Week09 Maths

This course material is now made available for public usage.Special acknowledgement to School of Computing, National University of Singapore

for allowing Steven to prepare and distribute these teaching materials.

CS3233C i i P iCompetitiveProgramming

Dr.StevenHalimWeek09 Mathematicsin Programming ContestsinProgrammingContests

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O tliOutline

MiniContest#7+Discussion+Break+Admins Mathematics Related Problems & Algorithms MathematicsRelatedProblems&Algorithms

AdHocMathematicsProblems(quickoverview) Those that do not need specific algorithm just basic coding/math skillThosethatdonotneedspecificalgorithm,justbasiccoding/mathskill

JavaBigInteger Class Number Theory, especially Prime Factors and Modulo ArithmeticNumberTheory,especiallyPrimeFactorsandModuloArithmetic Manyothertopicsareforselfreadingathome(CP2.9)

CS3233 CompetitiveProgramming,StevenHalim,SoC,NUS

M th ti CS d ICPC/IOI (1)Mathematics,CS,andICPC/IOI(1)

ComputerScienceisdeeplyrootedinMathsC t M th Compute=Math

ItisnotasurprisetoseemanyMathsproblemsinICPC(PS:IOItasksareusuallynotMathsspecific) Many of which, I do not have time to teach youManyofwhich,Idonothavetimetoteachyou Fewothers,IcannotteachyouasIdonotknowthemyetCS3233 is NOT a pure Mathematics module CS3233isNOTapureMathematicsmodule

Only1week(1.5hours)isdevotedforMathematicsrelatedtopic

It i i if i k b l i Itisniceifwecanimproveourranksbysolvingsomemathematicsproblemsinprogrammingcontest

CS3233 CompetitiveProgramming,StevenHalim,SoC,NUS

M th ti CS d ICPC/IOI (2)Mathematics,CS,andICPC/IOI(2)

Tips:R i hi h h l th ti Reviseyourhighschoolmathematics

(InNUS):TakeMAXXXXmodulesasCFM:D Readmorereferencesaboutpowerfulmathalgorithmsand/orinterestingnumbertheories,etc

StudyC++&Java.Util.Math/Java.MathLibrary TrymathsproblemsinUVa/otherOJandatprojecteulery p / p j

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Th L t Pl ( t h )TheLecturePlan(moreathome)

Today,wewilldiscussasmallsubset ofthisbigdomainPl Plan: Wewillskip/fastforwardthenotsointerestingstuffs IwillgiveseveralMathsrelatedpopquizzesusingclickersystemtoseehowfaryouknowthesetricks

Wewillfocusonseveralrelatedsubjects:BigInteger,PrimeFactors,andModuloArithmeticg g , ,

AllinvolveBig(Large)Integers...

You will then have to read Chapter 5 of CP2.9 on your ownYouwillthenhavetoreadChapter5ofCP2.9onyourown(itisahugechapterbtw)

CS3233 CompetitiveProgramming,StevenHalim,SoC,NUS

MathematicsRelated ProblemsMathematics RelatedProblems(ascurrentlylistedinCP2.9)

1. AdHocMathematics1. TheSimpleOnes2. Mathematical Simulation (Brute Force)

4. NumberTheory1. PrimeNumbers:SieveofEratosthenes2. GCD & LCM2. MathematicalSimulation(BruteForce)

3. FindingPatternorFormula4. Grid5. NumberSystemsorSequences

2. GCD&LCM3. Factorial4. PrimeFactors5. WorkingwithPrimeFactors

6. Logarithm,Exponentiation,Power7. Polynomial8. BaseNumberVariant9 Just Ad Hoc

6. FunctionsinvolvingPrimeFactors7. ModifiedSieve8. ModuloArithmetic9 Extended Euclid/Linear Diophantine Equation9. JustAdHoc

2. JavaBigInteger1. BasicFeatures2. BonusFeatures

9. ExtendedEuclid/LinearDiophantineEquation10. Others

5. ProbabilityTheory6 CycleFinding

3. Combinatorics1. FibonacciNumbers2. BinomialCoefficients

6. CycleFinding1. FloydsTortoiseHareAlgorithm

7. GameTheory1 Two Players Game Minimax

3. CatalanNumbers4. Others

1. TwoPlayersGame,Minimax2. Nim Game(SpragueGrundyTheorem)

8. AlsoseeChapter9

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Programmingproblemsthatarefromthedomainofmathematics,butwedonotneedspecializeddatastructure(s)oralgorithm(s)tosolvethemW ill d A QUICK SPLASH AND DASH L th d t il t h WewilldoAQUICKSPLASHANDDASHLearnthedetailsathomeSection5.2

