**This is an old revision of the document!**

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

When | Where | Start | Lecturer |
---|---|---|---|

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

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 The PDFs for lectures 1-13 are still available below, but are now outdated and will not be updated.

Date | Contents | Additional Material |
---|---|---|

April 11 | 1 Discrete Event Spaces and Probabilities 1.1 Discrete Probability Spaces 1.2 Independent Events | [MU05], pp. 3-6 |

April 13 | 1.2 (contd) Conditional Probability 1.3 Applications 1.3.1 The Minimum Cut Problem: Contract Alg. | [MU05], pp. 6-7 [MU05], pp. 12-13 |

April 18 | 1.3.1 (contd) The Minimum Cut Problem: Contract Alg., FastCut | [MU05], pp. 13-14, [MR95], pp. 289-294 |

April 20 | 1.3.1 (contd) The Minimum Cut Problem: FastCut 1.3.2 Reservoir Sampling | [MR95], pp. 294-295 |

April 25 | 2 Evaluating Outcomes of a Random Process 2.1 Random Variables and Expected Values | [MU05], pp. 20-23 |

April 27 | 2.1.1 Non-negative Integer Valued Random Variables 2.1.2 Conditional Expected Values | [MU05], pp. 25, 31, 26-27 |

May 02 | 2.2 Binomial Distribution and Geometric Distribution 2.3 Applications 2.3.1 Randomized QuickSort | [MU05], pp. 30-31, 34-38, 25-26 |

May 04 | 2.3.2 Randomized Approximation Algorithms 3 Concentration bounds: Markov's Inequality | [MU05], pp. 129-130, 44 |

May 09 | 3.1 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 11 | 3.3.2 Routing in Hypercubes | [MU05], pp. 72-74 [MR95], pp. 74-77 |

May 16 | no lecture (Pfingsten) | – |

May 18 | no lecture (Pfingsten) | – |

May 23 | 3.3.2 (contd) Routing in Hypercubes | [MR95], pp. 77-79 |

May 25 | no lecture (Dies Academicus) | – |

May 30 | 4 Random Walks 4.1 Applications 4.1.1 A local search algorithm for 2-SAT | [MU05], pp. 156-159 [MR95], pp. 128-129 |

June 01 | 4.1.2 Local Search algorithms for 3-SAT | [MU05], pp.159-163 |

June 06 | 6 Knapsack Problem and Multiobjective Optimization 6.1 Nemhauser-Ullmann Algorithm | |

June 08 | 6.2 Number of Pareto-optimal Solutions 6.2.1 Upper Bound |

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 |

- Problem Set 0 (to be discussed on April 19th)
- Problem Set 1 (hand in until April 20th, to be discussed April 26th)
- Problem Set 2 (hand in until April 27th, to be discussed May 3rd)
- Problem Set 3 (hand in until May 4th, to be discussed May 10th)
- Problem Set 4 (hand in until May 18th, to be discussed May 24th)
- Problem Set 5 (hand in until May 25th, to be discussed May 31th)
- Problem Set 6 (hand in until June 1st, to be discussed June 7th)
- Problem Set 7 (hand in until June 8th, to be discussed June 14th)
- Problem Set 8 (hand in until June 15th, to be discussed June 21th)

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!

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.

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.