Selecting the topic Why does the randomization help? Approximating the volume Finding the diameter Derandomization Phase transition in Discrete Mathematics, Randomized and Deterministic algorithms Miklós Simonovits Rényi Institute Fritz Jóska, 2013 május 23
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Phase transition in Discrete Mathematics, Randomized and ...math.bme.hu/~balazs/neqdyn/presentations/FJ70_simonovitsm3.pdf · bodies efficiently. Efficiently = in polynomially many
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Selecting the topic Why does the randomization help? Approximating the volume Finding the diameter Derandomization
Phase transition in Discrete Mathematics,Randomized and Deterministic algorithms
Miklós Simonovits
Rényi Institute
Fritz Jóska ,2013 május 23
Selecting the topic Why does the randomization help? Approximating the volume Finding the diameter Derandomization
Friends
Selecting the topic Why does the randomization help? Approximating the volume Finding the diameter Derandomization
Contents
„Phase transition” in Discrete Mathematics
The meaning of phase transition „here”
Random walks, the Polya story,
Random objects investigated for their own sake
using the ergodicity
Selecting the topic Why does the randomization help? Approximating the volume Finding the diameter Derandomization
Selecting the topic
Joska was always interested in Physics and AnalysisJóska started as a PhysisistHaving finished the university he started working on
Rudemo Entropy (?)Rényi, A.
On the dimension and entropy of probability distributions. ActaMath. Acad. Sci. Hungar. 10 1959 193–215
Rudemo, Mats
Dimension and entropy for a class of stochastic processes.(Russian summary) Magyar Tud. Akad. Mat. Kutató Int. Közl. 91964 73–88.On the dimension and entropy of probability distributions. ActaMath. Acad. Sci. Hungar. 10 1959 193–215
Selecting the topic Why does the randomization help? Approximating the volume Finding the diameter Derandomization
Happy Birthday, Jóska
Fritz, J.
Entropy of point processes. Studia Sci. Math. Hungar. 4 1969389–399.
Fritz, J.
An approach to the entropy of point processes. Collection of ar-ticles dedicated to the memory of Alfréd Rényi, II. Period. Math.Hungar. 3 (1973), 73–83.
Selecting the topic Why does the randomization help? Approximating the volume Finding the diameter Derandomization
Happy Birthday, Jóska
Selecting the topic Why does the randomization help? Approximating the volume Finding the diameter Derandomization
Meeting Jóska
Reiman, Olypiad, playing the piano
Selecting the topic Why does the randomization help? Approximating the volume Finding the diameter Derandomization
Physics
Both Jóska and I loved Physics very much in theHighschool.
Jóska started as a physisist, however, after a year hedecided that he will understand physics much better as amathematician.
Brave step „downgrading”
Selecting the topic Why does the randomization help? Approximating the volume Finding the diameter Derandomization
Rényi and Jóska
Jóska spent a long period of his time, in the Rényi Institute.Rényi invited him to the institute, Rényi gave him the firstresearch topic at the Institute
Selecting the topic Why does the randomization help? Approximating the volume Finding the diameter Derandomization
Phase transition for Pastur
Selecting the topic Why does the randomization help? Approximating the volume Finding the diameter Derandomization
Phase transition for us
Sudden changes in the behaviour of a system
TheoremKomlós-Szemerédi Iw we throw down edges at random, assoon as the min degree reaches 2, the graph will beHamiltonian.
The meaning of this theorem
Selecting the topic Why does the randomization help? Approximating the volume Finding the diameter Derandomization
Using randomized constructions
Theorem (Erdos: Ramsey, log n)Let c = 3. Most of the graphs on n vertices do not have c log ncomplete subgraphs,neither (induced) c log n-empty graphs.
Theorem (Shannon: typical codewords)≈ If the chanel has capacity c, then one can send over a littleless information with high security but if one tries to send overmore information, that will be lost. random codewords
Theorem (Pinsker: Expander)There are graphs with linear edge density having the expanderproperty
Selecting the topic Why does the randomization help? Approximating the volume Finding the diameter Derandomization
The three approaches:
• Erdos-Rényi:On the strength of connectednessasymmetry of graphs
• Gilbert
• Ehud Friedgut: sharp thresholdOn what does it depend if we have a slow or a
sudden phase-transition?
