The Probabilistic Method in Computer Science

1. The ProbabilisticMethod in Computer Science The ProbabilisticMethod in Computer Science

2. Combinatorics in Computer Science Combinatorics in Computer Science

3. Paul Erdós (1913-1996)

4. Mathematical Achievements • ~1500 papers • ~500 co-authors • Invented new branches of mathematics • Solved a LARGE number of beautiful problems • Posed a HUGE number of beautiful problems

5. Some Facts • Devoted his entire life to Math: • never had a woman • never had an house • never had an office • never had possessions (except for a small suitcase with silk shirts and underwears necessary for his skin) • never had much money (once he won 50000$ and he kept just 720$ for himself)

6. Some Other Facts • Took Amphetamines to be able to sleep 4 hours per day (clearly spending the remaining 20 hours doing math) • Waking you up at 4 o’clock: “Is your brain open?” • “Another roof, another proof” • Prizes for problems, for a total of 15000$: What if all problems are solved at once?

7. Number Theory • Branch of math studying integer numbers and, in particular, prime numbers • First Erdós success: at 17 years, a proof “from the Book” thatthereisalways a prime numberbetweenn and 2n (Bertrand’s postulate) • Prime NumberTheorem (simpleproofbyErdós and Selberg): • The numberofprimes up toxapproachesx/log(x) as x->∞.

8. Number Theory • Exercise: Prove that prime numbers are infinite • Exercise: Design a method for generating all prime numbers

9. Number Theory • Question 1: Does a closed formula exists generating (all and) only prime numbers?

10. Number Theory • Question 2: Does an infinite number of friendly numbers exist? • Two numbers are friendly if each one is the sum of the divisors of the other one. • 284=1+2+4+5+10+11+20+22+44+55+110 • 220=1+2+4+71+142 • Divisori(284)={1,2,4,71,142} • Divisori(220)={1,2,4,5,10,11,20,22,44, 55,110}

11. Number Theory • Question 3: Does an odd perfect number exists? • A perfect number is such that the sum of its divisors is equal to itself • 6=1+2+3 • 28=1+2+4+7+14 • ….

12. Erdós Vocabulary • Supreme Fascist -> God • Epsilon -> Children • Bosses -> Women • Slaves -> Men • Poison -> Alcohol • Preaching -> Lecturing • Dying -> Stop doing Math • Leaving -> Dying

13. Happy Ending Problem • Esther Klein: • Is it true that, for all n, there exists an integer g(n) such that any set of g(n) points in general position contains a convex n-gon?

14. Happy Ending Problem • g(4)=5

15. Happy Ending Problem • Erdós and Szekeres: • Szekeres and Klein got married 

16. Ramsey Theory • Aboutunavoidableoccurrencesofpatterns in a largeinstanceof the problem • In a party, every two persons either know each other or they don’t • Is it true that in a party with a sufficiently large number r(n) of persons, there are always n persons pairwise knowing each other or n persons pairwise not knowing each other?

17. Ramsey Theory • Exercise: r(3)=? • Exercise: translate the problem into a graph problem • hint: (if two persons know each other, then the graph has an edge…) • Exercise: translate the problem into a graph coloring problem • hint: (use complete graphs…)

18. Ramsey Theory • Actualboundsforr(n)

19. Ramsey Theory • Erdós LB: • There are waystobicolor • There are at most • colorings with a monochromatic • Therefore, a coloring with no monochromatic exists if

20. Ramsey Theory • Exercise: construct a sequenceof the first 100 integerscontaining no subsequenceof 11 decreasingnumbers and no subsequenceof 11 increasingnumbers • Exercise: construct a sequenceof the first 101 integerscontaining no subsequenceof 11 decreasingnumbers and no subsequenceof 11 increasingnumbers

21. Ramsey Theory • Everysequenceofn2 +1distinctnumberscontains a decreasingsequenceofn+1numbers or anincreasingsequenceofn+1 numbers

22. Ramsey Theory • Proof: • ConsideranysequenceSofn2 +1distinctnumbers, and associate to the i-thnumberofS a pair(ai,bi) suchthatai and biare the sizesof the longestincreasing and decreasingsubsequencesofStill the i-thnumber. • Then(ai,bi)≠(ak,bk) • Hence, an > n orbn > n

