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c Steven,Felix,Suhendry
(cid:13)
ii
Contents
Authors’ Profiles vii
5 Mathematics 273
5.1 OverviewandMotivation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 273
5.2 AdHocMathematicalProblems . . . . . . . . . . . . . . . . . . . . . . . . . 274
5.3 NumberTheory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 282
5.3.1 PrimeNumbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 282
5.3.2 ProbabilisticPrimeTesting(JavaOnly) . . . . . . . . . . . . . . . . 284
5.3.3 FindingPrimeFactorswithOptimizedTrialDivisions. . . . . . . . . 284
5.3.4 FunctionsInvolvingPrimeFactors . . . . . . . . . . . . . . . . . . . 286
5.3.5 ModifiedSieve. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 288
5.3.6 GreatestCommonDivisor&LeastCommonMultiple . . . . . . . . . 288
5.3.7 Factorial . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 289
5.3.8 WorkingwithPrimeFactors . . . . . . . . . . . . . . . . . . . . . . . 289
5.3.9 ModularArithmetic . . . . . . . . . . . . . . . . . . . . . . . . . . . 290
5.3.10 ExtendedEuclideanAlgorithm . . . . . . . . . . . . . . . . . . . . . 292
5.3.11 NumberTheoryinProgrammingContests . . . . . . . . . . . . . . . 293
5.4 Combinatorics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 298
5.4.1 FibonacciNumbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . 298
5.4.2 BinomialCoefficients . . . . . . . . . . . . . . . . . . . . . . . . . . . 299
5.4.3 CatalanNumbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 300
5.4.4 CombinatoricsinProgrammingContests . . . . . . . . . . . . . . . . 301
5.5 ProbabilityTheory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 305
5.6 Cycle-Finding . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 308
5.6.1 ProblemDescription . . . . . . . . . . . . . . . . . . . . . . . . . . . 308
5.6.2 SolutionsusingEfficientDataStructures . . . . . . . . . . . . . . . . 308
5.6.3 Floyd’sCycle-FindingAlgorithm . . . . . . . . . . . . . . . . . . . . 309
5.7 GameTheory(Basic) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 312
5.8 MatrixPower . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 315
5.8.1 SomeDefinitionsandSampleUsages . . . . . . . . . . . . . . . . . . 315
5.8.2 EfficientModularPower(Exponentiation) . . . . . . . . . . . . . . . 316
5.8.3 EfficientMatrixModularPower(Exponentiation) . . . . . . . . . . . 317
5.8.4 DPSpeed-upwithMatrixPower . . . . . . . . . . . . . . . . . . . . 318
5.9 SolutiontoNon-StarredExercises . . . . . . . . . . . . . . . . . . . . . . . . 321
5.10 ChapterNotes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 324
6 String Processing 325
6.1 OverviewandMotivation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 325
6.2 AdHocString(Harder) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 326
6.3 StringProcessingwithDP . . . . . . . . . . . . . . . . . . . . . . . . . . . . 329
iii
CONTENTS c Steven,Felix,Suhendry
(cid:13)
6.3.1 StringAlignment(EditDistance) . . . . . . . . . . . . . . . . . . . . 329
6.3.2 LongestCommonSubsequence. . . . . . . . . . . . . . . . . . . . . . 331
6.3.3 NonClassicalStringProcessingwithDP . . . . . . . . . . . . . . . . 331
6.4 StringMatching . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 333
6.4.1 LibrarySolutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 333
6.4.2 Knuth-Morris-Pratt(KMP)Algorithm . . . . . . . . . . . . . . . . . 333
6.4.3 StringMatchingina2DGrid . . . . . . . . . . . . . . . . . . . . . . 336
6.5 SuffixTrie/Tree/Array . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 338
6.5.1 SuffixTrieandApplications . . . . . . . . . . . . . . . . . . . . . . . 338
6.5.2 SuffixTree. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 340
6.5.3 ApplicationsofSuffixTree . . . . . . . . . . . . . . . . . . . . . . . . 341
