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Here's a lower bound from sorting. Given an input set $S$ of length $n$ to be sorted, create an input to your running median problem consisting of $n-1$ copies of a number smaller than the minimum of $S$, then $S$ itself, then $n-1$ copies of a number larger than the maximum of $S$, and set $k=2n-1$. The running medians of this input are the same as the ...
Let $f\colon \{0,1\}^n \to \{0,1\}$ be a boolean function. If it has a polynomial representation $P$ then it has a multilinear polynomial representation $Q$ of degree $\deg Q \leq \deg P$: just replace any power $x_i^k$, where $k \geq 2$, by $x_i$. So we can restrict our attention to multilinear polynomials. Claim: The polynomials $\{ \prod_{i \in S} x_i : ... 26 Let$p$be a polynomial such that for all$x\in \{0,1\}^n$,$p(x) = \sf{OR}(x)$. Consider the symmetrization of the polynomial$p$: $$q(k) = \frac{1}{\binom{n}{k}} \sum_{x: |x| = k} p(x).$$ Note that, since the OR function is a symmetric boolean function, we have that for$k = 1, 2, \ldots, n$,$q(k) = 1$, and$q(0) = 0$. Since$q-1$is a non-zero ... 21 Markov proved that any function of$n$inputs can be computed with only$\lceil \log (n+1)\rceil$negations. An efficient constructive version was described by Fisher. See also an exposition of the result from the GLL blog. More precisely: Theorem: Suppose$f : \{0,1\}^n \to \{0,1\}^m$is computed by a circuit$C$with$g$gates, then it is also computed by ... 19 A family of bitvectors is the class of solutions to a 2-SAT problem if and only if it has the median property: if you apply the bitwise majority function to any three solutions you get another solution. See e.g. https://en.wikipedia.org/wiki/Median_graph#2-satisfiability and its references. So if you can find three solutions for which this is not true, then ... 18 According to the paper A$5n − o(n)$Lower Bound on the Circuit Size over$U_2$of a Linear Boolean Function by Kulikov, Melanich, and Mihajlin, when$m=o(n)$there are no lower bounds known better than$3n - o(n)$. It also outlines a method for obtaining functions for which a$4n - o(n)$lower bound holds, when$m=n$, based on a result of Lamagne and ... 17 This question has been completely resolved (up to constant factors) by a recent result of Benjamin Rossman (http://eccc.hpi-web.de/report/2013/169/). As Kaveh points out above, a depth$d$, size$S$, circuit can be converted to a depth$d$, size$S^d$formula. Rossman shows that this is essentially tight. For any depth$d$, he exhibits a function that can ... 15 There are "natural" problems hard for Ackermannian time; in fact there is a growing body of literature on the subject. A place to start is the survey in Section 6 of Complexity Hierarchies Beyond Elementary. What is even more interesting is that there are complete problems for Ackermannian time, under say primitive-recursive many-one reductions: define for ... 15 Hmm, turns out there's actually an matching upper bound that isn't too hard: To produce the truth table$HALT_n$in a finite amount of time, the only information that is needed is the number of machines of description length at most$n$which halt. This number is not more than$2^{n}$, so it can be represented with about$n$bits. Then we can start all such ... 13 According to Theorem 3.1 in Alexis Maciel and Denis Therien Threshold Circuits of Small Majority-Depth there is indeed a depth-3 circuit for computing the addition of two numbers. The precise bound is$\Delta_2 \cdot \mathsf{NC}^0_1$where$\Delta_2 = \Sigma_2 \cap \Pi_2$are problems which have depth-2$\mathsf{AC}^0$circuits with both$\vee,\wedge$... 13 One approach on this question, related to complexity-theoretic questions, is due to Shub and Smale  who proved that if$n!$is ultimately hard to compute, then$\mathsf{VP}\neq\mathsf{VNP}$over some field. Their model of computation is the straight-line programs: The goal is to compute$n!$from the constant$1$, using only additions, subtractions and ... 13 The paper Counting Quantifiers, Successor Relations and Logarithmic Space, by Kousha Etessami proves that the problem$\mathbf{ORD}$(which is essentially checking if a vertex$s$precedes a vertex$t$in an outdegree one graph$G$, that is promised to be a path) is$\mathsf{L}$-hard under quantifier free projections. The problem$\mathbf{ORD}$can be seen ... 12 The bounds... We have in fact$NFA(L) \ge Cov(M) + Cov(N)$, see Theorem 4 in (Gruber & Holzer 2006). For an upper bound, we have$2^{Cov(M)+Cov(N)} \ge DFA(L) \ge NFA(L)$, see Theorem 11 in the same paper. ...cannot be substantially improved There can be a subexponential gap between$Cov(M)+Cov(N)$and$NFA(L)$. The following example, and the proof ... 11 here are new results on this said to be the 1st in ~3 decades and some brief commentary A better-than-3n lower bound for the circuit complexity of an explicit function / Find, Golovnev, Hirsch, Kulikov We consider Boolean circuits over the full binary basis. We prove a$(3+\frac{1}{86})n−o(n)$lower bound on the size of such a circuit for an explicitly ... 11 It depends what you count as "the GCT program." Consider the specific suggestion (GCT I, GCT II) to use the vanishing/nonvanishing of certain multiplicities in the orbit closures of the determinant and permanent to resolve the strong permanent versus determinant conjecture (i.e., that the permanent is not in the orbit closure of any polynomially larger ... 