computational complexity

, Volume 24, Issue 3, pp 645–694 | Cite as

The NOF Multiparty Communication Complexity of Composed Functions

  • Anil Ada
  • Arkadev Chattopadhyay
  • Omar Fawzi
  • Phuong Nguyen


We study the k-party “number on the forehead” communication complexity of composed functions \({f \circ \vec{g}}\), where \({f:\{0,1\}^n \to \{\pm 1\}}\), \({\vec{g} = (g_1,\ldots,g_n)}\), \({g_i : \{0,1\}^k \to \{0,1\}}\) and for \({(x_1,\ldots,x_k) \in (\{0,1\}^n)^k}\), \({f \circ \vec{g}(x_1,\ldots,x_k) = f(\ldots,g_i(x_{1,i},\ldots,x_{k,i}), \ldots)}\). When \({\vec{g} = (g, g,\ldots, g)}\), we denote \({f \circ \vec{g}}\) by \({f \circ g}\). We show that there is an \({O({\rm log}^3 n)}\) cost simultaneous protocol for SYM\({\circ g}\) when k >  1 + log  n, SYM is any symmetric function and g is any function. When k >  1 +  2 log  n, our simultaneous protocol applies to SYM\({\circ \, \vec{g}}\) with \({\vec{g}}\) being a vector of n arbitrary functions. We also get a non-simultaneous protocol for SYM\({\circ \, \vec{g}}\) of cost \({O(n/2^k \cdot {\rm log}\, n+ k {\rm log}\, n)}\) for any k ≥  2. In the setting of k ≤  1 + log  n, we study more closely functions of the form MAJORITY\({\circ g}\), MODm\({\circ g}\) and NOR\({\circ g}\), where the latter two are generalizations of the well-known and studied functions generalized inner product and disjointness, respectively. We characterize the communication complexity of these functions with respect to the choice of g. In doing so, we answer a question posed by Babai et al. (SIAM J Comput 33:137–166, 2003) and determine the communication complexity of MAJORITYQCSBk, where QCSBk is the “quadratic character of the sum of the bits” function.

In the second part of our paper, we utilize the connection between the ‘number on the forehead’ model and Ramsey theory to construct a large set without a k-dimensional corner (k-dimensional generalization of a k-term arithmetic progression) in \({(\mathbb{F}_{2}^{n})^k}\), thereby obtaining the first non-trivial bound on the corresponding Ramsey number. Furthermore, we give an explicit coloring of [N] ×  [N] without a monochromatic two-dimensional corner and use this to obtain an explicit three-party protocol of cost \({O(\sqrt{n})}\) for the EXACTN function. For x1,x2,x3n-bit integers, EXACTN(x1,x2,x3) = −1 iff x1x2x3 = N.


Number on the forehead model communication complexity Ramsey theory 

Subject Classification

68Q05 68Q17 05D10 


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Copyright information

© Springer Basel 2014

Authors and Affiliations

  • Anil Ada
    • 1
  • Arkadev Chattopadhyay
    • 2
  • Omar Fawzi
    • 3
  • Phuong Nguyen
    • 4
  1. 1.School of Computer ScienceMcGill UniversityMontrealCanada
  2. 2.School of Technology and Computer ScienceTata Institute of Fundamental ResearchMumbaiIndia
  3. 3.Institute for Theoretical Physics, ETHZürichSwitzerland
  4. 4.Dép. d’informatique et de recherche opérationnelleUniversité de MontréalMontrealCanada

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