6.5.4 SuffixArray . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 343
6.5.5 ApplicationsofSuffixArray . . . . . . . . . . . . . . . . . . . . . . . 351
6.6 StringMatchingwithHashing . . . . . . . . . . . . . . . . . . . . . . . . . . 355
6.6.1 HashingaString . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 355
6.6.2 RollingHash . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 355
6.6.3 Rabin-KarpStringMatchingAlgorithm. . . . . . . . . . . . . . . . . 357
6.6.4 CollisionsProbability . . . . . . . . . . . . . . . . . . . . . . . . . . . 358
6.7 AnagramandPalindrome . . . . . . . . . . . . . . . . . . . . . . . . . . . . 359
6.7.1 Anagram. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 359
6.7.2 Palindrome . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 359
6.8 SolutiontoNon-StarredExercises . . . . . . . . . . . . . . . . . . . . . . . . 363
6.9 ChapterNotes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 364
7 (Computational) Geometry 365
7.1 OverviewandMotivation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 365
7.2 BasicGeometryObjectswithLibraries . . . . . . . . . . . . . . . . . . . . . 368
7.2.1 0DObjects: Points . . . . . . . . . . . . . . . . . . . . . . . . . . . . 368
7.2.2 1DObjects: Lines . . . . . . . . . . . . . . . . . . . . . . . . . . . . 371
7.2.3 2DObjects: Circles . . . . . . . . . . . . . . . . . . . . . . . . . . . . 376
7.2.4 2DObjects: Triangles . . . . . . . . . . . . . . . . . . . . . . . . . . 378
7.2.5 2DObjects: Quadrilaterals. . . . . . . . . . . . . . . . . . . . . . . . 381
7.3 AlgorithmsonPolygonwithLibraries . . . . . . . . . . . . . . . . . . . . . . 384
7.3.1 PolygonRepresentation . . . . . . . . . . . . . . . . . . . . . . . . . 384
7.3.2 PerimeterofaPolygon . . . . . . . . . . . . . . . . . . . . . . . . . . 384
7.3.3 AreaofaPolygon. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 385
7.3.4 CheckingifaPolygonisConvex . . . . . . . . . . . . . . . . . . . . . 386
7.3.5 CheckingifaPointisInsideaPolygon . . . . . . . . . . . . . . . . . 387
7.3.6 CuttingPolygonwithaStraightLine . . . . . . . . . . . . . . . . . . 388
7.3.7 FindingtheConvexHullofaSetofPoints . . . . . . . . . . . . . . . 390
7.4 3DGeometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 396
7.5 SolutiontoNon-StarredExercises . . . . . . . . . . . . . . . . . . . . . . . . 398
7.6 ChapterNotes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 400
8 More Advanced Topics 401
8.1 OverviewandMotivation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 401
8.2 MoreAdvancedSearchTechniques . . . . . . . . . . . . . . . . . . . . . . . 402
8.2.1 BacktrackingwithBitmask . . . . . . . . . . . . . . . . . . . . . . . 402
8.2.2 State-SpaceSearchwithBFSorDijkstra’s . . . . . . . . . . . . . . . 405
8.2.3 MeetintheMiddle . . . . . . . . . . . . . . . . . . . . . . . . . . . . 407
iv
CONTENTS c Steven,Felix,Suhendry
(cid:13)
8.3 MoreAdvancedDPTechniques . . . . . . . . . . . . . . . . . . . . . . . . . 411
8.3.1 DPwithBitmask . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 411
8.3.2 CompilationofCommon(DP)Parameters . . . . . . . . . . . . . . . 412
8.3.3 HandlingNegativeParameterValueswithOffset. . . . . . . . . . . . 412
8.3.4 MLE/TLE?UseBetterStateRepresentation. . . . . . . . . . . . . . 414
8.3.5 MLE/TLE?DropOneParameter,RecoverItfromOthers . . . . . . 415
8.3.6 MultipleTestCases? NoMemoTableRe-initializations . . . . . . . . 416
8.3.7 MLE?UsebBSTorHashTableasMemoTable . . . . . . . . . . . . 417
8.3.8 TLE?UseBinarySearchTransitionSpeedup. . . . . . . . . . . . . . 417
8.3.9 OtherDPTechniques. . . . . . . . . . . . . . . . . . . . . . . . . . . 418
8.4 NetworkFlow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 420
8.4.1 OverviewandMotivation . . . . . . . . . . . . . . . . . . . . . . . . 420
8.4.2 Ford-FulkersonMethod. . . . . . . . . . . . . . . . . . . . . . . . . . 420
