4,634 research outputs found

    On the Computational Power of Radio Channels

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    Radio networks can be a challenging platform for which to develop distributed algorithms, because the network nodes must contend for a shared channel. In some cases, though, the shared medium is an advantage rather than a disadvantage: for example, many radio network algorithms cleverly use the shared channel to approximate the degree of a node, or estimate the contention. In this paper we ask how far the inherent power of a shared radio channel goes, and whether it can efficiently compute "classicaly hard" functions such as Majority, Approximate Sum, and Parity. Using techniques from circuit complexity, we show that in many cases, the answer is "no". We show that simple radio channels, such as the beeping model or the channel with collision-detection, can be approximated by a low-degree polynomial, which makes them subject to known lower bounds on functions such as Parity and Majority; we obtain round lower bounds of the form Omega(n^{delta}) on these functions, for delta in (0,1). Next, we use the technique of random restrictions, used to prove AC^0 lower bounds, to prove a tight lower bound of Omega(1/epsilon^2) on computing a (1 +/- epsilon)-approximation to the sum of the nodes\u27 inputs. Our techniques are general, and apply to many types of radio channels studied in the literature

    Algorithms and lower bounds for de Morgan formulas of low-communication leaf gates

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    The class FORMULA[s]GFORMULA[s] \circ \mathcal{G} consists of Boolean functions computable by size-ss de Morgan formulas whose leaves are any Boolean functions from a class G\mathcal{G}. We give lower bounds and (SAT, Learning, and PRG) algorithms for FORMULA[n1.99]GFORMULA[n^{1.99}]\circ \mathcal{G}, for classes G\mathcal{G} of functions with low communication complexity. Let R(k)(G)R^{(k)}(\mathcal{G}) be the maximum kk-party NOF randomized communication complexity of G\mathcal{G}. We show: (1) The Generalized Inner Product function GIPnkGIP^k_n cannot be computed in FORMULA[s]GFORMULA[s]\circ \mathcal{G} on more than 1/2+ε1/2+\varepsilon fraction of inputs for s=o ⁣(n2(k4kR(k)(G)log(n/ε)log(1/ε))2). s = o \! \left ( \frac{n^2}{ \left(k \cdot 4^k \cdot {R}^{(k)}(\mathcal{G}) \cdot \log (n/\varepsilon) \cdot \log(1/\varepsilon) \right)^{2}} \right). As a corollary, we get an average-case lower bound for GIPnkGIP^k_n against FORMULA[n1.99]PTFk1FORMULA[n^{1.99}]\circ PTF^{k-1}. (2) There is a PRG of seed length n/2+O(sR(2)(G)log(s/ε)log(1/ε))n/2 + O\left(\sqrt{s} \cdot R^{(2)}(\mathcal{G}) \cdot\log(s/\varepsilon) \cdot \log (1/\varepsilon) \right) that ε\varepsilon-fools FORMULA[s]GFORMULA[s] \circ \mathcal{G}. For FORMULA[s]LTFFORMULA[s] \circ LTF, we get the better seed length O(n1/2s1/4log(n)log(n/ε))O\left(n^{1/2}\cdot s^{1/4}\cdot \log(n)\cdot \log(n/\varepsilon)\right). This gives the first non-trivial PRG (with seed length o(n)o(n)) for intersections of nn half-spaces in the regime where ε1/n\varepsilon \leq 1/n. (3) There is a randomized 2nt2^{n-t}-time #\#SAT algorithm for FORMULA[s]GFORMULA[s] \circ \mathcal{G}, where t=Ω(nslog2(s)R(2)(G))1/2.t=\Omega\left(\frac{n}{\sqrt{s}\cdot\log^2(s)\cdot R^{(2)}(\mathcal{G})}\right)^{1/2}. In particular, this implies a nontrivial #SAT algorithm for FORMULA[n1.99]LTFFORMULA[n^{1.99}]\circ LTF. (4) The Minimum Circuit Size Problem is not in FORMULA[n1.99]XORFORMULA[n^{1.99}]\circ XOR. On the algorithmic side, we show that FORMULA[n1.99]XORFORMULA[n^{1.99}] \circ XOR can be PAC-learned in time 2O(n/logn)2^{O(n/\log n)}

    A Nearly Optimal Lower Bound on the Approximate Degree of AC0^0

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    The approximate degree of a Boolean function f ⁣:{1,1}n{1,1}f \colon \{-1, 1\}^n \rightarrow \{-1, 1\} is the least degree of a real polynomial that approximates ff pointwise to error at most 1/31/3. We introduce a generic method for increasing the approximate degree of a given function, while preserving its computability by constant-depth circuits. Specifically, we show how to transform any Boolean function ff with approximate degree dd into a function FF on O(npolylog(n))O(n \cdot \operatorname{polylog}(n)) variables with approximate degree at least D=Ω(n1/3d2/3)D = \Omega(n^{1/3} \cdot d^{2/3}). In particular, if d=n1Ω(1)d= n^{1-\Omega(1)}, then DD is polynomially larger than dd. Moreover, if ff is computed by a polynomial-size Boolean circuit of constant depth, then so is FF. By recursively applying our transformation, for any constant δ>0\delta > 0 we exhibit an AC0^0 function of approximate degree Ω(n1δ)\Omega(n^{1-\delta}). This improves over the best previous lower bound of Ω(n2/3)\Omega(n^{2/3}) due to Aaronson and Shi (J. ACM 2004), and nearly matches the trivial upper bound of nn that holds for any function. Our lower bounds also apply to (quasipolynomial-size) DNFs of polylogarithmic width. We describe several applications of these results. We give: * For any constant δ>0\delta > 0, an Ω(n1δ)\Omega(n^{1-\delta}) lower bound on the quantum communication complexity of a function in AC0^0. * A Boolean function ff with approximate degree at least C(f)2o(1)C(f)^{2-o(1)}, where C(f)C(f) is the certificate complexity of ff. This separation is optimal up to the o(1)o(1) term in the exponent. * Improved secret sharing schemes with reconstruction procedures in AC0^0.Comment: 40 pages, 1 figur

    An average-case depth hierarchy theorem for Boolean circuits

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    We prove an average-case depth hierarchy theorem for Boolean circuits over the standard basis of AND\mathsf{AND}, OR\mathsf{OR}, and NOT\mathsf{NOT} gates. Our hierarchy theorem says that for every d2d \geq 2, there is an explicit nn-variable Boolean function ff, computed by a linear-size depth-dd formula, which is such that any depth-(d1)(d-1) circuit that agrees with ff on (1/2+on(1))(1/2 + o_n(1)) fraction of all inputs must have size exp(nΩ(1/d)).\exp({n^{\Omega(1/d)}}). This answers an open question posed by H{\aa}stad in his Ph.D. thesis. Our average-case depth hierarchy theorem implies that the polynomial hierarchy is infinite relative to a random oracle with probability 1, confirming a conjecture of H{\aa}stad, Cai, and Babai. We also use our result to show that there is no "approximate converse" to the results of Linial, Mansour, Nisan and Boppana on the total influence of small-depth circuits, thus answering a question posed by O'Donnell, Kalai, and Hatami. A key ingredient in our proof is a notion of \emph{random projections} which generalize random restrictions
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