A nonlinear lower bound on the practical combinational complexity

  • Xaver Gubáš
  • Juraj Hromkovič
  • Juraj Waczulík
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 577)


An infinite sequence F={fn} n=1 of one-output Boolean functions with the following three properties is constructed:
  1. (1)

    f n can be computed by a Boolean circuit with O(n) gates.

  2. (2)

    For any positive, nondecreasing, and unbounded function h : N → R, each Boolean circuit having an n/h(n) separator requires nonlinear number of gates to compute f n .

  3. (3)

    Each planar Boolean circuit requires Ω(n2) gates to compute f n .


Thus, one can say that f n has linear combinational complexity and a nonlinear practical combinational complexity because the constant-degree parallel architectures used in practice have their separators in O(n/log2n).


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. AUY83.
    Aho, A.V.-Ullman, J.D.-Yannakakis, M.: On notions of information transfer in VLSI circuits. In: Proc. 15th ACM STOC, ACM Press 1983, pp. 133–139.Google Scholar
  2. B1l84.
    Blum, N.: A Boolean function requiring 3n networksize. Theoretical Computer Science 28 (1984), pp. 337–345.Google Scholar
  3. Br41.
    Brooks, R.L.: On coloring the nodes of network. In: Proc. Cambridge Philos. Soc. 37 (1941), 194–197.Google Scholar
  4. GG81.
    Gaber-Galil, Z.: Explicit constructions of linear-sized superconcentrators. J. Comp. Syst. Sci. 22 (1981), 407–420.Google Scholar
  5. HS73.
    Harper, L.H.-Savage, J.E.: Complexity made Simple. In: Proc. of the Intern. Symp. on Combinatorial Theory, Rom, Sept. 1973, pp. 2–15.Google Scholar
  6. HHS75.
    Harper, L.H.-Hsiek, W.N.-Savage, J.E.: A class of Boolean functions with linear combitional complexity. Theoretical Computer Science 1, No. 2 (1975), pp. 161–183.Google Scholar
  7. Hr88a.
    Hromkovič, J.: The advantages of a new approach to defining the communication complexity of VLSI. Theor. Comp. Science 57 (1988), 97–111.Google Scholar
  8. Hr88b.
    Hromkovič, J.: Some complexity aspects of VLSI computations. Part 1. A framework for the study of information transfer in VLSI circuits. Computers and Artificial Intelligence 7 (1988), No. 3, pp. 229–252.Google Scholar
  9. Hr89.
    Hromkovič, J.: Some complexity aspects of VLSI computations. Part 6. Communication complexity. Computers and Artificial Intelligence 8 (1989), No. 3, pp. 209–225.Google Scholar
  10. Hr90.
    Hr90 Hromkovič, J.: Nonlinear lower bounds on the number of processors of circuits with sublinear separators. Information and Computation, to appear.Google Scholar
  11. KJ90.
    Kumičáková-Jirásková, G.: Chomsky hierarchy and communication complexity. J. Inf. Process. Cybern. EIK 25 (1989), No. 4, 157–164.Google Scholar
  12. KM65.
    Kloss, B.M.-Malyshev, V.A.: Bounds on complexity of some classes of functions. Vest. Mosk. Univ., Seria matem., mech., 1965, No. 4, pp. 44–51 (in Russian).Google Scholar
  13. LS81.
    Lipton, R.J.-Sedgewick, R.: Lower bounds for VLSI. In: Proc. ACM STOC'81, ACM 1981, pp. 300–307.Google Scholar
  14. LT79.
    Lipton, R.J.-Tarjan, R.E.: A separator theorem for planar graphs. SIAM J. Appl. Math. 36 (1979), No. 2, pp. 177–189.Google Scholar
  15. LRSH88.
    Lozkin, C.A.-Rybko, A.N.-SapoŽenko, A.A.-Hromkovič, J.-škalikova, N.A.: On a approach to a bound on the area complexity of combinational circuits. In: Mathematical problems in computation theory. Proc. of the Banach Center Publications. Warsaw 1988, pp. 501–510 (in Russian), the full version of this paper is accepted for publication in Theor. Comp. Sci.Google Scholar
  16. Lu58.
    Lupanov, O.B.: Ob odnom metode siteza skhem (zv. VUZ radiofizika) 1 (1958), pp. 120–140 (in Russian).Google Scholar
  17. McC85.
    McColl: Planar circuits have short specifications. In: Proc. 2nd STACS'85, Lecture Notes in Computer Science 18, Springer-Verlag 1985, pp. 231–242.Google Scholar
  18. McCP84.
    McColl-Paterson, M.P.: The planar realization of Boolean functions. Technical Report, University of Warwick 1984.Google Scholar
  19. Pa77.
    Paul, W.J.: A 2,5n — lower bound on the combinational complexity of Boolean functions. SIAM J. Comp. 6 (1977), pp. 427–443.Google Scholar
  20. PS84.
    Papadimitriou, Ch.-Sipser, M.: Communication complexity. J. Computer System Sci. 28 (1984), pp. 260–269.Google Scholar
  21. Re81.
    Redkin, N.P.: Proof of minimality of circuits consisting of functional elements. Problemy kibernetiki 38 (1981), pp. 181–216 (in Russian).Google Scholar
  22. Sch74.
    Schnorr, G.P.: Zwei lineare untere Schranken für die Komplexität Boolescher Funktionen. Computing 13 (1974), pp. 155–171.Google Scholar
  23. Sch80.
    Schnorr, G.P.: A 3 · n lower bound on the network complexity of Boolean functions. Theor. Comp. Science 10 (1980), pp. 83–92.Google Scholar
  24. Sh49.
    Shannon, C.E.: The synthesis of two-terminal switching circuits. Bell System Techn. J. 28 (1949), pp. 59–98.Google Scholar
  25. So65.
    Sopranenko, E.P.: Minimal realizations of functions by circuits using functional elements. Probl. Kibernetiki 15 (1965), pp. 117–134 (in Russian).Google Scholar
  26. Tu89.
    Turán, G.: Lower bounds for synchronous circuits and planar circuits. Infor. Proc. Lett. 30 (1989), pp. 37–40.Google Scholar
  27. Ul84.
    Ullman, J.: Computational Aspects of VLSI. Principles of Computer Science Series. Computer Science Press 1984.Google Scholar
  28. We87.
    Wegener, I.: The Complexity of Boolean Functions. Wiley-Teubner Series in Computer Science, John Wiley and Sons Ltd., and B.G. Teubner, Stuttgart 1987.Google Scholar
  29. Ya81.
    Yao, A.C.: The entropic limitation of VLSI computations. In: Proc. 13th Annual ACM STOC, ACM 1981, pp. 308–311.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1992

Authors and Affiliations

  • Xaver Gubáš
    • 2
  • Juraj Hromkovič
    • 1
  • Juraj Waczulík
    • 3
  1. 1.Departement of Mathematics and Computer ScienceUniversity of PaderbornPaderbornGermany
  2. 2.Department of Computer ScienceComenius UniversityBratislavačSFR
  3. 3.Computer Science Institute of the Comenius UniversityBratislavačSFR

Personalised recommendations