8.4.3 Edmonds-KarpAlgorithm . . . . . . . . . . . . . . . . . . . . . . . . 422
8.4.4 Dinic’sAlgorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 423
8.4.5 FlowGraphModeling-Classic . . . . . . . . . . . . . . . . . . . . . 428
8.4.6 FlowGraphModeling-NonClassic. . . . . . . . . . . . . . . . . . . 432
8.4.7 NetworkFlowinProgrammingContests . . . . . . . . . . . . . . . . 433
8.5 GraphMatching. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 435
8.5.1 OverviewandMotivation . . . . . . . . . . . . . . . . . . . . . . . . 435
8.5.2 GraphMatchingVariants . . . . . . . . . . . . . . . . . . . . . . . . 435
8.5.3 UnweightedMCBM. . . . . . . . . . . . . . . . . . . . . . . . . . . . 436
8.5.4 WeightedMCBMandUnweighted/WeightedMCM . . . . . . . . . . 439
8.6 NP-hard/completeProblems . . . . . . . . . . . . . . . . . . . . . . . . . . . 441
8.6.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 441
8.6.2 Pseudo-Polynomial: Knapsack,Subset-Sum,Coin-Change . . . 442
8.6.3 Traveling-Salesman-Problem(TSP) . . . . . . . . . . . . . . . 443
8.6.4 Hamiltonian-Path/Tour . . . . . . . . . . . . . . . . . . . . . . . 445
8.6.5 Longest-Path. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 446
8.6.6 Max-Independent-SetandMin-Vertex-Cover . . . . . . . . . 447
8.6.7 Min-Set-Cover . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 453
8.6.8 Min-Path-Cover . . . . . . . . . . . . . . . . . . . . . . . . . . . . 454
8.6.9 Satisfiability(SAT). . . . . . . . . . . . . . . . . . . . . . . . . . 455
8.6.10 Steiner-Tree . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 457
8.6.11 Graph-Coloring . . . . . . . . . . . . . . . . . . . . . . . . . . . . 459
8.6.12 Min-Clique-Cover . . . . . . . . . . . . . . . . . . . . . . . . . . . 460
8.6.13 OtherNP-hard/completeProblems . . . . . . . . . . . . . . . . . . . 461
8.6.14 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 462
8.7 ProblemDecomposition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 465
8.7.1 TwoComponents: BinarySearchtheAnswerandOther . . . . . . . 465
8.7.2 TwoComponents: InvolvingEfficientDataStructure . . . . . . . . . 467
8.7.3 TwoComponents: InvolvingGeometry . . . . . . . . . . . . . . . . . 468
8.7.4 TwoComponents: InvolvingGraph . . . . . . . . . . . . . . . . . . . 468
8.7.5 TwoComponents: InvolvingMathematics . . . . . . . . . . . . . . . 468
8.7.6 TwoComponents: GraphPreprocessingandDP . . . . . . . . . . . . 469
8.7.7 TwoComponents: Involving1DStaticRSQ/RMQ . . . . . . . . . . 470
8.7.8 Three(orMore)Components . . . . . . . . . . . . . . . . . . . . . . 470
8.8 SolutiontoNon-StarredExercises . . . . . . . . . . . . . . . . . . . . . . . . 478
8.9 ChapterNotes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 480
v
CONTENTS c Steven,Felix,Suhendry
(cid:13)
9 Rare Topics 481
9.1 OverviewandMotivation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 481
9.2 SlidingWindow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 483
9.3 SparseTableDataStructure . . . . . . . . . . . . . . . . . . . . . . . . . . . 485
9.4 SquareRootDecomposition . . . . . . . . . . . . . . . . . . . . . . . . . . . 488
9.5 Heavy-LightDecomposition . . . . . . . . . . . . . . . . . . . . . . . . . . . 493
9.6 TowerofHanoi . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 496
9.7 MatrixChainMultiplication . . . . . . . . . . . . . . . . . . . . . . . . . . . 497
9.8 LowestCommonAncestor . . . . . . . . . . . . . . . . . . . . . . . . . . . . 499
9.9 TreeIsomorphism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 501
9.10 DeBruijnSequence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 505
9.11 FastFourierTransform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 508
9.12 Pollard’srhoAlgorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 528
9.13 ChineseRemainderTheorem. . . . . . . . . . . . . . . . . . . . . . . . . . . 530
