Typically-Correct Derandomization for Small Time and Space

Suppose a language L can be decided by a bounded-error randomized algorithm that runs in space S and time n * poly(S). We give a randomized algorithm for L that still runs in space O(S) and time n * poly(S) that uses only O(S) random bits; our algorithm has a low failure probability on all but a negligible fraction of inputs of each length. As an immediate corollary, there is a deterministic algorithm for L that runs in space O(S) and succeeds on all but a negligible fraction of inputs of each length. We also give several other complexity-theoretic applications of our technique

Preserving Randomness for Adaptive Algorithms

Suppose Est is a randomized estimation algorithm that uses n random bits and outputs values in R^d. We show how to execute Est on k adaptively chosen inputs using only n + O(k log(d + 1)) random bits instead of the trivial nk (at the cost of mild increases in the error and failure probability). Our algorithm combines a variant of the INW pseudorandom generator [Impagliazzo et al., 1994] with a new scheme for shifting and rounding the outputs of Est. We prove that modifying the outputs of Est is necessary in this setting, and furthermore, our algorithm\u27s randomness complexity is near-optimal in the case d {-1, 1} using O(n log n) * poly(1/theta) queries to F and O(n) random bits (independent of theta), improving previous work by Bshouty et al. [Bshouty et al., 2004]

The Adversarial Noise Threshold for Distributed Protocols

We consider the problem of implementing distributed protocols, despite adversarial channel errors, on synchronous-messaging networks with arbitrary topology. In our first result we show that any $n$-party $T$-round protocol on an undirected communication network $G$ can be compiled into a robust simulation protocol on a sparse ($\mathcal{O}(n)$ edges) subnetwork so that the simulation tolerates an adversarial error rate of $\Omega\left(\frac{1}{n}\right)$; the simulation has a round complexity of $\mathcal{O}\left(\frac{m \log n}{n} T\right)$, where $m$ is the number of edges in $G$. (So the simulation is work-preserving up to a $\log$ factor.) The adversary's error rate is within a constant factor of optimal. Given the error rate, the round complexity blowup is within a factor of $\mathcal{O}(k \log n)$ of optimal, where $k$ is the edge connectivity of $G$. We also determine that the maximum tolerable error rate on directed communication networks is $\Theta(1/s)$ where $s$ is the number of edges in a minimum equivalent digraph. Next we investigate adversarial per-edge error rates, where the adversary is given an error budget on each edge of the network. We determine the exact limit for tolerable per-edge error rates on an arbitrary directed graph. However, the construction that approaches this limit has exponential round complexity, so we give another compiler, which transforms $T$-round protocols into $\mathcal{O}(mT)$-round simulations, and prove that for polynomial-query black box compilers, the per-edge error rate tolerated by this last compiler is within a constant factor of optimal.Comment: 23 pages, 2 figures. Fixes mistake in theorem 6 and various typo

Near-Optimal Pseudorandom Generators for Constant-Depth Read-Once Formulas

We give an explicit pseudorandom generator (PRG) for read-once AC^0, i.e., constant-depth read-once formulas over the basis {wedge, vee, neg} with unbounded fan-in. The seed length of our PRG is O~(log(n/epsilon)). Previously, PRGs with near-optimal seed length were known only for the depth-2 case [Gopalan et al., 2012]. For a constant depth d > 2, the best prior PRG is a recent construction by Forbes and Kelley with seed length O~(log^2 n + log n log(1/epsilon)) for the more general model of constant-width read-once branching programs with arbitrary variable order [Michael A. Forbes and Zander Kelley, 2018]. Looking beyond read-once AC^0, we also show that our PRG fools read-once AC^0[oplus] with seed length O~(t + log(n/epsilon)), where t is the number of parity gates in the formula. Our construction follows Ajtai and Wigderson\u27s approach of iterated pseudorandom restrictions [Ajtai and Wigderson, 1989]. We assume by recursion that we already have a PRG for depth-d AC^0 formulas. To fool depth-(d + 1) AC^0 formulas, we use the given PRG, combined with a small-bias distribution and almost k-wise independence, to sample a pseudorandom restriction. The analysis of Forbes and Kelley [Michael A. Forbes and Zander Kelley, 2018] shows that our restriction approximately preserves the expectation of the formula. The crux of our work is showing that after poly(log log n) independent applications of our pseudorandom restriction, the formula simplifies in the sense that every gate other than the output has only polylog n remaining children. Finally, as the last step, we use a recent PRG by Meka, Reingold, and Tal [Meka et al., 2019] to fool this simpler formula

Child and Parent Predictors of Perceptions of Parent–Child Relationship Quality

Objective/Method: Predictors of perceptions of parent–child relationship quality were examined for 175 children with ADHD, 119 comparison children, and parents of these children, drawn from the follow-up phase of the Multimodal Treatment Study of Children with ADHD. Results/Conclusion: Children with ADHD perceived their mothers and fathers as more power assertive than comparison children. Children higher on depressive symptomatology also perceived their mothers and fathers as less warm and more power assertive. Mothers perceived themselves as more power assertive and fathers perceived themselves as less warm if they were higher on depressive symptomatology themselves or had children with ADHD or higher levels of depressive symptomatology. Several interactions indicated that the association between child factors and parental perceptions of warmth and power assertion often depended on parental depressive symptomatology. The findings resolve a previous contradiction in the literature regarding the relationship between child depressive symptoms and parental perceptions of parent–child relationship quality