Robert Šámal, Martin Tancer

Lecture on Probabilistic techniques (Přednáška Pravděpodobnostní techniky (NTIN022)) 2018/19

Lecture: Thu, 15:40, S6, Exercise classes: Thu, 17:20, S6 (TA Matěj Konečný). Webpage for exercise classes: web of Matěj Konečný.

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


Exam times are scheduled in the IS. If you have special requiests, let us know by email: samal@iuuk and tancer@kam, where `iuuk' stands for and `kam' stands for


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 the lecture, you may check .

Organization remark

Exercise classes will be based on individual work of the attendants (homeworks; they are supposed to be solved during the semester).

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.


Topics covered

Will be added during the term.
Oct 3 (MT)
Overview of basic notions in probability: probability space, examples, union bound. Probabilistic approach for a proof of existence of certain graphs: $R(k,k) > 2^{k/2}$, 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, $1 + x \leq e^x$. Continuation with basic notions: Independent events, conditional probability. [MV 1, 2.1, AS 1.1]
Oct 10 (MT)
Further applications of probabilistic approach: CNF formulas where each clause has $k$ distinct literals and number of clauses is less than $2^k$ is satisfiable; Erdős-Ko-Rado theorem. Further basic notions from probability theory: random variables; expected value; independent variables; linearity of expected value; for independent variables, product of expected values is the expected value of product. [MV 1, 2.3, 3.1, AS 2.1, your own notes (CNF formulas)]
Oct 17 (MT)
Applications of linearity of expected value: Number of fixed points in a random permutation. 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, your own notes (MAXCUT, derandomization)]
Oct 24 (MT)
Markov's inequality. An existence of graphs with arbitrarily high chromatic number and girth. Bayes theorem. Application: testing $AB = C$ for three given $n \times n$ matrices faster than matrix multiplication. Geometric/continuous application of alterations: Points in unit square and triangles of small area. (Not finished yet.) [MV 4.0, 4.2, AS 3.3, your own notes (Bayes, multiplication testing)]
Oct 31 (MT)
Finishing the proof from the last time. Variance, standard deviation, covariance. Variance of a sum of random variables. Chebyshev's inequality. Lower bound for a middle binomial coefficient. Threshold functions. The function $1/n$ is a threshold function for triangle containment. (A proof will be given on the next lecture.) [MV 5.1, 5.2, 5.3]
Nov 7 (MT)
Proof that $1/n$ is a threshold function for triangle containment. Threshold function for containment of a balanced graph $H$. Hardy-Ramanujan theorem (the number of primes dividing an integer $n$ is typically close to $\ln \ln n$); final computations in the proof only sketched. [MV 5.3, AS 4.2]
Nov 14 (RS)
Lovász local lemma, symmetric version. Three applications: two-colorability of a sparse hypergraph, routing in a graph by edge disjoint paths, and cycles of length divisible by $k$ in directed graphs.
Nov 21 (RS)
Statement of the general Local Lemma. General implies symmetric. (But proof of general LL is skipped.) Remarks about algorithmic version. Basic Chernoff inequality. Comparison to Chebyshev inequality. Several easy applications. Comparison to Central Limit Theorem. [MU 7.2 or MV 7.1-7.2.]
Nov 28 (RS)
More variants of the Chernoff inequality. [MU, O'D starting on page 84] Randomized routing in a hypercube [MU 4.5.1].
Dec 5 (RS)
Markov chains: definition, representation by finite automata and by matrices. Algorithms for 2-SAT and for 3-SAT. Markov chain on a finite path and on an infinite path. [MU 7.1]
Dec 12 (RS)
More Markov chains: accesible/communicating states, states transient/recurrent, and null/positive recurrent, examples. Periodic/aperiodic states/MC. Ergodic states/MC. [MU 7.2, 7.3]
Dec 18 (RS)
Finite irreducible aperiodic Markov chain is ergodic (with a proof). Theorem about stationary distribution of a Markov chain. We omitted the proof. Motivation of the balls-and-bins: birthday paradox, bucket-sort, maximum in a bin (upper bound). [MU 5.1]
Jan 8 (RS)
Balls-and-bins: lower bound on the max-load. Poisson random variables, Poisson approximation for balls-and-bins. Power of two choices (without a proof). [MU 5.2-5.4] board (Corrected version -- version before Jan 21 had a typo in the formula for Poisson distribution!)