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Related Concept Videos

Uniform Distribution01:19

Uniform Distribution

The uniform distribution is a continuous probability distribution of events with an equal probability of occurrence. This distribution is rectangular.Two essential properties of this distribution are The area under the rectangular shape equals 1. There is a correspondence between the probability of an event and the area under the curve.Further, the mean and standard deviation of the uniform distribution can be calculated when the lower and upper cut-offs, denoted as a and b,...
Propagation of Uncertainty from Random Error00:59

Propagation of Uncertainty from Random Error

An experiment often consists of more than a single step. In this case, measurements at each step give rise to uncertainty. Because the measurements occur in successive steps, the uncertainty in one step necessarily contributes to that in the subsequent step. As we perform statistical analysis on these types of experiments, we must learn to account for the propagation of uncertainty from one step to the next. The propagation of uncertainty depends on the type of arithmetic operation performed on...
Second Uniqueness Theorem01:16

Second Uniqueness Theorem

Consider a region consisting of several individual conductors with a definite charge density in the region between these conductors. The second uniqueness theorem states that if the total charge on each conductor and the charge density in the in-between region are known, then the electric field can be uniquely determined.
In contrast, consider that the electric field is non-unique and apply Gauss's law in divergence form in the region between the conductors and the integral form to the surface...
Randomized Experiments01:13

Randomized Experiments

The randomization process involves assigning study participants randomly to experimental or control groups based on their probability of being equally assigned. Randomization is meant to eliminate selection bias and balance known and unknown confounding factors so that the control group is similar to the treatment group as much as possible. A computer program and a random number generator can be used to assign participants to groups in a way that minimizes bias.
Simple randomization
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Unusual Results01:16

Unusual Results

Unusual results are those that have a very low chance of occurring. Unusual results can be identified using probabilities and the range rule of thumb. In problems involving probability, unusual results can be observed in 2 instances – an unusually high number of successes or an unusually low number of successes.
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Random Variables01:09

Random Variables

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Related Experiment Videos

Uniform derandomization from pathetic lower bounds.

Eric Allender1, V Arvind, Rahul Santhanam

  • 1Department of Computer Science, Rutgers University, New Brunswick, NJ 08855, USA. allender@cs.rutgers.edu

Philosophical Transactions. Series A, Mathematical, Physical, and Engineering Sciences
|June 20, 2012
PubMed
Summary

This study shows that even weak circuit size lower bounds can derandomize probabilistic algorithms. Specifically,

Related Experiment Videos

Area of Science:

  • Theoretical Computer Science
  • Computational Complexity Theory
  • Algorithm Design

Background:

  • Probabilistic computation, dating back to Turing, faces implementation challenges with limited randomness.
  • Derandomization research explores minimizing or eliminating randomness in algorithms.
  • Efficient derandomization typically requires strong circuit size lower bounds, which are currently lacking.

Purpose of the Study:

  • To demonstrate that modest circuit size lower bounds can achieve significant derandomization.
  • To connect 'pathetic' lower bounds to the efficient simulation of probabilistic algorithms.
  • To explore two specific instances where such derandomization is possible.

Main Methods:

  • Investigating the implications of n(1+ε) lower bounds for circuit complexity.
  • Analyzing the word problem over S(5) in the context of constant-depth threshold circuits.
  • Examining matrix multiplication and its relation to arithmetic circuit identity testing.

Main Results:

  • A lower bound of n(1+ε) for constant-depth threshold circuits solving the S(5) word problem implies derandomization of uniform polynomial-size probabilistic threshold circuits.
  • The absence of constant-depth arithmetic circuits of size n(1+ε) for n 3x3 matrix multiplication implies subexponential time derandomization for black-box identity testing in arithmetic circuits.

Conclusions:

  • Weak circuit size lower bounds can be surprisingly powerful for derandomization.
  • These findings bridge the gap between existing lower bound capabilities and the requirements for efficient probabilistic algorithm simulation.
  • The research opens new avenues for understanding the power of randomness in computation.