### Sidebar

#### Current Semester (SoSe 20)

##### Master

This is an old revision of the document!

# MA-INF 1213: Randomized Algorithms & Probabilistic Analysis 2016

If you have questions or remarks to Part I or the tutorials, contact Melanie Schmidt.

## Lecture

When Where Start Lecturer
Monday, 10:00-11:30
Wedn., 12:00-13:30
LBH / Hörsaal III.03a April 11 Röglin (June-),
Schmidt (-June)

## Schedule

Notice: The Lecture Notes for the complete lecture (Part I+II) appeared! This PDF is now the main course material and will be updated irregularly. The PDFs for lectures 1-13 are still available below, but are now outdated and will not be updated.

April 111 Discrete Event Spaces and Probabilities
1.1 Discrete Probability Spaces
1.2 Independent Events
[MU05], pp. 3-6
April 131.2 (contd) Conditional Probability
1.3 Applications
1.3.1 The Minimum Cut Problem: Contract Alg.
[MU05], pp. 6-7
[MU05], pp. 12-13
April 181.3.1 (contd) The Minimum Cut Problem:
Contract Alg., FastCut

[MU05], pp. 13-14,
[MR95], pp. 289-294
April 201.3.1 (contd) The Minimum Cut Problem: FastCut
1.3.2 Reservoir Sampling

[MR95], pp. 294-295
April 252 Evaluating Outcomes of a Random Process
2.1 Random Variables and Expected Values

[MU05], pp. 20-23
April 272.1.1 Non-negative Integer Valued Random Variables
2.1.2 Conditional Expected Values

[MU05], pp. 25, 31, 26-27
May 022.2 Binomial Distribution and Geometric Distribution
2.3 Applications
2.3.1 Randomized QuickSort

[MU05], pp. 30-31, 34-38, 25-26
May 042.3.2 Randomized Approximation Algorithms
3 Concentration bounds: Markov's Inequality

[MU05], pp. 129-130, 44
May 093.1 Variance and Chebyshev's Inequality
3.2 Chernoff/Rubon bounds
3.3 Applications
3.3.1 Parameter Estimation
Lecture Notes
[MU05], pp. 45, 47-49, 64, 66-68
May 113.3.2 Routing in Hypercubes
[MU05], pp. 72-74
[MR95], pp. 74-77
May 16no lecture (Pfingsten)
May 18no lecture (Pfingsten)
May 233.3.2 (contd) Routing in Hypercubes
[MR95], pp. 77-79
May 304 Random Walks
4.1 Applications
4.1.1 A local search algorithm for 2-SAT

[MU05], pp. 156-159
[MR95], pp. 128-129
June 014.1.2 Local Search algorithms for 3-SAT
[MU05], pp.159-163
June 066 Knapsack Problem and Multiobjective Optimization
6.1 Nemhauser-Ullmann Algorithm
June 086.2 Number of Pareto-optimal Solutions
6.2.1 Upper Bound

## Tutorials

When Where Start Lecturer
Tuesday, 15:15-16:00 LBH, E.08 April 19 Schmidt
Tuesday, 16:15-17:00 LBH, E.08 April 19 Schmidt

## Course Material

The Lecture Notes cover the lecture. Part I is largely based on the following two books:

• [MR95] Rajeev Motwani and Prabhakar Raghavan. Randomized Algorithms. ISBN: 978-0521474658, Cambridge University Press, 1995.
• [MU05] Michael Mitzenmacher and Eli Upfal. Probability and Computing. ISBN: 978-0521835404, Cambridge University Press, 2005.

The following PDFs correspond to the lectures in Part I. They are now outdated and will not be updated!

## Content

The lecture has two parts. First, we consider the design and analysis of randomized algorithms. Many algorithmic problems can be solved more efficiently when allowing randomized decisions. Additionally, randomized algorithms are often easier to design and analyze than their (known) deterministic counterparts. For example, we will see an elegant algorithm for the minimum cut problem. Randomized algorithms can also be more robust on average, like randomized Quicksort.

The analysis of randomized algorithms builds on a set of powerful tools. We will get to know basic tools from probabily theory, very useful tail inequalities and techniques to analyze random walks and Markov chains. We apply these techniques to develop and analyze algorithms for important algorithmic problems like sorting and k-SAT.

Statements on randomized algorithms are either proven to hold on expectation or with high probability over the random choices. This deviates from the classical algorithm analysis but is still a worst-case analysis in its core. In the second part of the lecture, we learn about probabilistic analysis of algorithms. There are a number of important problems and algorithms for which worst-case analysis does not provide useful or empirically accurate results. One prominent example is the simplex method for linear programming whose worst-case running time is exponential while in fact it runs in near-linear time on almost all inputs of interest. Another example is the knapsack problem. While this problem is NP-hard, it is a very easy optimization problem in practice and even very large instances with millions of items can be solved efficiently. The reason for this discrepancy between worst-case analysis and empirical observations is that for many algorithms worst-case instances have an artificial structure and hardly ever occur in practical applications.

In smoothed analysis, one does not study the worst-case behavior of an algorithm but its (expected) behavior on random or randomly perturbed inputs. We will prove, for example, that there are algorithms for the knapsack problem whose expected running time is polynomial if the profits or weights are slightly perturbed at random. This shows that instances on which these algorithms require exponential running time are fragile with respect to random perturbations and even a small amount of randomness suffices to rule out such instances with high probability. Hence, it can be seen as an explanation for why these algorithms work well in practice. We will also apply smoothed analysis to the simplex method, clustering problems, the traveling salesman problem, etc.

## Exams

Even though there is no formal requirement to participate in the tutorials and to submit the homework problems, it is strongly recommended to do so. Oral exams can be taken on July 27, July 28, and July 29. Please schedule your exam with Antje Bertram until June 30.