11 It depends on the precise definition of RAM being used, but (for most reasonable definitions of RAMs) this would also imply that SAT is not solvable in$O(n^{2-e})$time by multitape TMs, a longstanding open problem. The reason is that there is a very efficient reduction from linear time on RAMs to SAT (in general, nondeterministic quasi linear time on RAMs ... 10 Let$P(x_1,\ldots,x_n)$be a property on$n$variables. Suppose that there is a 2CNF formula$\varphi(x_1,\ldots,x_n,y_1,\ldots,y_m)$such that $$P(x_1,\ldots,x_n) \Leftrightarrow \exists y_1 \cdots \exists y_m \varphi(x_1,\ldots,x_n,y_1,\ldots,y_m).$$ We claim that$\varphi$is equivalent to a 2CNF formula$\psi$involving only$x_1,\ldots,x_n$. To prove ... 9 It made my day when my friend James told me that this thread from long ago was rekindled. Thank you for that. Also, I had an urge to share some interesting references that are relevant to L vs Log(DCFL) vs Log(CFL). Have a great day! http://link.springer.com/chapter/10.1007%2F978-3-642-14031-0_35#page-1 http://link.springer.com/chapter/10.1007/3-540-... 9 Alon, Matias, and Szegedy proved that finding the frequency of the most frequent of$n$items requires$\Omega(n)$space in the worst case in the streaming model, even if you allow a constant number of passes over the input. Since finding the most frequent item gives you a two-pass algorithm for computing its frequency (find in the first pass, track its ... 9 The tight bound is$ n(n^n -(n-1)^n) $, which was given in Removing Bidirectionality from Nondeterministic Finite Automata by Christos Kapoutsis (2005). I am also putting a figure from this paper giving a clear picture: To get more, I think this paper is a good starting point. To check the related recent developments, I can also recommend to check the ... 9 This question has just been resolved! As I mentioned, it was known that$Pn(f) \leq D(f) \leq (Pn(f))^2$, but it was a major open problem to show that either$Pn(f) = \Theta(D(f))$or that there exists a function for which$Pn(f) = o(D(f))$. A few days ago this was resolved by Mika Göös, Toniann Pitassi, Thomas Watson (http://eccc.hpi-web.de/report/... 9 Edit: This algorithm is now presented here: http://arxiv.org/abs/1406.1717 Yes, to solve this problem it is sufficient to perform the following operations: Sort$n/k$vectors, each with$k$elements. Do linear-time post-processing. Very roughly, the idea is this: Consider two adjacent blocks of input,$a$and$b$, both with$k$elements; let the elements ... 9 I don't know any nontrivial lower bounds other than the$\Omega(n\log n)$algebraic-decision-tree lower bound for two dimensions, which also extends to larger$d$. As for upper bounds: In 3d, the whole hull can be found in$O(n\log n)$; in four or higher dimensions that doesn't work because the hull complexity grows exponentially with$d$, but nevertheless ... 9 The following is a proof over any ring of characteristic zero that the Hamiltonian cycle polynomial is not a polynomial-size monotone projection of the permanent. The basic idea is that monotone projections of polynomials with nonnegative coefficients lead to the Newton polytope of one being an extended formulation of the Newton polytope of the other, and ... 9$N^1(DISJ_n) \geq n$. The fooling set technique actually provides lower bounds on the cover number.$C^1(f)$is the minimum number of monochromatic rectangles needed to cover the$1$-inputs of$f$. In the fooling set technique, you create a set of input pairs and show that each must lie in distinct monochromatic rectangles. In this case, look at the set ... 9$S^p_2$and$PP$are both known not to have$n^k$-circuits for any fixed k and there is no known containment between them. Details in my blog post. Update: As Rickey Demer points out, these results do not necessarily give a language with a lower bound for all$n$in$S_2^p$. I think the$\Delta^p_3$is probably the best known. Since$PP$has complete sets ... 9 The permanent would seem to qualify, at least conditionally (that is, assuming$\mathsf{VP}^0 \neq \mathsf{VNP}^0$). Note that the Boolean version of the permanent is just to decide whether a given bipartite graph has a perfect matching, which has poly-size circuits. [Summarizing the comments below:] Despite this example being conditional, nothing more than ... 9 The first one is average-case analysis, for sets of keys that are already somewhat randomly distributed (chosen either before or after the choice of hash function but with a probability distribution that is independent of the hash function). The second one is worst-case analysis, for sets of keys that are not random but are instead specially chosen to make ... 9 If you just need any code$E : \{0,1\}^n \to \{0,1\}^m$where$m=O(n)$and where the distance is linear in$m$, then what you are looking for is called an "asymptotically good code". There are many explicit constructions of such codes, and you can find the basic ones in lecture notes of courses about coding theory. For example, you can find a description of ... 8 The class of functions computable by formulas of polynomial size is equivalent to the (nonuniform) class$\mathsf{NC}^1\$ of functions computable by (bounded fanin/fanout) circuits of logarithmic depth. Proving the two implications is a nice exercise. In one direction, you can recursively "untangle" each level of a circuit by creating copies of gates. This ...