**CS 294-202 Pseudorandomness**

**Course Description:**

Randomized algorithms give a broad and rich algorithmic toolkit (e.g., sampling, Monte Carlo methods). Randomness is an essential resource in distributed computing, cryptography, and interactive proofs. In this course, we would explore the role of randomness in computation: Can we derandomize algorithms without changing their time or space complexity? Can we "purify" randomness from a weak source of randomness?

Pseudo-randomness is the property of "appearing random" while having little or no randomness. Pseudo-randomness plays a significant role in error-correcting codes, expander graphs, randomness extractors, and pseudo-random generators. In this course, we will see all these beautiful applications. In the second part of the course, we would focus on the question of derandomization of small-space computation, also known as the "**RL** versus **L**" question. It asks whether all problems that can be decided in randomized logarithmic space (RL) can also be decided in deterministic logarithmic space (L). We would cover recent approaches towards showing that RL = L.

Undergraduate students who wish to take this class should fill out the following Google Form.

**General Information: **

**Time and Place: **Tuesday, Thursday 3:30-5:00 PM, Soda 405

**Office Hours:** Monday 3:00-4:00 PM (via Zoom).

**Instructor: **Avishay Tal, atal "at" berkeley.edu

**TA: **Kewen Wu, shlw_kevin [at] hotmail [dot] com

**Grading: **Homeworks - 40% (4 assignments), Final Project & Presentation - 50%, Scribes - 10%.

**Prereqs:** CS170 or equivalent is required.

**Textbook:**

Salil Vadhan - Pseudorandomness

**Problem Sets:**

Homework 1 - Due Monday, Sep 20.

**Topics:**

k-wise Independence

ε-Biased Distributions

Error Correcting Codes

Expander Graphs - Cheeger's Inequality, Zig-Zag Product, SL=L

Pseudorandom Generators - NW Construction

Random Extractors

Connections between all the above

Derandomization of Small-Space Computation

**Lectures****:**

Introduction + derandomizing an approximation algorithm Max-Cut

k-wise independence

Error reduction + Intro to PRGs Lecture note

Construction of small biased distributions [AGHP] + Brief introduction to discrete Fourier analysis

Error correcting codes, Code concatenation, Connections between small biased distributions and error correcting codes

Expander Graphs - Combinatorial Definition, Existence using the Probabilistic Method

Spectral Expansion Lecture note

Expander Mixing Lemma

Random Walks on Expanders

Zig-Zag Product

SL=L

**Additional Reading****:**

Simple Construction of Almost k-wise Independent Random Variables - Alon, Goldreich, Håstad, Peralta

Expander Graphs and Their Applications -- Hoory, Linial, Wigderson