9.14 Lucas’Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 534
9.15 RareFormulasorTheorems . . . . . . . . . . . . . . . . . . . . . . . . . . . 536
9.16 CombinatorialGameTheory . . . . . . . . . . . . . . . . . . . . . . . . . . . 538
9.17 GaussianEliminationAlgorithm. . . . . . . . . . . . . . . . . . . . . . . . . 543
9.18 ArtGalleryProblem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 546
9.19 ClosestPairProblem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 547
9.20 A*andIDA*: InformedSearch . . . . . . . . . . . . . . . . . . . . . . . . . 548
9.21 PancakeSorting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 551
9.22 EggDroppingPuzzle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 554
9.23 DynamicProgrammingOptimization . . . . . . . . . . . . . . . . . . . . . . 558
9.24 Push-RelabelAlgorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 566
9.25 MinCost(Max)Flow. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 571
9.26 Hopcroft-KarpAlgorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . 573
9.27 Kuhn-MunkresAlgorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . 574
9.28 Edmonds’MatchingAlgorithm . . . . . . . . . . . . . . . . . . . . . . . . . 577
9.29 ChinesePostmanProblem . . . . . . . . . . . . . . . . . . . . . . . . . . . . 580
9.30 ConstructiveProblem. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 582
9.31 InteractiveProblem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 585
9.32 LinearProgramming . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 586
9.33 GradientDescent . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 589
9.34 ChapterNotes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 590
Bibliography 593
vi
Authors’ Profiles
Steven Halim, PhD1
[email protected]
StevenHalimisaseniorlecturerinSchoolofComput-
ing,NationalUniversityofSingapore(SoC,NUS).He
teaches several programming courses in NUS, rang-
ingfrombasicprogrammingmethodology,intermedi-
atetoharddatastructuresandalgorithms,webpro-
gramming, and also the ‘Competitive Programming’
modulethatusesthisbook. Heisthecoachofboth
the NUS ICPC teams and the Singapore IOI team. He participated in several ICPC Re-
gionals as a student (Singapore 2001, Aizu 2003, Shanghai 2004). So far, he and other
trainers@NUShavesuccessfullygroomedvariousICPCteamsthatwontendifferentICPC
Regionals(seebelow),advancedtoICPCWorldFinalseleventimes(2009-2010;2012-2020)
with current best result of Joint-14th in ICPC World Finals Phuket 2016 (see below), as
wellassevengold,nineteensilver,andfifteenbronzeIOImedalists(2009-2019). Heisalso
the Regional Contest Director of ICPC Asia Singapore 2015+2018 and is the Deputy Di-
rector+InternationalCommitteememberfortheIOI2020+2021inSingapore. Hehasbeen
invited to give international workshops about ICPC/IOI at various countries, e.g., Bolivia
ICPC/IOIcampin2014,SaudiArabiaIOIcampin2019,CambodiaNOIcampin2020.
Steven is happily married to Grace Suryani Tioso and has two daughters and one son:
JaneAngelinaHalim,JoshuaBenHalim,andJemimahCharissaHalim.
ICPC Regionals # Year(s)
AsiaJakarta 5 2013(ThanQ),2014(ThanQ+),2015(RRwatameda),
2017(DomiNUS),2019(SendBobstoAlice)
AsiaManila 2 2017(Pandamiao),2019(7Halim)
AsiaNakhonPathom 1 2018(Pandamiao)
AsiaYangon 1 2018(3body2)
AsiaKualaLumpur 1 2019(3body3)
Table1: NUSICPCRegionalsWinsin2010s
ICPCWorldFinals Team Name Rank Year
Phuket,Thailand RRwatameda Joint-14/128 2016
Ekaterinburg,Russia ThanQ+ Joint-19/122 2014
RapidCity,USA TeamTam Joint-20/133 2017
Table2: NUSICPCWorldFinalsTop3Resultsin2010s
1PhDThesis: “AnIntegratedWhite+BlackBoxApproachforDesigningandTuningStochasticLocal
SearchAlgorithms”,2009.
vii
AUTHORS’PROFILES c Steven,Felix,Suhendry
(cid:13)
Felix Halim, PhD2
[email protected]
Felix Halim is a senior software engineer at Google.
