251 research outputs found

    Separating NOF communication complexity classes RP and NP

    Full text link
    We provide a non-explicit separation of the number-on-forehead communication complexity classes RP and NP when the number of players is up to \delta log(n) for any \delta<1. Recent lower bounds on Set-Disjointness [LS08,CA08] provide an explicit separation between these classes when the number of players is only up to o(loglog(n))

    Hadamard Tensors and Lower Bounds on Multiparty Communication Complexity

    Get PDF
    We develop a new method for estimating the discrepancy of tensors associated with multiparty communication problems in the ``Number on the Forehead\u27\u27 model of Chandra, Furst and Lipton. We define an analogue of the Hadamard property of matrices for tensors in multiple dimensions and show that any kk-party communication problem represented by a Hadamard tensor must have Omega(n/2k)Omega(n/2^k) multiparty communication complexity. We also exhibit constructions of Hadamard tensors, giving Omega(n/2k)Omega(n/2^k) lower bounds on multiparty communication complexity for a new class of explicitly defined Boolean functions

    Simplified Lower Bounds on the Multiparty Communication Complexity of Disjointness

    Get PDF
    We show that the deterministic number-on-forehead communication complexity of set disjointness for k parties on a universe of size n is Omega(n/4^k). This gives the first lower bound that is linear in n, nearly matching Grolmusz\u27s upper bound of O(log^2(n) + k^2n/2^k). We also simplify the proof of Sherstov\u27s Omega(sqrt(n)/(k2^k)) lower bound for the randomized communication complexity of set disjointness

    Disentangling causal webs in the brain using functional Magnetic Resonance Imaging: A review of current approaches

    Get PDF
    In the past two decades, functional Magnetic Resonance Imaging has been used to relate neuronal network activity to cognitive processing and behaviour. Recently this approach has been augmented by algorithms that allow us to infer causal links between component populations of neuronal networks. Multiple inference procedures have been proposed to approach this research question but so far, each method has limitations when it comes to establishing whole-brain connectivity patterns. In this work, we discuss eight ways to infer causality in fMRI research: Bayesian Nets, Dynamical Causal Modelling, Granger Causality, Likelihood Ratios, LiNGAM, Patel's Tau, Structural Equation Modelling, and Transfer Entropy. We finish with formulating some recommendations for the future directions in this area

    Simultaneous Multiparty Communication Protocols for Composed Functions

    Get PDF
    In the Number On the Forehead (NOF) multiparty communication model, kk players want to evaluate a function F:X1××XkYF : X_1 \times\cdots\times X_k\rightarrow Y on some input (x1,,xk)(x_1,\dots,x_k) by broadcasting bits according to a predetermined protocol. The input is distributed in such a way that each player ii sees all of it except xix_i. In the simultaneous setting, the players cannot speak to each other but instead send information to a referee. The referee does not know the players' input, and cannot give any information back. At the end, the referee must be able to recover F(x1,,xk)F(x_1,\dots,x_k) from what she obtained. A central open question, called the logn\log n barrier, is to find a function which is hard to compute for polylog(n)polylog(n) or more players (where the xix_i's have size poly(n)poly(n)) in the simultaneous NOF model. This has important applications in circuit complexity, as it could help to separate ACC0ACC^0 from other complexity classes. One of the candidates belongs to the family of composed functions. The input to these functions is represented by a k×(tn)k\times (t\cdot n) boolean matrix MM, whose row ii is the input xix_i and tt is a block-width parameter. A symmetric composed function acting on MM is specified by two symmetric nn- and ktkt-variate functions ff and gg, that output fg(M)=f(g(B1),,g(Bn))f\circ g(M)=f(g(B_1),\dots,g(B_n)) where BjB_j is the jj-th block of width tt of MM. As the majority function MAJMAJ is conjectured to be outside of ACC0ACC^0, Babai et. al. suggested to study MAJMAJtMAJ\circ MAJ_t, with tt large enough. So far, it was only known that t=1t=1 is not enough for MAJMAJtMAJ\circ MAJ_t to break the logn\log n barrier in the simultaneous deterministic NOF model. In this paper, we extend this result to any constant block-width t>1t>1, by giving a protocol of cost 2O(2t)log2t+1(n)2^{O(2^t)}\log^{2^{t+1}}(n) for any symmetric composed function when there are 2Ω(2t)logn2^{\Omega(2^t)}\log n players.Comment: 17 pages, 1 figure; v2: improved introduction, better cost analysis for the 2nd protoco
    corecore