MA-INF 1213: Randomized Algorithms & Probabilistic Analysis

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


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


Notice: The lecture notes have been updated on 28th of 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 The PDFs for lectures 1-13 are still available below, but are now outdated and will not be updated.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 Course Materials
April 111 Discrete Event Spaces and Probabilities
1.1 Discrete Probability Spaces
1.2 Independent Events
Lecture Notes
[MU05], pp. 3-6
April 131.2 (contd) Conditional Probability
1.3 Applications
1.3.1 The Minimum Cut Problem: Contract Alg.
Lecture Notes
[MU05], pp. 6-7
[MU05], pp. 12-13
April 181.3.1 (contd) The Minimum Cut Problem:
Contract Alg., FastCut
Lecture Notes
[MU05], pp. 13-14,
[MR95], pp. 289-294
April 201.3.1 (contd) The Minimum Cut Problem: FastCut
1.3.2 Reservoir Sampling
Lecture Notes
[MR95], pp. 294-295
April 252 Evaluating Outcomes of a Random Process
2.1 Random Variables and Expected Values
Lecture Notes
[MU05], pp. 20-23
April 272.1.1 Non-negative Integer Valued Random Variables
2.1.2 Conditional Expected Values
Lecture Notes
[MU05], pp. 25, 31, 26-27
May 022.2 Binomial Distribution and Geometric Distribution
2.3 Applications
2.3.1 Randomized QuickSort
Lecture Notes
[MU05], pp. 30-31, 34-38, 25-26
May 042.3.2 Randomized Approximation Algorithms
3 Concentration bounds: Markov's Inequality
Lecture Notes
[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 Lecture Notes
[MU05], pp. 72-74
[MR95], pp. 74-77
May 16no lecture (Pfingsten)
May 18no lecture (Pfingsten)
May 233.3.2 (contd) Routing in Hypercubes Lecture Notes
[MR95], pp. 77-79
May 25no lecture (Dies Academicus)
May 304 Random Walks
4.1 Applications
4.1.1 A local search algorithm for 2-SAT
Lecture Notes
[MU05], pp. 156-159
[MR95], pp. 128-129
June 014.1.2 Local Search algorithms for 3-SAT Lecture Notes
[MU05], pp.159-163
June 066 Knapsack Problem and Multiobjective Optimization
6.1 Nemhauser-Ullmann Algorithm
Lecture Notes
June 086.2 Number of Pareto-optimal Solutions
6.2.1 Upper Bound
Lecture Notes
June 136.2.1 (contd) Upper Bound
6.3 Multiobjective Optimization
6.2.2 Lower Bound
Lecture Notes
June 156.2.2 (contd) Lower Bound
6.4 Core Algorithms
Lecture Notes
June 206.4 (contd) Core Algorithms
7 Smoothed Complexity of Binary Optimization Problems
Lecture Notes
June 227 (contd) Smoothed Complexity of Binary Optimization Problems Lecture Notes
June 277 (contd) Smoothed Complexity of Binary Optimization Problems
8 Successive Shortest Path Algorithm
8.1 The SSP Algorithm
8.1.1 Elementary Properties
8.2 Smoothed Analysis
Lecture Notes
June 298.2.1 Terminology and Notation
8.2.2 Proof of the Upper Bound
8.2.3 Further Properties of the SSP Algorithm
Lecture Notes
July 48.2.3 (contd) Further Properties of the SSP Algorithm
8.2.4 Proof of Lemma 8.7
Lecture Notes
July 69 The 2-Opt Algorithm for the TSP
9.1 Overview of Results
9.2 Polynomial Bound for phi-Perturbed Graphs
9.3 Improved Analysis
Lecture Notes
July 119.3 (contd) Improved Analysis
10 The k-Means Method
Lecture Notes
July 1310.1 Potential Drop in an Iteration of k-Means
10.2 Iterations with Large Cluster Changes
10.3 Iterations with Small Cluster Changes
Lecture Notes
July 18 10.4 Proof of Theorem 10.1 Lecture Notes


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 Sets

Course Material

The Lecture Notes cover the complete 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, July 29, and August 11. Please schedule your exam with Antje Bertram until June 30. The re-exams will be on September 26.

lehre/ss16/vl-randalg.txt · Zuletzt geändert: 2016/07/13 15:44 von roeglin