While in Google, he worked on distributed system
problems,dataanalysis,indexing,internaltools,and
database related stuff. Felix has a passion for web
development. He created uHunt to help UVa on-
line judge users find the next problems to solve.
He also developed a crowdsourcing website, https:
//kawalpemilu.org, to let the Indonesian public to
overseeandactivelykeeptrackoftheIndonesiagen-
eralelectionin2014and2019.
Asacontestant,FelixparticipatedinIOI2002Ko-
rea(representingIndonesia),ICPCManila2003-2005,
Kaohsiung2006, andWorldFinalsTokyo2007(rep-
resenting Bina Nusantara University). He was also
one of Google India Code Jam 2005 and 2006 final-
ists. Asaproblemsetter,FelixsetproblemsforICPC
Jakarta2010,2012,2013,ICPCKualaLumpur2014,
andseveralIndonesiannationalcontests.
FelixishappilymarriedtoSiskaGozali. ThepictureontherightisoneoftheirEurope
honeymoontravelphotos(inSwitzerland)afterICPCWorldFinals@Porto2019. Formore
informationaboutFelix,visithiswebsiteathttps://felix-halim.net.
Suhendry Effendy, PhD3
suhendry.eff[email protected]
Suhendry Effendy is a research fellow in the School
of Computing of the National University of Singa-
pore (SoC, NUS). He obtained his bachelor degree
inComputerSciencefromBinaNusantaraUniversity
(BINUS),Jakarta, Indonesia, andhisPhDdegreein
ComputerSciencefromNationalUniversityofSinga-
pore, Singapore. BeforecompletinghisPhD,hewas
a lecturer in BINUS specializing in algorithm anal-
ysis and served as the coach for BINUS competitive
programmingteam(nicknamedas“Jollybee”).
Suhendry is a recurring problem setter for the
ICPCAsiaJakartasincetheveryfirstin2008. From
2010 to 2016, he served as the chief judge for the
ICPC Asia Jakarta collaborating with many other
problemsetters. Healsosetproblemsinmanyother
contests, such as the ICPC Asia Kuala Lumpur, the
ICPCAsiaSingapore,andOlimpiadeSainsNasional
bidangKomputer(IndonesiaNationalScienceOlympiadinInformatic)tonamebutafew.
2PhDThesis: “SolvingBigDataProblems: fromSequencestoTablesandGraphs”,2012.
3PhDThesis: “GraphPropertiesandAlgorithmsinSocialNetworks: Privacy, SybilAttacks, andthe
ComputerScienceCommunity”,2017.
viii
Chapter 5
Mathematics
Weallusematheveryday;topredictweather,totelltime,tohandlemoney.
Mathismorethanformulasorequations;it’slogic,it’srationality,
it’susingyourmindtosolvethebiggestmysteriesweknow.
— TV show NUMB3RS
5.1 Overview and Motivation
Theappearanceofmathematics-relatedproblemsinprogrammingcontestsisnotsurprising
sinceComputerScienceisdeeplyrootedinMathematics. Manyinterestingreallifeproblems
canbemodeledasmathematicalproblemsasyouwillfrequentlyseeinthischapter.
RecentICPCproblemsets(basedonourexperienceinAsianRegionals)usuallycontain
one or two mathematical problems. Recent IOIs usually do not contain pure mathematics
tasks, but many tasks do require mathematical insights. This chapter aims to prepare
contestantsindealingwithmanyofthesemathematicalproblems.
We are aware that different countries place different emphases in mathematics training
in pre-University education. Thus, some contestants are familiar with the mathematical
terms listed in Table 5.1. But for others, these mathematical terms do not ring a bell,
perhapsbecausethecontestanthasnotlearntitbefore,orperhapsthetermisdifferentin
the contestant’s native language. In this chapter, we want to make a more level-playing
field for the readers by listing as many common mathematical terminologies, definitions,
problems,andalgorithmsthatfrequentlyappearinprogrammingcontestsaspossible.
