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Notes on algorithms | Lecture notes on algorithms

Lecture notes on algorithms. Notes on topics related to algorithms. Examples (Max Cut, Turán’s theorem). Method of conditional probabilities (Max Cut). Pessimistic estimators (Turáns theorem). Bounds related to random stopping times. Wald’s equation for dependent increments. Markov bound for super-martingales. Expected maximum (or minimum) of many sums. Expected deviation of a sum. Modeling Set Cover and Multicommodity Flow. Rounding an LP relaxation. Set Cover / greedy algorithm. Set Cover / alterations.

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Notes on algorithms | Lecture notes on algorithms | algnotes.info Reviews

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Lecture notes on algorithms. Notes on topics related to algorithms. Examples (Max Cut, Turán’s theorem). Method of conditional probabilities (Max Cut). Pessimistic estimators (Turáns theorem). Bounds related to random stopping times. Wald’s equation for dependent increments. Markov bound for super-martingales. Expected maximum (or minimum) of many sums. Expected deviation of a sum. Modeling Set Cover and Multicommodity Flow. Rounding an LP relaxation. Set Cover / greedy algorithm. Set Cover / alterations.

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4 table of contents

5 misc background material

6 probabilistic method

7 basic bounds

8 chernoff bound

9 chernoff bound proof

10 wald’s equation

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Notes on algorithms | Lecture notes on algorithms | algnotes.info Reviews

https://algnotes.info

Lecture notes on algorithms. Notes on topics related to algorithms. Examples (Max Cut, Turán’s theorem). Method of conditional probabilities (Max Cut). Pessimistic estimators (Turáns theorem). Bounds related to random stopping times. Wald’s equation for dependent increments. Markov bound for super-martingales. Expected maximum (or minimum) of many sums. Expected deviation of a sum. Modeling Set Cover and Multicommodity Flow. Rounding an LP relaxation. Set Cover / greedy algorithm. Set Cover / alterations.

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Chernoff bound proof | Notes on algorithms

http://algnotes.info/on/background/probabilistic-method/chernoff-proof

Lecture notes on algorithms. The statement and proof of a typical Chernoff bound. Click for background material. A gentle introduction to Chernoff bounds. A Chernoff bound, and proof. Chernoff bound [ 1. Let Y= sum {t=1} T y t. Be a sum of independent random variables, where each y t. Is in [0,1]. And T gt 0. Is fixed. Let varepsilon , mu gt 0. With varepsilon le 1. A) If textrm{E}[y {t}] le mu. For all t le T. Then Pr [Y ge (1 varepsilon ) mu T] , gt , exp ({- varepsilon 2} mu T/3). For all t le T.

2

Expected maximum (or minimum) of many sums | Notes on algorithms

http://algnotes.info/on/background/stopping-times/expected-maximum-sum

Lecture notes on algorithms. Expected maximum (or minimum) of many sums. Bounds on the expected maximum (or minimum) among a collection of sums. It can be technically convenient to work with expectations directly, instead of working with probabilities. Here, given a. Of sums of 0/1 random variables, we bound the expected maximum (or minimum) sum in the collection. For example, suppose Alice throws balls randomly into n. Bins just until the first bin has n. The bound differs from Chernoff in a few ways:.

3

Chernoff bound | Notes on algorithms

http://algnotes.info/on/background/probabilistic-method/chernoff

Lecture notes on algorithms. A brief introduction to Chernoff bounds. If you’re already familiar with Chernoff bounds, you may prefer to skip directly to the statement and proof of a typical Chernoff bound. Chernoff bounds (a.k.a. tail bounds, Hoeffding/Azuma/Talagrand inequalities, the method of bounded differences, etc. [ 1. Are used to bound the probability that some function (typically a sum) of many small random variables falls in the tail of its distribution (far from its expectation). Here are a c...

4

Linear programming | Notes on algorithms

http://algnotes.info/on/background/linear-programming

Lecture notes on algorithms. Notes on Linear Programming and Integer Linear Programming. Linear programming and integer linear programming. An instance of Linear Programming requires finding a vector x. That optimizes a linear objective function subject to linear constraints. For example, given a matrix A in { mathbb R} {n times m}. And vectors b in { mathbb R} n. C in { mathbb R} m. Min x { c cdot x x in { mathbb R} m, Ax ge b } . ]. Is any vector x. Satisfying the constraints x in { mathbb R} m. The op...

5

basic bounds | Notes on algorithms

http://algnotes.info/on/background/probabilistic-method/basic-bounds

Lecture notes on algorithms. Existence proofs, the naive union bound, linearity of expectation, Markov bound. Here is the basic principle underlying the probabilistic method:. Fix any random experiment. Let X. Be any random variable and let mu = textrm{E}[X]. Then there exists at least one outcome where X ge mu. And at least one where X le mu. Here are three simple but fundamental bounds:. For any two random events A. Pr [A text { or } B] le Pr [A] Pr [B]. ]. Random events A 1,A 2, ldots ,A n.

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Notes on algorithms | Lecture notes on algorithms

Lecture notes on algorithms. Notes on topics related to algorithms. Examples (Max Cut, Turán’s theorem). Method of conditional probabilities (Max Cut). Pessimistic estimators (Turáns theorem). Bounds related to random stopping times. Wald’s equation for dependent increments. Markov bound for super-martingales. Expected maximum (or minimum) of many sums. Expected deviation of a sum. Modeling Set Cover and Multicommodity Flow. Rounding an LP relaxation. Set Cover / greedy algorithm. Set Cover / alterations.

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