Theory of Computing Systems

, Volume 61, Issue 2, pp 322–351 | Cite as

Processing Succinct Matrices and Vectors



We study the complexity of algorithmic problems for matrices that are represented by multi-terminal decision diagrams (MTDD). These are a variant of ordered decision diagrams, where the terminal nodes are labeled with arbitrary elements of a semiring (instead of 0 and 1). A simple example shows that the product of two MTDD-represented matrices cannot be represented by an MTDD of polynomial size. To overcome this deficiency, we extended MTDDs to + MTDDs by allowing componentwise symbolic addition of variables (of the same dimension) in rules. It is shown that accessing an entry, equality checking, matrix multiplication, and other basic matrix operations can be solved in polynomial time for + MTDD-represented matrices. On the other hand, testing whether the determinant of a MTDD-represented matrix vanishes is P S P A C E-complete, and the same problem is N P-complete for + MTDD-represented diagonal matrices. Computing a specific entry in a product of MTDD-represented matrices is # P-complete.


Multi-terminal decision diagrams Succinct matrix representations Matrix computation 


  1. 1.
    Allender, E., Balaji, N., Datta, S.: Low-depth uniform threshold circuits and the bit-complexity of straight line programs. In: Proceedings of MFCS 2014, LNCS 8635, pp 13–24. Springer, Berlin Heidelberg New York (2014)Google Scholar
  2. 2.
    Àlvarez, C., Jenner, B.: A very hard log-space counting class. Theor. Comput. Sci. 107, 3–30 (1993)MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    Beaudry, M., Holzer, M.: The complexity of tensor circuit evaluation. Comput. Complex. 16(1), 60–111 (2007)MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Berman, P., Karpinski, M., Larmore, L.L., Plandowski, W., Rytter, W.: On the complexity of pattern matching for highly compressed two-dimensional texts. J. Comput. Syst. Sci. 65, 332–350 (2002)MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Bertoni, A., Choffrut, C., Radicioni, R.: Literal shuffle of compressed words. In: Proceedings of IFIP TCS 2008, volume 273 of IFIP, pp 87–100. Springer, Berlin Heidelberg New York (2008)Google Scholar
  6. 6.
    Blumensath, A., Grädel, E.: Finite presentations of infinite structures: automata and interpretations. Theor. Comput. Syst. 37, 641–674 (2004)MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Bryant, R.E.: Graph-based algorithms for boolean function manipulation. IEEE Trans. Comput. 35(8), 677–691 (1986)CrossRefMATHGoogle Scholar
  8. 8.
    Caussinus, H., McKenzie, P., Thérien, D., Vollmer, H.: Nondeterministic NC1 computation. J. Comput. Syst. Sci. 57(2), 200–212 (1998)CrossRefMATHGoogle Scholar
  9. 9.
    Cook, S.A.: A taxonomy of problems with fast parallel algorithms. Inf. Control. 64, 2–22 (1985)MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    Damm, C., Holzer, M., McKenzie, P.: The complexity of tensor calculus. Comput. Complex. 11(1–2), 54–89 (2002)MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    Eppstein, D., Goodrich, M.T., Sun, J.Z.: Skip quadtrees: dynamic data structures for multidimensional point sets. Int. J. Comput. Geom. Appl. 18, 131–160 (2008)MathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    Feigenbaum, J., Kannan, S., Vardi, M.Y., Viswanathan, M.: The complexity of problems on graphs represented as obdds Chicago Journal of Theoretical Computer Science (1999)Google Scholar
  13. 13.
    Fujii, H., Ootomo, G., Hori, C.: Interleaving based variable ordering methods for ordered binary decision diagrams. IEEE Computer Society, pp 38–41 (1993)Google Scholar
  14. 14.
    Fujita, M., McGeer, P.C., Yang, J.C.-Y.: Multi-terminal binary decision diagrams: an efficient data structure for matrix representation. Form. Method Syst. Des. 10(2/3), 149–169 (1997)CrossRefGoogle Scholar
  15. 15.
    Galota, M., Vollmer, H.: Functions computable in polynomial space. Inf. Comput. 198(1), 56–70 (2005)MathSciNetCrossRefMATHGoogle Scholar
  16. 16.
    Galperin, H., Wigderson, A.: Succinct representations of graphs. Inf. Control. 56, 183–198 (1983)MathSciNetCrossRefMATHGoogle Scholar
  17. 17.
    Geffert, V., Mereghetti, C., Palano, B.: More concise representation of regular languages by automata and regular expressions. Inf. Comput. 208(4), 385–394 (2010)MathSciNetCrossRefMATHGoogle Scholar
  18. 18.
    Grenet, B., Koiran, P., Portier, N.: On the complexity of the multivariate resultant. J. Complex. 29(2), 142–157 (2013)MathSciNetCrossRefMATHGoogle Scholar