ArithmeticProgression GeometricProgression Polynomial
Algebra Logarithm/Power BigInteger
NumberTheory PrimeNumber SieveofEratosthenes
Miller-Rabin GreatestCommonDivisor LowestCommonMultiple
Factorial EulerPhi ModifiedSieve
ExtendedEuclidean LinearDiophantine ModularInverse
Combinatorics Fibonacci GoldenRatio
Binet’sFormula Zeckendorf’sTheorem PisanoPeriod
BinomialCoefficients Fermat’slittletheorem Lucas’Theorem
CatalanNumbers Inclusion-Exclusion ProbabilityTheory
Cycle-Finding GameTheory Zero-SumGame
DecisionTree PerfectPlay Minimax
NimGame Sprague-GrundyTheorem MatrixPower
Table5.1: Listofsome mathematicaltermsdiscussedinthischapter
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5.2. ADHOCMATHEMATICALPROBLEMS c Steven,Felix,Suhendry
(cid:13)
5.2 Ad Hoc Mathematical Problems
We start this chapter with something light: the Ad Hoc mathematical problems. These
areprogrammingcontestproblemsthatrequirenomorethanbasicprogrammingskillsand
some fundamental mathematics. As there are still too many problems in this category, we
furtherdividethemintosub-categories,asshownbelow. Theseproblemsarenotplacedin
Book1astheyareAdHocproblemswith(heavier)mathematicalflavor. Butrememberthat
manyoftheseAdHocmathematicalproblemsaretheeasierones. Todowellintheactual
programmingcontests,contestantsmustalsomastertheothersections ofthischapter.
Finding(Simple)FormulaorPattern
•
These problems require the problem solver to read the problem description carefully
togetasimplifiedformulaortospotthepattern. Attackingthemdirectlywillusually
result in a TLE verdict. The actual solutions are usually short and do not require
loopsorrecursions. Example: LetsetS beaninfinitesetofsquareintegers: 1,4,9,
{
16, 25, ... . Given an integer X (1 X 1018), count how many integers in S are
} ≤ ≤
lessthanX. Theanswerissimply: √X 1 . ThisisanO(1)solution.
b − c
Note that in Section 5.4, we will discuss Combinatorics problems that will also end
up with some (not necessarily simple) formula. We also have Section 9.15 where we
discussafewknownbutveryraremathematicalformulas.
BaseNumberConversionorVariants
•
Thesearethemathematicalproblemsinvolvingbasenumbers. Themostfrequenttype
involvesthestandard conversionproblemsthatcanbeeasilysolvedmanuallyorwith
C/C++/Python/OCaml(limited)orJavaInteger/BigInteger(mostgeneric)library.
Forexample,toconvert132inbase8(octal)intobase2(binary),wecanusebase10
(decimal)astheintermediatestep: (132) is1 82+3 81+2 80=64+24+2=(90)
8 × × × 10
and (90) is 90 45(0) 22(1) 11(0) 5(1) 2(1) 1(0) 0(1) =
10 → → → → → → →
(1011010) (thatis,divideby2until0,thenreadtheremaindersfrombackwards).
2
However,wecanalsousebuilt-inlibraries:
– C/C++:
int v; scanf("%o", &v); // read v in octal
bitset<32> bin(v); // use bitset
printf("%s\n", bin.to_string().c_str()); // print in binary
– Python:
print("{0:b}".format(int(str(input()), 8))) # octal to binary
– OCaml:
Printf.sprintf "%X" (int_of_string "0o374");; # octal to hexa
– Java:
If we know Java Integer/BigInteger class, we can actually construct an instance
ofInteger/BigIntegerclassinanybase(radix)anduseitstoString(int radix)
methodtoprintthevalueofthatinstanceinanybase(radix). Thisisamuchmore
flexiblelibrarysolutionthanC/C++orPythonsolutionsearlierthatarelimited
to popular bases = 2/8/10/16. See an example below for Kattis - basicremains
(also available at UVa 10551 - Basic Remains). Given a base b and two non-
negativeintegerspandm—bothinbaseb,computep%mandprinttheresult
asabasebinteger. Thesolutionisasfollows:
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