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Symbolic Graphs: Linear Solutions to Connectivity Related Problems

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Abstract

The importance of symbolic data structures such as Ordered Binary Decision Diagrams (OBDD) is rapidly growing in many areas of Computer Science where the large dimensions of the input models is a challenging feature: OBDD based graph representations allowed to define truly new standards in the achievable dimensions for the Model Checking verification technique. However, OBDD representations pose strict constraints in the algorithm design issue. For example, Depth-First Search (DFS) is not feasible in a symbolic framework and, consequently, many state-of-the-art DFS based algorithms (e.g., connectivity procedures) cannot be easily rearranged to work on symbolically represented graphs. We devise here a symbolic algorithmic strategy, based on the new notion of spine-set, that is general enough to be the engine of linear symbolic step algorithms for both strongly connected components and biconnected components. Our procedures improve on previously designed connectivity symbolic algorithms. Moreover, by an application to the so-called “bad cycle detection problem”, our technique can be used to efficiently solve the emptiness problem for various kinds of ω-automata.

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References

  1. Aho, A.V., Hopcroft, J.E., Ullman, J.D.: The Design and Analysis of Computer Algorithms. Addison–Wesley, Reading (1974)

    MATH  Google Scholar 

  2. Akers, S.B.: Binary decision diagrams. IEEE Trans. Comput. C-27(6), 509–516 (1978)

    Article  Google Scholar 

  3. Bloem, R.P.: Search Techniques and Automata for Symbolic Model Checking. University of Colorado (2001)

  4. Bloem, R.P., Gabow, H.N., Somenzi, F.: An algorithm for strongly connected component analysis in nlog n symbolic steps. In: Hunt, W.A. Jr., Johnson, S.D. (eds.) Proc. of Int. Conference on Formal Methods in Computer-Aided Design (FMCAD’00). LNCS, vol. 1954, pp. 37–54. Springer, Berlin (2000)

    Chapter  Google Scholar 

  5. Bloem, R.P., Gabow, H.N., Somenzi, F.: An algorithm for strongly connected component analysis in nlog n symbolic steps. Form. Methods Syst. Des. 28, 1–20 (2005)

    Google Scholar 

  6. Bollig, B., Wegener, I.: Improving the variable ordering of OBDDs is NP-complete. IEEE Trans. Comput. 45(9), 993–1002 (1996)

    Article  MATH  Google Scholar 

  7. Bouali, A., de Simone, R.: Symbolic bisimulation minimization. In: von Bochmann, G., Probst, D.K. (eds.) Proc. of Int. Conference on Computer Aided Verification (CAV’92). LNCS, vol. 663, pp. 96–108. Springer, Berlin (1992)

    Google Scholar 

  8. Brayton, R.K., Hachtel, G.D., Sangiovanni-Vincentelli, A., Somenzi, F., Aziz, A., Cheng, S.T., Edwards, S., Khatri, S., Kukimoto, Y., Pardo, A., Qadeer, S., Ranjan, R.K., Sarwary, S., Shiple, T.R., Swamy, G., Villa, T.: VIS: a system for verification and synthesis. In: Alur, R., Henzinger, T.A. (eds.) Proc. of Int. Conference on Computer Aided Verification (CAV’96). LNCS, vol. 1102, pp. 428–432. Springer, Berlin (1996)

    Google Scholar 

  9. Bryant, R.E.: Graph-based algorithms for boolean function manipulation. IEEE Trans. Comput. C-35(8), 677–691 (1986)

    Article  Google Scholar 

  10. Bryant, R.E.: Symbolic boolean manipulation with ordered binary decision diagrams. ACM Comput. Surv. 24(3), 293–318 (1992)

    Article  Google Scholar 

  11. Burch, J.R., Clarke, E.M., McMillan, K.L., Dill, D.L., Hwang, L.J.: Symbolic model checking: 1020 states and beyond. In: Proc. of IEEE Symp. on Logic in Computer Science (LICS’90), pp. 1–33. IEEE Computer Society, Los Alamitos (1990)

    Google Scholar 

  12. Clarke, E.M., Grumberg, O., Peled, D.A.: Model Checking. MIT Press, Cambridge (1999)

    Google Scholar 

  13. Cormen, T.H., Leiserson, C.E., Rivest, R.L.: Introduction to Algorithms. MIT Press, Cambridge (1990)

    Google Scholar 

  14. Drechsler, R., Sieling, D.: Binary decision diagrams in theory and practice. Softw. Tools Technol. Transf. 3, 103–112 (2001)