  19. 19.
    Hayashida, M., Ruan, P., Akutsu, T.: A quadsection algorithm for grammar-based image compression. Integr. Comput. Aided Eng. 19(1), 23–38 (2012)Google Scholar
  20. 20.
    Ibarra, O.H., Moran, S.: Probabilistic algorithms for deciding equivalence of straight-line programs. J. Assoc. Comput. Mach. 30(1), 217–228 (1983)MathSciNetCrossRefMATHGoogle Scholar
  21. 21.
    König, D., Lohrey, M.: Evaluating matrix circuits. Technical report. arXiv:1502.03540, to appear in Proceedings of COCOON 2015 (2015)
  22. 22.
    Ladner, R.E.: Polynomial space counting problems. SIAM J. Comput. 18, 1087–1097 (1989)MathSciNetCrossRefMATHGoogle Scholar
  23. 23.
    Lengauer, T., Wagner, K.W.: The correlation between the complexities of the nonhierarchical and hierarchical versions of graph problems. J. Comput. Syst. Sci. 44, 63–93 (1992)MathSciNetCrossRefMATHGoogle Scholar
  24. 24.
    Lenstra, H.: Integer programming with a fixed number of variables. Math. Oper. Res. 8, 538–548 (1983)MathSciNetCrossRefMATHGoogle Scholar
  25. 25.
    Lohrey, M.: Leaf languages and string compression. Inf. Comput. 209, 951–965 (2011)MathSciNetCrossRefMATHGoogle Scholar
  26. 26.
    Lohrey, M.: Algorithmics on SLP-compressed strings: a survey. Groups, Complexity, Cryptology 4, 241–299 (2012)MathSciNetCrossRefMATHGoogle Scholar
  27. 27.
    Lohrey, M., Maneth, S.: The complexity of tree automata and XPath on grammar-compressed trees. Theor. Comput. Sci. 363(2), 196–210 (2006)MathSciNetCrossRefMATHGoogle Scholar
  28. 28.
    Lohrey, M., Mathissen, C.: Isomorphism of regular trees and words. Inf. Comput. 224, 71–105 (2013)MathSciNetCrossRefMATHGoogle Scholar
  29. 29.
    Malod, G.: Succinct algebraic branching programs characterizing non-uniform complexity classes. In: Proceedings of FCT 2011, LNCS 6914, pp 205–216. Springer, Berlin Heidelberg New York (2011)Google Scholar
  30. 30.
    Meinel, C., Theobald, T.: Algorithms and Data Structures in VLSI Design: OBDD - Foundations and Applications. Springer, Berlin Heidelberg New York (1998)CrossRefMATHGoogle Scholar
  31. 31.
    Mereghetti, C., Palano, B.: Threshold circuits for iterated matrix product and powering. Informatique Théorique et Applications 34(1), 39–46 (2000)MathSciNetCrossRefMATHGoogle Scholar
  32. 32.
    Plandowski, W.: Testing equivalence of morphisms in context-free languages. In: Proceedings of ESA 1994, LNCS 855, pp 460–470. Springer, Berlin Heidelberg New York (1994)Google Scholar
  33. 33.
    Samet, H.: The Design and Analysis of Spatial Data Structures. Addison-Wesley, Reading, MA (1990)Google Scholar
  34. 34.
    Storjohann, A., Mulders, T.: Fast algorithms for linear algebra modulo N. In: Proceedings of ESA 1998, LNCS 1461, pp 139–150. Springer, Berlin Heidelberg New York (1998)Google Scholar
  35. 35.
    Taĭclin, M.A.: Algorithmic problems for commutative semigroups. Dokl. Akad. Nauk SSSR 9(1), 201–204 (1968)MathSciNetGoogle Scholar
  36. 36.
    Toda, S.: Counting problems computationally equivalent to computing the determinant. Technical Report CSIM 91-07, Tokyo University of Electro-Communications (1991)Google Scholar
  37. 37.
    Toda, S.: PP is as hard as the polynomial-time hierarchy. SIAM J. Comput. 20, 865–877 (1991)MathSciNetCrossRefMATHGoogle Scholar
  38. 38.
    Torfah, H., Zimmermann, M.: The complexity of counting models of linear-time temporal logic. In: Proceedings of FSTTCS 2014, volume 29 of LIPIcs, Schloss Dagstuhl - Leibniz-Zentrum für Informatik, pp 241–252 (2014)Google Scholar
  39. 39.
    Valiant, L.G.: Completeness classes in algebra. In: Proceedings of STOC 1979, ACM, pp 249–261 (1979)Google Scholar
  40. 40.
    Veith, H.: How to encode a logical structure by an OBDD. In: Proceedings of 13th Annual IEEE Conference on Computational Complexity, IEEE Computer Society, pp 122–131 (1998)Google Scholar
  41. 41.
    Wegener, I.: The size of reduced OBDD’s and optimal read-once branching programs for almost all boolean functions. IEEE Trans. Comput. 43(11), 1262–1269 (1994)CrossRefMATHGoogle Scholar
  42. 42.
    Weibel, C.: The K-book: an Introduction to Algebraic K-theory. Graduate Studies in Mathematics, vol. 145. AMS (2013)Google Scholar

Copyright information

© Springer Science+Business Media New York 2015

Authors and Affiliations

  1. 1.Department für Elektrotechnik und InformatikUniversität SiegenSiegenGermany
  2. 2.Institut für InformatikGoethe-UniversitätFrankfurtGermany

Personalised recommendations