Martin Tancer

Lecture on Probabilistic techniques (Přednáška Pravděpodobnostní techniky (NTIN022)) 2021/22

Lecture: Tue, 12:20, S10, Exercise classes: Thu, 12:20, S7 (TA Denys Bulavka and Matěj Konečný). Webpage for exercise classes: web of Matěj Konečný.

List of topics covered by lectures is at the bottom of this webpage.

At the moment the plan is teaching with presence of the students in the lecture room without a hybrid form. In case of difficulties, send me an email and we will try to find some solution. You can find my email on my homepage.

Lecture contents


If you are interested in exams: I have scheduled three dates for exams in the study information system (SIS). However, it is also possible to agree on a different date. Send me an email, if you need a different date.


Probabilistic techniques are a major tool of discrete mathematics, they are also frequently used in design and analysis of algorithms and other areas of computer science. The lecture introduces basic notions, methods, and estimates and illustrates them on examples from computer science and discrete mathematics.
If you are interested in more details about expected contents of the lecture, you may check one of the previous lectures (some changes may occur).

Lecture notes

[MV] J. Matoušek, J. Vondrák: The Probabilistic Method ITI Series (preprints Institute of Computer Science, MFF UK), 2002-087; available in the library of MFF UK or you may get a revised version online


Basic notions and methods: events, expected value and its linearity, conditional probability, Bayes' rule.
Basic inequalities and estimates: Markov's a Chebyshev's inequality, Chernoff-type estimates.
Probabilistic method: basic method and alteration method, Lovász local lemma.
Advanced techniques: balls and bins model, basic estimates and applications, Markov chains, stationary distribution, basic continuous distributions as limits of discrete ones, properties and applications. Illustration of the methods on examples from various areas of discrete mathematics and theoretical computer science.


Slides from 2020

In case that you would find them useful, here are some slides from 2020 (not for all lectures). However, they were intended for online teaching and it is not guaranteed that they will match the lectures this year. Use on your own risk: Lecture 1 , Lecture 2, Lecture 3, Lecture 4, Lecture 5, Lecture 6, Lecture 7.

Topics covered

Oct 5 (taught by Matěj Konečný)
Motivation. Overview of basic notions in probability: probability space, examples, union bound. Probabilistic approach for a proof of existence of certain graphs: $R(k,k) > 2^{k/2-1}$, where $R(k,k)$ is the Ramsey number, the minimal number of vertices that guarantees that a graph has an independent set or a clique with $k$ vertices. Estimates of factorial and binomial coefficients, $e^x \geq 1+x$. Continuation with basic notions: Independent events, conditional probability. [MV 1, 2.1, AS 1.1]
Oct 12
Satisfiability of CNF formulas. Erdős-Ko-Rado theorem. Further notions: random variables, expectation, linearity of expectation, independent random variables, product of expectation for independent random variables, indicators. Application: number of fixed points in a random permutation. [MV 1, 2.3, 3.1, AS 2.1, slides above (CNF formulas)]
Oct 19
Further applications of linearity of expected value: tournaments with many Hamiltonian paths; MAXSAT - satisfying many clauses in a formula; MAXCUT - existence of a cut in a graph with many edges (derandomization of MAXCUT, sketched also MAXSAT); balancing vectors, $\pm$ sums of unit vectors can be achieved both high and low. Alterations: weak Turán theorem. [MV 3, 4.1, AS 2.1, 2.2, 2.4, slides above (MAXCUT, derandomization)]
Oct 26
Markov's inequality. Showing the existence of graphs with arbitrarily high chromatic number and girth. Bayes theorem. Geometric/continuous application of alterations: Points in unit square and triangles of small area (proof started but not yet finished). [MV 4.0, 4.2, AS 3.3, slides above (Bayes)]
Nov 2
Finishing the proof from the last time. Testing $AB = C$ for three given $n \times n$ matrices faster than matrix multiplication. Variance, standard deviation, covariance. Variance of a sum of random variables. Chebyshev's inequality. Lower bound for the middle binomial coefficient. [MV 5.1, 5.2, slides for matrix multiplication]
Nov 9
Threshold functions. The function $1/n$ is a threshold function for triangle containment. An auxiliary lemma for harder implication. Threshold function for containment of a balanced graph $H$. [MV 5.3]
Nov 16
Motivation. Dependency graph. Lovász local lemma, symmetric an general version. Shown how general implies symmetric but otherwise left without proof. Two applications of Lovász local lemma: two-colorability of a sparse hypergraph, and cycles of length divisible by $k$ in directed graphs. (In the last case, the bound on the degree in the dependency graph only sketched.) [MV 6.1--6.3]
Nov 23
Chernoff's inequality for binomial distribution. Application: combinatorial discrepancy. More general version of Chernoff's inequality for sum of independent random variable attaining values in [0,1]. Application: Randomized rounding (not finished). [MV, 7.1, 7.2]
Nov 30
Randomized rounding (proof finished). Markov chains: definition, representation via transition matrix and via directed graph. Relation between mth power of the transition matrix and probability of transition from one state to another in m steps. [MV 7.2, MU 7.1]
Dec 7
Randomized algorithms for 2-SAT and for 3-SAT. [MU 7.1]
Dec 14
More Markov chains: accesible/communicating states, transient/recurrent, and null/positive recurrent states, examples. Gambler's ruin. Periodic/aperiodic states/MC. Ergodic states/MC. Theorem about stationary distribution of a Markov chain (proof next time). [MU 7.2, 7.3]
Dec 21
Theorem about stationary distribution of a Markov chain (sketch of a proof based on the Perron-Frobenius theorem and Jordan decomposition; an alternative proof can be found here). Balls and bins: birthday paradox, maximum load. [MU 5.1]
Jan 4
Bucket sort. Poisson distribution, its expected value. Sum of independent Poisson random variables is a Poisson random variable. Poisson distribution is a limit of the binomial distribution. Approximation for balls-and-bins. [MU 5.2-5.3]