    Google Scholar 

  15. Emerson, E.A., Lei, C.L.: Efficient model checking in fragments of the propositional mu-calculus. In: Proc. of IEEE Symp. on Logic in Comuter Science (LICS’86), pp. 267–278. IEEE Computer Society, Los Alamitos (1986)

    Google Scholar 

  16. Feigenbaum, J., Kannan, S., Vardi, M.Y., Viswanathan, M.: The complexity of problems on graphs represented as OBDDs. Chic. J. Theor. Comput. Sci. 47, 1–25 (1999)

    MathSciNet  Google Scholar 

  17. Fisler, K., Vardi, M.Y.: Bisimulation and model checking. In: Pierre, L., Kropf, T. (eds.) Proc. of Correct Hardware Design and Verification Methods (CHARME’99). LNCS, vol. 1703, pp. 338–341. Springer, Berlin (1999)

    Google Scholar 

  18. Fisler, K., Vardi, M.Y.: Bisimulation minimization and symbolic model checking. Form. Methods Syst. Des. 21(1), 39–78 (2002)

    Article  MATH  Google Scholar 

  19. Francez, N.: Fairness. Springer, Berlin (1986)

    MATH  Google Scholar 

  20. Frank, A.: Connectivity and network flows. In: Handbook of combinatorics, vol. 1, pp. 875–917. Elsevier, Amsterdam (1995)

    Google Scholar 

  21. Gabbay, D.M., Pnueli, A., Shelah, S., Stavi, J.: On the temporal basis of fairness. In: Proc. of ACM Sym. on Principles of Programming Languages (POPL’80), pp. 163–173. ACM, New York (1980)

    Google Scholar 

  22. Gentilini, R., Policriti, A.: Biconnectivity on symbolically represented graphs: a linear solution. In: Proc. of Int. Symposium on Algorithms and Computation (ISAAC’03). LNCS, vol. 2906, pp. 554–564. Springer, Berlin (2003)

    Google Scholar 

  23. Gentilini, R., Piazza, C., Policriti, A.: Computing strongly connected components in a linear number of symbolic steps. In: Proc. of Int. Symposium on Discrete Algorithms (SODA’03), pp. 573–582. ACM, New York (2003)

    Google Scholar 

  24. Hachtel, G.D., Somenzi, F.: A symbolic algorithms for maximum flow in 0-1 networks. Form. Methods Syst. Des. 10(2/3), 207–219 (1997)

    Article  Google Scholar 

  25. Hojati, R., Touati, H.J., Kurshan, R.P., Brayton, R.K.: Efficient mega-regular language containment. In: von Bochmann, G., Probst, D.K. (eds.) Proc of Int. Conference on Computer Aided Verification (CAV’92). LNCS, vol. 663, pp. 396–409. Springer, Berlin (1992)

    Google Scholar 

  26. Hopcroft, J.E., Tarjan, R.E.: Efficient algorithms for graph manipulation [h] (algorithm 447). Commun. ACM 16(6), 372–378 (1973)

    Article  Google Scholar 

  27. Lei, C.Y.: Representation of switching circuits by binary-decision programs. Bell Syst. Tech. J. 38, 985–999 (1959)

    Google Scholar 

  28. McMillan, K.L.: Symbolic Model Checking: An Approach to the State Explosion Problem. Kluwer Academic, Dordrecht (1993)

    Google Scholar 

  29. Meinel, C., Theobald, T.: Algorithms and Data Structures in VLSI Design. OBDD—Foundations and Applications. Springer, Berlin (1998)

    Google Scholar 

  30. Nunkesser, R., Woelfel, P.: Representation of Graphs by OBDDs. In: Proc. of Int. Symposium on Algorithms and Computation (ISAAC’05). LNCS, vol. 3827, pp. 1132–1142. Springer, Berlin (2005)

    Google Scholar 

  31. Ravi, K., Bloem, R.P., Somenzi, F.: A comparative study of symbolic algorithms for the computation of fair cycles. In: Hunt, W.A. Jr., Johnson, S.D. (eds.) Proc. of Int. Conference on Formal Methods in Computer-Aided Design (FMCAD’00). LNCS, vol. 1954, pp. 143–160. Springer, Berlin (2000)

