Dynamic data structures for series parallel digraphs

preliminary version
  • Giuseppe F. Italiano
  • Alberto Marchetti Spaccamela
  • Umberto Nanni
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 382)


We consider the problem of dynamically maintaining minimal series parallel directed acyclic graphs (MSP dags), general series parallel directed acyclic graphs (GSP dags), two-terminal series parallel directed acyclic graphs (TTSP dags) and looped series parallel directed graphs (looped SP digraphs). We present data structures for updating (by both inserting and deleting either a group of edges or vertices) MSP dags, GSP dags, TTSP dags and looped SP digraphs of m edges and n vertices in O(log n) worst-case time. The time required to check whether there is a path between two given vertices is O(log n), while a path of length k can be traced out in O(k+log n) time. For MSP and TTSP dags, our data structures are able to report a regular expression describing all the paths between two vertices x and y in O(h+log n), where hn is the total number of vertices which are contained in paths from x to y. Although MSP and GSP dags can have as many as O(n2) edges, we use an implicit representation which requires only O(n) space. Motivations for studying dynamic graphs arise in several areas, such as communication networks, incremental compilation environments and the design of very high level languages, while the dynamic maintenance of series parallel graphs is also relevant in incremental data flow analysis.


Minimum Span Tree Transitive Closure Parallel Composition Decomposition Tree Dynamic Graph 
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]
    F. Afrati, An efficient parallel algorithm for directed reachability in series parallel graphs, manuscript 1988.Google Scholar
  2. [2]
    F. Afrati, D. Goldin, and P. Kanellakis, Efficient parallelism for structured data: directed reachability in S-P dags, Technical Report, Brown University, 1988.Google Scholar
  3. [3]
    A. V. Aho, M. R. Garey, and J. D. Ullman, The transitive reduction of a directed graph, SIAM J. Comput. 1 (1972), 131–137.CrossRefGoogle Scholar
  4. [4]
    A. V. Aho, J. E. Hopcroft, and J. D. Ullman, The Design and Analysis of Computer Algorithms, Addison-Wesley, Reading, MA, 1974.Google Scholar
  5. [5]
    M. Ajtai and R. Fagin, Reachability is harder for directed than for undirected graphs, Proc. 29th Annual Symp. on Foundations of Computer Science, 1988, 358–367.Google Scholar
  6. [6]
    G. Ausiello, G. F. Italiano, A. Marchetti Spaccamela, and U. Nanni, On-line computation of minimal and maximal length paths, in preparation.Google Scholar
  7. [7]
    G. Ausiello, A. Marchetti Spaccamela, and U. Nanni, Dynamic maintenance of paths and path expressions in graphs, Proc. 1st Internat. Joint Conf. ISSAC 88 (Int. Symp. on Symbolic and Algebraic Computation) and AAECC 6 (6th Int. Conf. on Applied Algebra, Algebraic Algorithms and Error Correcting Codes), Lecture Notes in Computer Science, Springer-Verlag, Berlin, 1989.Google Scholar
  8. [8]
    M. W. Bern, E. L. Lawler, and A. L. Wong, Why certain subgraph computations require only linear time, Proc. 26th Annual Symp. on Foundations of Computer Science, 1985, 117–125.Google Scholar
  9. [9]
    M. Burke, and B. G. Ryder, Incremental iterative data flow analysis algorithms, Technical Report LCSR-TR-96, Department of Computer Science, Rutgers University, August 1987.Google Scholar
  10. [10]
    M. D. Carrol and B. G. Ryder, Incremental data flow analysis via dominator and attribute updates, Proc. 15th Annual ACM SIGACT-SIGPLAN Symposium on Principles of Programming Languages, 1988, 274–284.Google Scholar
  11. [11]
    F. Chin and D. Houk, Algorithms for updating minimum spanning trees, J. Comput. System. Sci. 16 (1978), 333–344.CrossRefGoogle Scholar
  12. [12]
    R. J. Duffin, Topology of series parallel networks, Journal of Mathematical Analysis and Applications 10 (1965), 303–318.CrossRefGoogle Scholar
  13. [13]
    D. Eppstein, Parallel recognition of series-parallel graphs, manuscript, 1989.Google Scholar
  14. [14]
    D. Eppstein, G. F. Italiano, M. Yung, Minimum spanning trees in dynamic planar graphs, Tech. Rep. CUCS-434-89, Department of Computer Science, Columbia University, 1989.Google Scholar
  15. [15]
    S. Even, and H. Gazit, Updating distances in dynamic graphs, Methods of Operations Research 49 (1985), 371–387.Google Scholar
  16. [16]
    S. Even, and Y. Shiloach, An on-line edge deletion problem, J. Assoc. Comput. Mach. 28 (1981), 1–4.Google Scholar
  17. [17]
