A Generalization of Spira’s Theorem and Circuits with Small Segregators or Separators

  • Anna Gál
  • Jing-Tang Jang
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7147)


Spira [28] showed that any Boolean formula of size s can be simulated in depth O(logs). We generalize Spira’s theorem and show that any Boolean circuit of size s with segregators of size f(s) can be simulated in depth O(f(s)logs). If the segregator size is at least s ε for some constant ε > 0, then we can obtain a simulation of depth O(f(s)). This improves and generalizes a simulation of polynomial-size Boolean circuits of constant treewidth k in depth O( k 2 logn) by Jansen and Sarma [17]. Since the existence of small balanced separators in a directed acyclic graph implies that the graph also has small segregators, our results also apply to circuits with small separators. Our results imply that the class of languages computed by non-uniform families of polynomial-size circuits that have constant size segregators equals non-uniform NC 1.

Considering space bounded Turing machines to generate the circuits, for f(s)log2 s-space uniform families of Boolean circuits our small-depth simulations are also f(s)log2 s-space uniform. As a corollary, we show that the Boolean Circuit Value problem for circuits with constant size segregators (or separators) is in deterministic SPACE (log2 n). Our results also imply that the Planar Circuit Value problem, which is known to be P-Complete [16], can be solved in deterministic \(SPACE (\sqrt{n} \log n)\).


Boolean Function Directed Acyclic Graph Turing Machine Boolean Formula Arithmetic Circuit 
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.
    Alon, A., Seymour, P., Thomas, R.: A Separator Theorem for Graphs with an Excluded Minor and its Applications. In: Proceedings of STOC, pp. 293–299 (1990)Google Scholar
  2. 2.
    Barrington, D., Lu, C., Miltersen, P.B., Skyum, S.: On monotone planar circuits. In: Proceedings of IEEE Conference on Computational Complexity, pp. 24–33 (1999)Google Scholar
  3. 3.
    Bonet, M., Buss, S.R.: Size-depth tradeoffs for Boolean formulae. Information Processing Letters 49(3), 151–155 (1994)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Borodin, A.: On Relating Time and Space to Size and Depth. SIAM Journal on Computing 6(4), 733–744 (1977)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Brent, R.P.: The Parallel Evaluation of General Arithmetic Expressions. Journal of the ACM 21(2), 201–206 (1974)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Bshouty, N., Cleve, R., Eberly, W.: Size-Depth Tradeoffs for Algebraic Formulas. SIAM Journal on Computing 24(4), 682–705 (1995)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Bürgisser, P., Clausen, M., Shokrollahi, M.A.: Algebraic complexity theory. Springer, Heidelberg (1997)CrossRefzbMATHGoogle Scholar
  8. 8.
    Buss, S.R.: The Boolean formula value problem is in ALOGTIME. In: Proceedings of STOC, pp. 123–131 (1987)Google Scholar
  9. 9.
    Buss, S., Cook, S., Gupta, A., Ramachandran, V.: An Optimal Parallel Algorithm for Formula Evaluation. SIAM Journal on Computing 21(4), 755–780 (1992)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Chakraborty, T., Datta, S.: One-Input-Face MPCVP Is Hard for L, But in LogDCFL. In: Arun-Kumar, S., Garg, N. (eds.) FSTTCS 2006. LNCS, vol. 4337, pp. 57–68. Springer, Heidelberg (2006)CrossRefGoogle Scholar
  11. 11.
    Downey, R.G., Fellows, M.R.: Parameterized Complexity. Springer, Heidelberg (1999)CrossRefzbMATHGoogle Scholar
  12. 12.
    Dymond, P., Tompa, M.: Speedups of Deterministic Machines by Synchronous Parallel Machines. J. Comp. and Sys. Sci. 30(2), 149–161 (1985)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Flum, J., Grohe, M.: Parameterized Complexity Theory. Springer, Heidelberg (2006)zbMATHGoogle Scholar
  14. 14.
    Gál, A., Jang, J.: The Size and Depth of Layered Boolean Circuits. In: López-Ortiz, A. (ed.) LATIN 2010. LNCS, vol. 6034, pp. 372–383. Springer, Heidelberg (2010)CrossRefGoogle Scholar
  15. 15.
    Gilbert, J.R., Hutchinson, J.P., Tarjan, R.E.: A Separator Theorem for Graphs of Bounded Genus. Journal of Algorithms 5(3), 391–407 (1984)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Goldschlager, L.: The monotone and planar circuit value problem is complete for P. SIGACT News, 25–27 (1977)Google Scholar
  17. 17.
    Jansen, M., Sarma, J.: Balancing Bounded Treewidth Circuits. In: Ablayev, F., Mayr, E.W. (eds.) CSR 2010. LNCS, vol. 6072, pp. 228–239. Springer, Heidelberg (2010)CrossRefGoogle Scholar
  18. 18.
    Ladner, R.E.: The circuit value problem is log-space complete for P. SIGACT News 6(2), 18–20 (1975)CrossRefGoogle Scholar
  19. 19.
    Limaye, N., Mahajan, M., Sarma, J.: Upper bounds for monotone planar circuit value and variants. Computational Complexity 18, 377–412 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Lipton, R., Tarjan, R.E.: A Separator Theorem for Planar Graphs. SIAM J. Appl. Math. 36, 177–189 (1979)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Lynch, N.A.: Log space recognition and translation of parenthesis languages. J. Assoc. Comput. Mach. 24, 583–590 (1977)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Papadimitriou, C.H.: Computational complexity. Addison-Wesley (1994)Google Scholar
  23. 23.
    Paterson, M.S., Valiant, L.G.: Circuit Size is Nonlinear in Depth. Theoretical Computer Science 2(3), 397–400 (1976)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Paul, W.J., Pippenger, N., Szemerédi, E., Trotter, W.T.: On determinism versus non-determinism and related problems. In: Proceedings of FOCS, pp. 429–438 (1983)Google Scholar
  25. 25.
    Robertson, N., Seymour, P.D.: Graph Minors II. Algorithmic aspects of tree width. Journal of Algorithms 7, 309–322 (1986)MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Santhanam, R.: On separators, segregators and time versus space. In: Proceedings of the Sixteenth Annual Conference on Computational Complexity, pp. 286–294 (2000)Google Scholar
  27. 27.
    Savitch, W.J.: Relationships Between Nondeterministic and Deterministic Tape Complexities. Journal of Computer and System Sciences 4(2), 177–192 (1970)MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    Spira, P.M.: On time-hardware complexity tradeoffs for Boolean functions. In: Proc. 4th Hawaii Symp. on System Sciences, pp. 525–527 (1971)Google Scholar
  29. 29.
    Valiant, L., Skyum, S., Berkowitz, S., Rackoff, C.: Fast Parallel Computation of Polynomials Using Few Processors. SIAM J. Comp. 12(4), 641–644 (1983)MathSciNetCrossRefzbMATHGoogle Scholar
  30. 30.
    Wegener, I.: Relating monotone formula size and monotone depth of Boolean functions. Information Processing Letters 16(1), 41–42 (1983)MathSciNetCrossRefzbMATHGoogle Scholar
  31. 31.
    Wegener, I.: The Complexity of Boolean Functions (1987)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  • Anna Gál
    • 1
  • Jing-Tang Jang
    • 1
  1. 1.Dept. of Computer ScienceUniversity of Texas at AustinAustinUSA

Personalised recommendations