ELSEVIER Theoretical Computer Science 143 (1995) 335-342 Theoretical Computer Science Note A nonlinear lower bound on the practical combinational complexity* Xaver GubS, Juraj HromkoviEb,*, Juraj Waczulik” “Department of Computer Science, Comenius University, 842 15 Bratislava, Slovak Republic bInstitut fiir Informatik und Praktische Mathematik, Christian-Albrechts-Universitiit zu Kiel. OlshausenstraJe 40. 24098 Kiel. Germany ‘Computer Science Institute of the Comenius University, 842 15 Bratislava, Slovak Republic Received August 1992; revised October 1994 Communicated by MS. Paterson zyxwvutsrqponmlkjihgfedcbaZYXWVUTS Abstract An infinite sequence F = { f”}z= 1 of one-output Boolean functions with the following two properties is constructed: (1) fn can be computed by a Boolean circuit with O(n) gates. (2) For any positive, nondecreasing, and unbounded function h : N + R, each Boolean circuit having an m/h(m) separator requires a nonlinear number Q(nh(n)) of gates to computef, (e.g., each planar Boolean circuit requires Q(n’) gates to computef,). Thus, one can say that f” has linear combinational complexity and a nonlinear practical combinational complexity because the constant-degree parallel architectures used in practice have separators in O(m/log, m). 1. Introduction One of the most challenging problems in complexity theory is to prove a nonlinear lower bound on the combinational complexity (the number of gates in Boolean circuits) of a specific Boolean function. The highest lower bounds are only linear ones (for the base of all Boolean functions of two variables in [2,5,6,11,21,24], for some special complete bases in [22,24,27]) despite the well-known fact that almost all Boolean funGiions of n variables require Q(2”/n) combinational complexity [17,26]. “Supported in part DFG-Grant DI 412/2-l and by the Leibniz program of DFG. * Corresponding author. 0304-3975/95/%09.50 0 1995-Elsevier Science B.V. All rights reserved SSDI 0304-3975(94)00269-X