    G. N. Frederickson, Data structures for on-line updating of minimum spanning trees, SIAM J. Comput. 14 (1985), 781–798.CrossRefGoogle Scholar
  18. [18]
    G. N. Frederickson and M. A. Srinivas, On-line updating of degree-constrained minimu spanning trees, Proc. 22nd Annual Allerton Conference on Communication, Control and Computing, 1984.Google Scholar
  19. [19]
    D. Harel, On-line maintenance of the connected components of dynamic graphs, Unpublished manuscript, 1982.Google Scholar
  20. [20]
    N. Horspool, Incremental generation of LR parsers, Technical Report, Department of Computer Science, University of Victoria, March 1988.Google Scholar
  21. [21]
    T. Ibaraki, and N. Katoh, On-line computation of transitive closure for graphs, Inform. Process. Lett. 16 (1983), 95–97.Google Scholar
  22. [22]
    G. F. Italiano, Amortized efficiency of a path retrieval data structure, Theoret. Comput. Sci. 48 (1986), 273–281.Google Scholar
  23. [23]
    G. F. Italiano, Finding paths and deleting edges in directed acyclic graphs, Inform. Process. Lett. 28 (1988), 5–11.Google Scholar
  24. [24]
    T. Kikuno, N. Yoshida, and Y. Kakuda, A linear time algorithm for the domination number of a series parallel graph, Disc. Appl. Math. 5 (1983), 299–311.Google Scholar
  25. [25]
    J. A. La Poutré, and J. van Leeuwen, Maintenance of transitive closure and transitive reduction of graphs, Proc. International Workshop on Graph-Theoretic Concepts in Computer Science, Lecture Notes in Computer Science, vol. 314, Springer-Verlag, Berlin, 1988, 106–120.Google Scholar
  26. [26]
    E. L. Lawler, Sequencing jobs to minimize total weight completion time subject to precedence constraints, Annals of Discrete Math. 2 (1978), 75–90.Google Scholar
  27. [27]
    C. L. Monma, and J. B. Sidney, Sequencing with series-parallel precedence constraints, Math. of Operations Research 4, 215–224.Google Scholar
  28. [28]
    F. P. Preparata, and R. Tamassia, Fully dynamic techniques for point location and transitive closure in planar structures, Proc. 29th Annual Symp. on Foundations of Computer Science, 1988, 558–567.Google Scholar
  29. [29]
    J. H. Reif, A topological approach to dynamic graph connectivity, Inform. Process. Lett. 25 (1987), 65–70.Google Scholar
  30. [30]
    H. Rohnert, A dynamization of the all pairs least cost path problem, Proc. 2nd Annual Symp. on Theoretical Aspects of Computer Science (STACS 85), Lecture Notes in Computer Science, vol. 182, Springer-Verlag, Berlin, 1985, 279–286.Google Scholar
  31. [31]
    D. D. Sleator, and R. E. Tarjan, A data structure for dynamic trees, J. Comput. System Sci. 24 (1983), 362–381.Google Scholar
  32. [32]
    P. M. Spira and A. Pan, On finding and updating spanning trees and shortest paths, SIAM J. Comput. 4 (1975), 375–380.Google Scholar
  33. [33]
    K. Takamizawa, T. Nishizeki, and N. Saito, Linear time computability of combinatorial problems on series parallel graphs, J. Assoc. Comput. Mach. 29 (1982), 623–641.Google Scholar
  34. [34]
    R. E. Tarjan, Data structures and network algorithms, CBMS-NSF Regional Conference Series in Applied Mathematics, vol. 44, SIAM, 1983.Google Scholar
  35. [35]
    R. E. Tarjan, Amortized computational complexity, SIAM J. Alg. Disc. Meth. 6 (1985), 306–318.Google Scholar
  36. [36]
    R. E. Tarjan, and J. van Leeuwen, Worst-case analysis of set union algorithms, J. Assoc. Comput. Mach. 31 (1984), 245–281.MathSciNetGoogle Scholar
  37. [37]
    J. Valdes, R. E. Tarjan, and E. L. Lawler, The recognition of series parallel digraphs, SIAM J. Comput. 11 (1982), 298–313.Google Scholar
  38. [38]
    H. Whitney, Non-separable and planar graphs, Trans. Amer. Math. Soc. 34 (1932), 339–362.MathSciNetGoogle Scholar
  39. [39]
    D. M. Yellin, A dynamic transitive closure algorithm, Research Report, IBM Research Division, T. J. Watson Research Center, Yorktown Heights, NY 10598, February 1988.Google Scholar
  40. [40]
    D. M. Yellin, and R. Strom, INC: a language for incremental computations, Proc. ACM SIGPLAN '88 Conference on Programming Language Design and Implementation, 1988, 115–124.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1989

Authors and Affiliations

  • Giuseppe F. Italiano
    • 1
    • 2
  • Alberto Marchetti Spaccamela
    • 3
  • Umberto Nanni
    • 2
  1. 1.Department of Computer ScienceColumbia UniversityNew York
  2. 2.Dipartimento di Informatica e SistemisticaUniversità di Roma “La Sapienza”RomaItaly
  3. 3.Dipartimento di Matematica Pura e ApplicataUniversità di L'AquilaL'AquilaItaly

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