Selecting the topic Why does the randomization help? Approximating the volume Finding the diameter Derandomization
Supersaturated graphs
.
.......
Selecting the topic Why does the randomization help? Approximating the volume Finding the diameter Derandomization
Universal path
Aleliunas, Karp, Lipton, Lovász, Rackoff,Random walks, universal traversal sequences, and the complex-ity of maze problems. 20th Annual Symposium on Foundationsof Computer Science, pp. 218–223, IEEE, New York, 1979.
From the introduction: "It is well known that the reachability prob-lem for directed graphs is logspace-complete for the complex-ity class NSPACE(logn), and thus holds the key to the openquestion of whether DSPACE(logn)=NSPACE(logn). Here asusual DSPACE(logn) is the class of languages that are ac-cepted in logn space by deterministic Turing machines, whileNSPACE(logn) is the class of languages that are accepted inlogn space by nondeterministic ones. . . .
The reachability problem for undirected graphs has also beenconsidered [N. D. Jones, Y. E. Lien and W. T. Laasev, Math.Systems Theory 10 (1976), no. 1, 1â17; MR0443429 (56
Selecting the topic Why does the randomization help? Approximating the volume Finding the diameter Derandomization
QuickSort
Wikipedia, shortened:
Quicksort is a sorting algorithm. On average, it makes O(n log n)comparisons to sort n items. It is also known as partition-exchange sort. In the worst case, it makes O(n2) comparisons,though this behavior is rare.Quicksort is often faster in practice than other O(n log n) al-gorithms .
Selecting the topic Why does the randomization help? Approximating the volume Finding the diameter Derandomization
Finding the median
Seems to be trivial, but highly non-trivialThe randomized algorithm is „completely trivial”
• Take a random subset of n2/3 elements
• Sort them
• Choose two of them defining an interval containing themedian „very probably”
• compare each element with them: expected number ofcomparisions is ≈ 3
2n + o(n).
• find the median
Selecting the topic Why does the randomization help? Approximating the volume Finding the diameter Derandomization
What is the problem?
We wish to calculate the volume of high-dimensionalbodies efficiently.
Efficiently = in polynomially many steps.
Is it possible deterministically?
Is it possible by randomized algorithms?
Selecting the topic Why does the randomization help? Approximating the volume Finding the diameter Derandomization
The problem of estimating the volume
Given ε > 0,Find an algorithm which outputs a ζ with
(1 − ε)ζ < vol (K ) < (1 + ε)ζ
Includes several combinatorial problems
E.g. full extensions of partial orderings
No exact solution is possible
Khachiyan: even to write down the result takes exponentialtime.
Permanent problem = counting the 1-factors in a bipartite graph
= hardest problem (Valiant) / This was the first case which wasapproximated by randomized a algorithm
Selecting the topic Why does the randomization help? Approximating the volume Finding the diameter Derandomization
Does the randomization really help?
If P = NP (???) then not as we would like to have it, however,then we do not need this „help”
OtherwiseDyer, M.; Frieze, A.: Computing the volume of convex bodies: acase where randomness provably helps, (1991)
It does help in the volume-estimation forhigh dimensional convex bodies given by oracles
Important:
High dimensionalconvexgiven by oracles
Selecting the topic Why does the randomization help? Approximating the volume Finding the diameter Derandomization
What is an oracle?
„Black box:” We ask questions and it provides some answersabout K
The oracles vary according to which questions can be askedand what types of answers do we get
Selecting the topic Why does the randomization help? Approximating the volume Finding the diameter Derandomization
The model: Oracle
NO / YES
x
SEPARATION ORACLE
Convex body ⇔ separation oracleAnswer: x ∈ K ,x 6∈ K , and a separating plane
Selecting the topic Why does the randomization help? Approximating the volume Finding the diameter Derandomization
Why to use oracles?
Several arguments for it:
There are combinatorial problems where the attached convexbody can be described only by oracles.Matching Polytope, . . . After Khachiyan
Object oriented programming
Mathematical Logic: P = NP? can be asked in various modelswith oracles
In some of them this holds, in others it does not.
Selecting the topic Why does the randomization help? Approximating the volume Finding the diameter Derandomization