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Give an efficient randomized algorithm with bounded one-sided error (false positive), for testing isomorphism between rooted trees with <math>n</math> vertices ...h>x_1,x_2,\ldots,x_n</math> is an unsorted list of <math>n</math> distinct numbers. We sample (with replacement) <math>t</math> items uniformly at random from ...

...to introduce the concepts of the <math>k</math>-cascade representation of numbers and the colex order of sets. :Given positive integers <math>m</math> and <math>k</math>, there exists a unique represent ...

...to introduce the concepts of the <math>k</math>-cascade representation of numbers and the colex order of sets. :Given positive integers <math>m</math> and <math>k</math>, there exists a unique represent ...

...other other for +1. Each <math>c_i</math> gives the difference between the numbers of subjects with feature <math>i</math> in the two groups. By minimizing <m ...less than <math>\frac{k}{2}-\sqrt{2m\ln n}=\left(1-\delta\right)\mu</math> positive <math>b_i</math>'s, where <math>\delta=\frac{2\sqrt{2m\ln n}}{k}</math>. Ap ...

...the independence of random variables, such as the Chernoff bounds. On the positive side, there are tools that require less independence. ...that makes <math>A(x,r)=1</math> a '''witness''' for <math>x</math>. For a positive <math>x</math>, at least half of <math>[p]</math> are witnesses. The random ...

...other other for +1. Each <math>c_i</math> gives the difference between the numbers of subjects with feature <math>i</math> in the two groups. By minimizing <m ...less than <math>\frac{k}{2}-\sqrt{2m\ln n}=\left(1-\delta\right)\mu</math> positive <math>b_i</math>'s, where <math>\delta=\frac{2\sqrt{2m\ln n}}{k}</math>. Ap ...

...other other for +1. Each <math>c_i</math> gives the difference between the numbers of subjects with feature <math>i</math> in the two groups. By minimizing <m ...less than <math>\frac{k}{2}-\sqrt{2m\ln n}=\left(1-\delta\right)\mu</math> positive <math>b_i</math>'s, where <math>\delta=\frac{2\sqrt{2m\ln n}}{k}</math>. Ap ...

...eild (you may think of it as the filed <math>\mathbb{Q}</math> of rational numbers, or the finite field <math>\mathbb{Z}_p</math> of integers modulo prime <ma ...ath>r \in\{0, 1\}^n</math>, thus the algorithm will return a "yes" for any positive instance (<math>AB=C</math>). ...

...ximation ratio of 2 to being 2-universal. The proof uses the fact that odd numbers are relative prime to a power of 2. We exploit that C-multiplication (*) of unsigned u-bit numbers is done <math>\bmod 2^u</math>, and have a one-line C-code for computing th ...

and hence this probability is positive if <math>\Pr[A_i]<1</math> for all <math>i</math>. which is positive if <math>\sum_{i=1}^m\Pr\left[A_i\right]<1</math>. This is a very loose bou ...

...that makes <math>A(x,r)=1</math> a '''witness''' for <math>x</math>. For a positive <math>x</math>, at least half of <math>[p]</math> are witnesses. The random Sampling <math>t</math> mutually independent random numbers from <math>[p]</math> can be quite expensive since it requires <math>\Omega ...

...ject, we define a probability space of objects in which the probability is positive that a randomly selected object has the required property. *If an object chosen randomly from a universe satisfies a property with positive probability, then there must be an object in the universe that satisfies th ...

...to introduce the concepts of the <math>k</math>-cascade representation of numbers and the colex order of sets. :Given positive integers <math>m</math> and <math>k</math>, there exists a unique represent ...

...to introduce the concepts of the <math>k</math>-cascade representation of numbers and the colex order of sets. :Given positive integers <math>m</math> and <math>k</math>, there exists a unique represent ...

:n = [[integer|positive integer]]. [[Category:Numbers]] ...