23. Ramsey Theory • The largestvalueofksuchthatr(k)isknownis… • Erdós, aliens, and Ramseynumbers…

24. Erdós skills • Neverdonelaundry • Neverpayedbills • Nevercooked (“I can makeverygoodcoldcereals and I couldprobablyboilaneggbut I nevertried”) • Neverdrived • JánosPach: “I entered the kitchen and sawpoolsofblood-likeredliquidtrailingto the refrigerator, wheretherewas a tomatojuicecartonwith a largehole on its side. Erdósmusthavebeenthirsty…”

25. Random Graph Theory • Exercise: design a methodtoconstruct a randomgraphwith n vertices and m edges • Exercise: design a methodtoconstruct a randomgraphwith n vertices

26. Random Graph Theory • Start withGhavingnvertices and no edge. At eachstepchoose a randomedgeamongthosenotbelongingtoG. Stop whenGhasmedges. • For a givenp, 0 ≤ p ≤ 1, eachpotentialedgeischosenwithprobabilityp, independentof the otheredges. The obtainedgraphisGn,p .

27. Random Graph Theory • RandomGraphs are usefulfor: • Analyzingdeterministicalgorithms • Designing randomsolutions

28. Random Graph Theory • Whichis the right valueofp? • p=0? • p=1? • p=1/2? • p=1/2n? • p=1/n?

29. Random Graph Theory • As Gacquires more and more edges, variousproperties and substructures emerge. The problemof interest istostudy the (sudden) appearanceofgraphpropertiesaspvaries. • A randomgraphhaspropertyA, if the probabilitythatGn,phaspropertyAapproachesto1, asnapproachesinfinity.

30. Random Graph Theory • Erdós-Rényiphases • Phase 1: • Gn,p is the disjointunionoftrees on kvertices • Trees on kverticesappearwhen

31. Random Graph Theory • Phase 2: • Gn,p containscyclesofanygivensizewithprobabilitytendingto a positive limit • Almostallvertices are in trees or in connectedcomponentswith a single cycle • The largestcomponentis a treewithΘ(log n)vertices

32. Random Graph Theory • Phase 3: the doublejump • The behaviorofGn,p whenisdramaticallydifferentfromwhen • Mostof the vertices are into a giantcomponentwhichhasΘ(n)vertices • Allothervertices are in trees or in connectedcomponentswith a single cycle. Eachsmallcomponenthas O(log n) vertices.

33. Random Graph Theory • Phase 4: • Allcomponentsotherthan the giantone are verysmall and are trees • Phase 5: • Thereisoneconnectedcomponent • Phase 6: • Thereisoneconnectedcomponent and the degreesofallvertices are asymptoticallyequal.

34. Erdós Number • Erdós had so many co-authors that they started classifying mathematicians by their Erdós number: • Erdós has Erdós number 0 • Its co-authors have Erdós number 1 • The co-authors of its co-authors have Erdós number 2 • … • Erdós number is either ≤7 or ∞

35. The Probabilistic Method • Tryingto prove that a structurewithcertaindesiredpropertiesexists, onedefinesan appropriate spaceofstructures and thenshowsthat the desiredpropertyholds in thisspacewith positive probability.

36. The Probabilistic Method • Consider a random 2-coloring of . • Forany set R of k vertices, let ARbe the eventthat R ismonochromatic. • Then, • Sincethere are choicesfor R, the probabilitythatoneofevents ARoccursis at most

37. The Probabilistic Method • If , then • Hence,

38. The Probabilistic Method • Tournament T: orientationof the edgesofKn. Vertices are players. An edge (u,v) meansthat a player u beats a player v. • T haspropertySkifforevery set K of k playersthereis a player not in K thatbeatsallplayers in K. • Isittruethatforevery k thereis a tournament T withpropertySk ?

39. The Probabilistic Method • Erdóssolution: • If ,thenthereis a tournament on n playerswhichhaspropertySk • Take a randomtournament on Kn=(V,E). • Foranyfixed set K of k players, let AKbe the eventthat no player in V-K beatsthem all.

40. The Probabilistic Method • Then, , asforevery player v in V-K, the probabilitythat v doesnot beat allplayers in K is and all the n-k eventscorrespondingto the differentchoicesof v are independent. Itfollowsthat: • Therefore, with positive probabilitythereis a tournament on n playerswithpropertySk

41. Finally I’m becomingstupider no more