, Volume 19, Issue 3, pp 403–435

Separation of the Monotone NC Hierarchy

  • Ran Raz
  • Pierre McKenzie
Original Paper

DOI: 10.1007/s004930050062

Cite this article as:
Raz, R. & McKenzie, P. Combinatorica (1999) 19: 403. doi:10.1007/s004930050062

, for the monotone depth of functions in monotone-P. As a result we achieve the separation of the following classes.

1. monotone-NC ≠ monotone-P.

2. For every i≥1, monotone-\(\)≠ monotone-\(\).

3. More generally: For any integer function D(n), up to \(\) (for some ε>0), we give an explicit example of a monotone Boolean function, that can be computed by polynomial size monotone Boolean circuits of depth D(n), but that cannot be computed by any (fan-in 2) monotone Boolean circuits of depth less than Const·D(n) (for some constant Const).

Only a separation of monotone-\(\) from monotone-\(\) was previously known.

Our argument is more general: we define a new class of communication complexity search problems, referred to below as DART games, and we prove a tight lower bound for the communication complexity of every member of this class. As a result we get lower bounds for the monotone depth of many functions. In particular, we get the following bounds:

1.  For st-connectivity, we get a tight lower bound of \(\). That is, we get a new proof for Karchmer–Wigderson's theorem, as an immediate corollary of our general result.

2.  For the k-clique function, with \(\), we get a tight lower bound of Ω(k log n). This lower bound was previously known for k≤ log n [1]. For larger k, however, only a bound of Ω(k) was previously known.

AMS Subject Classification (1991) Classes:  68Q15, 68Q25, 68R99

Copyright information

© János Bolyai Mathematical Society, 1999

Authors and Affiliations

  • Ran Raz
    • 1
  • Pierre McKenzie
    • 2
  1. 1.Department of Applied Mathematics and Computer Science, Weizmann Institute; Rehovot, 76100 Israel; E-mail:
  2. 2.Département d'informatique et recherche opérationnelle, Université de Montréal; C.P. 6128, succursale Centre-ville, Montréal (Québec), H3C 3J7 Canada; E-mail: mckenzie@iro.umontreal.caCA