ADHOCMATHEMATICS

CS3233 CompetitiveProgramming,StevenHalim,SoC,NUS

Th Si l OTheSimplerOnes

NothingtoteachTh t i l ll Theyaretoosimple,really

Youcanget~10ACsin

M th ti l Si l ti (B t F )MathematicalSimulation(BruteForce)

Nothingtoteachotherthantheonesalreadypresented during iterative/recursivepresentedduringiterative/recursiveCompleteSearchtopic Justremembertoprunethesearchspacewheneverpossible

Note:Problemsthatrequireothertechnique(likenumber theory knowledge) and cannot be solvednumbertheoryknowledge)andcannotbesolvedwithbruteforceareNOTclassifiedinthiscategory

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Fi di P tt F lFindingPatternorFormula

Thisrequiresyourmathematicalinsightstoobtainthose patterns/formulas as soon as possible tothosepatterns/formulasassoonaspossible toreducethetimepenalty(inICPCsetting)

Usefultrick: Solvesomesmallinstancesbyhandy Listthesolutionsandseeifthereis/areanypattern(s)?

Lets do a q ick e ercise Letsdoaquickexercise

CS3233 CompetitiveProgramming,StevenHalim,SoC,NUS

G idGrid

AlsoaboutfindingpatternIt i ti it i l ti th id Itrequirescreativity onmanipulatingthegridorconvertingittosimplerones

Example:

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N b S t SNumberSystemsorSequences

Nothingmuchtoteach:OM t f th ti f ll f ll i th bl Mostofthetime,carefullyfollowingtheproblemdescriptionissufficient

CS3233 CompetitiveProgramming,StevenHalim,SoC,NUS

L ith E ti ti PLogarithm,Exponentiation,Power

InC/C++,wehavelog (basee)andlog10 (base 10)log10 (base10)

InJava.lang.Math,weonlyhavelog (basee) Todologb(a) (baseb),wecanuse:log(a) / log(b)log(a) / log(b)

Btw,whatdoesthiscodesnippetdo?(i )fl (1 l 10((d bl ) ))(int)floor(1 + log10((double)a))

Andhowtocomputethenthrootofa?p

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P l i lPolynomial

Representation:Usuallythecoefficientsofthetermsin some sorted order (based on power)insomesortedorder(basedonpower)

Polynomialformatting,evaluation,derivation(Hornersrule),division,remainder,roots(Ruffinisrule))

Thesolutionusuallyrequirescarefulloops

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B N b V i tBaseNumberVariants

Doyouknowthatbasenumberconversionisnowsuper easy with Java BigInteger?supereasy withJavaBigInteger?

However,forsomevariants,westillhavetogotothebasicsmethod Thesolutionusuallyusebase10(decimal)y ( )asanintermediatestep

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A d A F OthAndAFewOthers

Bestwaytolearnthese:Viapractice

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S ti 5 3Section5.3APowerfulAPIforProgrammingContests

JAVABIGINTEGERCLASS

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J Bi I tJavaBigInteger

1. IamaJavauserbutIhavenever used it beforeneveruseditbefore

2. Iama(pure)C++usersoI never used it beforeInever useditbefore

3. IamaJavauserandIhaveused it beforeuseditbefore

4. Iambilingual(Java/C++)d I h d it b f andIhaveuseditbefore

0 000

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1 2 3 40 of 5

Bi I t (1)BigInteger(1)

Rangeofdefaultintegerdatatypes(C++)unsigned int = unsigned long: 232 (9 10 digits) unsignedint=unsignedlong:232(910digits)

unsignedlonglong:264 (1920digits)

Q ti Question: Whatis777!,i.e.factorialof777?

Solution? Big Integer: Use string to represent numberBigInteger:Usestring torepresentnumber

~numbercanbeaslongascomputermemorypermits FYI this is similar to how basic data types are stored in computerFYI,thisissimilartohowbasicdatatypesarestoredincomputermemory.Justthatthistimewedonothavelimitationofthenumberofbits(digits)used

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Bi I t (2)BigInteger(2)

OperationsonBigIntegerB i dd bt t lti l di id t Basic:add,subtract,multiply,divide,etc

Usehighschoolmethod Someexamplesbelow:

2181 carry21845

21845--- x

--- +263

1090 (218*5)872 (218*4)*10----- +9810

CS3233 CompetitiveProgramming,StevenHalim,SoC,NUS

Bi I t (3)BigInteger(3)

Note: Writing these high school methods during stressful contestWritingthese highschoolmethods duringstressfulcontest

environmentisnotagoodstrategy!

