Monotone circuits for connectivity require super-logarithmic depth

  • Authors:
  • Mauricio Karchmer;Avi Wigderson

  • Affiliations:
  • Inst. of Mathematics and Computer Science, Hebrew University, Jerusalem, Israel;Inst. of Mathematics and Computer Science, Hebrew University, Jerusalem, Israel

  • Venue:
  • STOC '88 Proceedings of the twentieth annual ACM symposium on Theory of computing
  • Year:
  • 1988

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Abstract

We prove that every monotone circuit which tests st-connectivity of an undirected graph on n nodes has depth &OHgr;(log2n). This implies a superpolynomial (n&OHgr;(log n)) lower bound on the size of any monotone formula for st-connectivity.The proof draws intuition from a new characterization of circuit depth in terms of communication complexity. It uses counting arguments and Extremal Set Theory.Within the same framework, we also give a very simple and intuitive proof of a depth analogue of a theorem of Krapchenko concerning formula size lower bounds.