: Random generation and enumeration of proper interval graphs. IEICE Transactions 93-D(7), 1816–1823 (2010)Google Scholar
  25. 25.
    Sawitzki, D.: Implicit flow maximization by iterative squaring. In: Van Emde Boas, P., Pokorný, J., Bieliková, M., Štuller, J. (eds.) SOFSEM 2004. LNCS, vol. 2932, pp. 301–313. Springer, Heidelberg (2004)CrossRefGoogle Scholar
  26. 26.
    Sawitzki, D.: The complexity of problems on implicitly represented inputs. In: Wiedermann, J., Tel, G., Pokorný, J., Bieliková, M., Štuller, J. (eds.) SOFSEM 2006. LNCS, vol. 3831, pp. 471–482. Springer, Heidelberg (2006)CrossRefGoogle Scholar
  27. 27.
    Sawitzki, D.: Exponential lower bounds on the space complexity of OBDD-based graph algorithms. In: Correa, J.R., Hevia, A., Kiwi, M. (eds.) LATIN 2006. LNCS, vol. 3887, pp. 781–792. Springer, Heidelberg (2006)CrossRefGoogle Scholar
  28. 28.
    Sieling, D., Wegener, I.: NC-algorithms for operations on binary decision diagrams. Parallel Processing Letters 3, 3–12 (1993)MathSciNetCrossRefGoogle Scholar
  29. 29.
    Wegener, I.: Branching programs and binary decision diagrams. SIAM Monographs on Discrete Mathematics and Applications (2000)Google Scholar
  30. 30.
    Woelfel, P.: Symbolic topological sorting with OBDDs. Journal of Discrete Algorithms 4, 51–71 (2006)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  • Marc Gillé
    • 1
  1. 1.LS2 InformatikTU DortmundGermany

Personalised recommendations