Fortunately,JavahasBigIntegerlibraryy, g g y Theyareallowedtobeusedincontests(ICPCandCS3233)

Souseit

Note:IOIdoesnotallowJavayet,andanyway,IhavenotseeBigIntegerrelatedtasksinIOI

Or,ifyouinsist,buildyourownBigIntlibraryandbringitshardcopytofuturecontests!py

CS3233 CompetitiveProgramming,StevenHalim,SoC,NUS

J Bi I t ClJavaBigIntegerClass

ThisclassisratherpowerfulN t j t it ll f b i th ti l ti Notjustitallowsforbasicmathematicaloperationsinvolvingbigintegers(addition,subtraction,multiplication,division mod or remainder and power)division,modorremainder,andpower)

Italsoprovidessupportfor: FindingGCDofbignumbers Findingthesolutionofxy modm(moduloarithmetic)V E B N b C i it f l VeryEasyBaseNumberConversion,quiteuseful

NEWinCP2.9:IsProbablePrime See various examples in the book Seevariousexamplesinthebook

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Section5.4

COMBINATORICS

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C bi t iCombinatorics

Givenproblemdescription,find some nice formula to count somethingfindsomeniceformulatocountsomething Codingis(usuallyvery)short Findingtheformulaisnotstraightforward

Ifformulahasoverlappingsubproblems useDP Ifformulayieldhugenumbers useJavaBigInteger

Memorize/studythebasicones:Fibonaccibased/ yformulas,BinomialCoefficients,CatalanNumbers

PS: On Line Encyclopedia of Integer Sequences PS:OnLineEncyclopediaofIntegerSequences Canbeagoodreference:http://oeis.org/

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Programmingproblemsthatrequirestheknowledgeofnumbertheory,otherwiseyouwilllikelygetTimeLimitExceeded(TLE)responsef l i th i lforsolvingthemnaivelySection5.5

NUMBERTHEORY

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P i N bPrimeNumbers

Firstprimeandtheonlyevenprime:2Fi t 10 i {2 3 5 7 11 13 17 19 23 29} First10primes:{2,3,5,7,11,13,17,19,23,29}

Primesinrange:g 1to100:25primes 1to1,000:168primes 1to7,919:1,000primes 1to10,000:1,229primes

Largestprimeinsigned32bitint=2,147,483,647U d/ i Used/appearin: Factoring Cryptography ManyotherproblemsinICPC,etc

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O ti i d P i T tiOptimizedPrimeTesting

AlgorithmsfortestingifNisprime:isPrime(N)Fi t t h k if N i di i ibl b i [2 N 1]? Firsttry:checkifNisdivisiblebyi [2..N1]?

O(N)

I d 1 I N di i ibl b i [2 (N)]? Improved1:IsNdivisiblebyi [2..sqrt(N)]? O(sqrt(N))

Improved2:IsNdivisiblebyi [3,5,..sqrt(N)]? Onetestfori=2,noneedtotestotherevennumbers! O(sqrt(N)/2)=O(sqrt(N))

Improved3:IsNdivisiblebyi primessqrt(N) O((sqrt(N)))=O(sqrt(N)/log(sqrt(N)))

(M)=numofprimesuptoMF thi d ll i b f h d Forthis,weneedsmallerprimesbeforehand

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P i G tiPrimeGeneration

Whatifwewanttogeneratealistofprime numbers between [0 N]?primenumbersbetween[0N]?

Slownavealgorithm:Loop i from [0 N]if (isPrime(i))

print i Canwedobetter?

Yes:SieveofEratosthenes

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Si f E t th Al ithSieveofEratosthenesAlgorithm

Generateprimesbetween[0N]:f Usebitset ofsizeN,setalltrueexceptindex0&1

Startfromi=2untilk*i>N Ifbitsetatindexiison,crossallmultipleofI(i.e.turnoffbitatindexi)startingfromi*i

Finally,whatevernotcrossedareprimes

Example:Example: 0,1,2,3,4,5,6,7,8,9,10,11,,51,52,53,54,55,,75,76,77, 0,1,2,3,4,5,6,7,8,9,10,11,,51,52,53,54,55,,75,76,77,, , , , , , , , , , , , , , , , , , , , , , 0,1,2,3,4,5,6,7,8,9,10,11,,51,52,53,54,55,,75,76,77, 0,1,2,3,4,5,6,7,8,9,10,11,,51,52,53,54,55,,75,76,77, 0,1,2,3,4,5,6,7,8,9,10,11,,51,52,53,54,55,,75,76,77,

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C d i & i P iCode:sieve&isPrime#include // compact STL for Sieve, better than vector!ll _sieve_size; // ll is defined as: typedef long long ll;bitset bs; // 10^7 should be enough for most casesvi primes; // compact list of primes in form of vector

void sieve(ll upperbound) { // create list of primes in [0..upperbound]sieve size = upperbound + 1; // add 1 to include upperbound_sieve_size = upperbound + 1; // add 1 to include upperbound

bs.set(); // set all bits to 1bs[0] = bs[1] = 0; // except index 0 and 1for (ll i = 2; i

G t t C Di i (GCD)GreatestCommonDivisor(GCD)

NaveAlgorithm:Fi d ll di i f d b ( l ) Findalldivisorsofaandb(slow)

Findthosethatarecommon Pickthegreatestone

Better&Famousalgorithm:D&CEuclidalgorithmetter & Famous algorithm: & C uclid algorithm GCD(a,0)=aGCD(a b) GCD(b a % b) // problem size decreases a lot!! GCD(a,b)=GCD(b,a%b)//problemsizedecreasesalot!!