    Chapter  Google Scholar 

  32. Sanghavi, J.V., Ranjan, R.K., Brayton, R.K., Sangiovanni-Vincentelli, A.: High performance BDD package based on exploiting memory hierarchy. In: Proc. of ACM/IEEE Design Automation Conference, 1996

  33. Sawitzki, D.: Implicit flow maximization by iterative squaring. In: Proc. of Int. Conference on Current Trends in Theory and Practice of Computer Science (SOFSEM2004). LNCS, vol. 2932, pp. 301–313. Springer, Berlin (2004)

    Google Scholar 

  34. Sawitzki, D.: A symbolic approach to the all-pairs shortest-paths problem. In: Proc. of Int. Conference on Graph-Theoretic Concepts in Computer Science (WG 2004). LNCS, vol. 3357, pp. 154–167. Springer, Berlin (2004)

    Google Scholar 

  35. Schneider, K.: Verification of Reactive Systems. Springer, Berlin (2004)

    MATH  Google Scholar 

  36. Sieling, D.: The nonapproximability of OBDD minimization. Inf. Comput. 172(2), 103–138 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  37. Somenzi, F.: Binary decision diagrams. In: Calculational System Design, Nato Science Series F: Computer and Systems Sciences, vol. 173, pp. 303–366. IOS Press, Amsterdam (1999)

    Google Scholar 

  38. Somenzi, F.: CUDD: CU Decision Diagram Package Release 2.3.1 (2001). Available at http://vlsi.colorado.edu/~fabio/CUDD/cuddIntro.html

  39. Tarjan, R.E.: Depth first search and linear graph algorithms. SIAM J. Comput. 1(2), 146–160 (1972)

    Article  MATH  MathSciNet  Google Scholar 

  40. Thomas, W.: Automata on infinite objects. In: Handbook of Theoretical Computer Science, vol. B, pp. 133–191. MIT Press, Cambridge (1990)

    Google Scholar 

  41. Touati, J.H., Brayton, R.K., Kurshan, R.P.: Testing language containment for omega-automata using BDD’s. Inf. Comput. 118(1), 101–109 (1995)

    Article  MATH  MathSciNet  Google Scholar 

  42. Vardi, M.Y.: An automata-theoretic approach to linear temporal logic. In: Logics for Concurrency: Structure versus Automata. LNCS, vol. 1043, pp. 238–266. Springer, Berlin (1996)

    Google Scholar 

  43. Wegener, I.: BDDs design, analysis, complexity, and applications. Discrete Appl. Math. 138, 229–251 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  44. Woelfel, R.: Symbolic topological sorting with OBDDs. J. Discrete Algorithms (2006, to appear) (also in MFCS’03, LNCS, vol. 2747, pp. 671–680)

  45. Xie, A., Beerel, P.: Implicit enumeration of strongly connected components and an application to formal verification. IEEE Trans. Comput. Aided Design Integrated Circuits Syst. 19, 1225–1230 (2000)

    Article  Google Scholar 

  46. Yuan, L., Gui, C., Chuah, C., Mohapatra, P.: Applications and design of heterogeneous and or broadband sensor networks. In: Proc. of IEEE First Workshop on Broadband Advanced Sensor Networks. IEEE Press, New York (2003)

    Google Scholar 

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Authors and Affiliations

  1. Università di Udine (DIMI), Via Le Scienze 206, 33100, Udine, Italy

    Raffaella Gentilini, Carla Piazza & Alberto Policriti

  2. Department of Computer Science Reactive Systems Group, Kaiserslautern University, Kaiserslautern, Germany

    Raffaella Gentilini

  3. Institute of Applied Geonomics, Odine, Italy

    Alberto Policriti

Authors
  1. Raffaella Gentilini
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  2. Carla Piazza
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  3. Alberto Policriti
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Correspondence to Alberto Policriti.

Additional information

This work is a revised and extended version of [22,23]. It is partially supported by the projects PRIN 2005015491 and BIOCHECK.

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Gentilini, R., Piazza, C. & Policriti, A. Symbolic Graphs: Linear Solutions to Connectivity Related Problems. Algorithmica 50, 120–158 (2008). https://doi.org/10.1007/s00453-007-9079-5

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