Itsrecursivecodeiseasytowrite:int gcd(int a, int b) { return (b == 0 ? a : gcd(b, a % b)); }

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L t C M lti l (LCM)LowestCommonMultiple(LCM)

lcm(a,b)=a*b/gcd(a,b)int lcm(int a, int b) { return (a / gcd(a, b) * b); }// Q: why we write the lcm code this way?

Noteforgcd/lcmofmorethan2numbers: gcd(a,b,c)=gcd(a,gcd(b,c));

BothgcdandlcmrunsinO(log10 n)g ( g10 )wheren=max(a,b)

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F t i lFactorial

Whatisthehighestn sothatfactorial(n) stillfitsin64 bits unsigned long long?64bitsunsignedlonglong? Answer:n=20

20! = 2432902008176640000 ull = 1844674407370955161521! 51090942171709440000 21! = 51090942171709440000

Hm...soalmostallfactorialrelatedquestionsrequireJavaBigInteger? Not really, if you use prime power factorizationNotreally,ifyouuseprimepowerfactorization Example:7!=2x3x4x5x6x7=2x3x2x2x5x2x3x7=24x32x5x7Now: is 7! divisible by 35? (Revisited Soon) Now:is7!divisibleby35?(RevisitedSoon)

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P i F tPrimeFactors

Directalgorithm:Generatelistofprimes(usesieve),check how many of them can divide integer NcheckhowmanyofthemcandivideintegerN Thiscanbeimproved!

Betteralgorithm:DivideandConquer! An integer N can be expressed as:AnintegerNcanbeexpressedas:

N=PF*N' PF=aprimefactor

But if integer I is a large prime, then thi i till l N'=anothernumberwhichisN/PF

IfN'=1,stop;otherwise,repeat

this is still slow.

This fact is the basis Nisreducedeverytimewefindadivisor for cryptography

techniques

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Okthatsenoughforthissemester

THEOTHERMATHSPROBLEMSIN PROGRAMMING CONTESTINPROGRAMMINGCONTEST

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N t C d i L t thi SNotCoveredinLecturethisSem

LinearDiophantineEquation(Section5.5) Probability Theory (Section 5 6) ProbabilityTheory(Section5.6) CycleFinding(Section5.7)

h ( ) GameTheory(Section5.8) GaussianElimination(Section9.4) MatrixPower(Section9.13) RomanNumerals(Section9.20)( ) TheyarealreadywritteninCP2.9

They are good readTheyaregoodread Readthemonyourown These problems will not appear as problem A or B in mini contest 8TheseproblemswillnotappearasproblemAorBinminicontest8

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M M Still N t i CP2 9 Y tManyMoreStillNotinCP2.9Yet

Mathematicsisalargefield Pollards Rho integer factoring algorithmPollard sRhointegerfactoringalgorithm ManyotherPrimetheorems,hypotheses,conjectures ChineseRemainderTheorem LotsofDivisibilityProperties CombinatorialGames,etc,

AgainCS3233!=MathModule Chapter 5 of CP2 9 has a collection of ~372 UVa programming Chapter5ofCP2.9hasacollectionof 372 UVa programming

exercisesthehighestamongthe9chaptersinCP2.9!

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Th P f thi L tThePaceofthisLecture

1. Tooslow,Ialreadyk ll thknowalltheseIwanttoknowmore

2. Fine;weareusedtoitnow

3 Crazy as always3. Crazyasalwaysandwestillhave

0 00lotsofreadingmaterial at home

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1 2 3

materialathome0 of 30

SSummary

Wehaveseensomemathematicsrelatedproblemsand algorithmsandalgorithms Toomanytobelearnedinonenight Evenso,manyothersareleftuncovered(someareinsideCP2.9foryoutoreadonyourownpace)

Bestwaytolearn:Lotsofpractice In the next two weeks, two more new topics:Inthenexttwoweeks,twomorenewtopics:

Week10:StringProcessing(FocusonSA)W k 11 (C t ti l) G t (F P l ) Week11:(Computational)Geometry(FocusonPolygons)

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R fReferences

CP2.9,Chapter5 andpartsofChapter9I t d ti t Al ith Ch 31 A di A/B/C IntroductiontoAlgorithms,Ch31,AppendixA/B/C

ProjectEuler,http://projecteuler.net/j , p //p j /

CS3233 CompetitiveProgramming,StevenHalim,SoC,NUS