Algorithms - Computer Science and Engineering

374 Pages·2011·9.22 MB·English

Algorithms
Department of Computer Science
University of Illinois at Urbana-Champaign
Instructor: Jeff Erickson
Teaching Assistants:
• Spring 1999: MitchHarrisandShripadThite
• Summer 1999 (IMCS):MitchHarris
• Summer 2000 (IMCS):MitchHarris
• Fall 2000: ChrisNeihengen,EktaManaktala,andNickHurlburt
• Spring 2001: BrianEnsink,ChrisNeihengen,andNickHurlburt
• Summer 2001 (I2CS):AshaSeetharamandDanBullok
• Fall 2002: Erin Wolf, Gio Kao, Kevin Small, Michael Bond, Rishi Talreja, Rob McCann, and
YasutakaFurakawa
• Spring 2004: DanCranston,JohnathonFischer,KevinMilans,andLanChen
• Fall 2005: ErinChambers,IgorGammer,andAdityaRamani
• Fall 2006: DanCranston,NitishKorula,andKevinMilans
• Spring 2007: KevinMilans
• Fall 2008: RezaZamani-Nasab
• Spring 2009: AlinaEne,BenMoseley,andAmirNayyeri
• Spring 2010: DavidMorrison,KyleFox,andRachitAgarwal
• Fall 2010: AlinaEne
(cid:13)c Copyright1999–2011JeffErickson.LastupdateJuly8,2011.
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Forthemostrecenteditionofthiswork,seehttp://www.cs.illinois.edu/~jeffe/teaching/algorithms/.

ShallItellyou,myfriend,howyouwillcometounderstandit?
Goandwriteabookonit.
—HenryHome,LordKames(1696–1782),toSirGilbertElliot
Youknow,Icouldwriteabook.
Andthisbookwouldbethickenoughtostunanox.
—LaurieAnderson,“LetX=X”,BigScience(1982)
I’mwritingabook.I’vegotthepagenumbersdone,
sonowIjusthavetofillintherest.
—StephenWright
About These Notes
Thiscoursepacketincludeslecturenotes,homeworkquestions,andexamquestionsfromalgorithms
courses I taught at the University of Illinois at Urbana-Champaign in Spring 1999, Fall 2000, Spring
2001,Fall2002,Spring2004,Fall2005,Fall2006,Spring2007,Fall2008,Spring2009,Spring2010,
andFall2010. TheselecturenotesandmyvideotapedlectureswerealsoofferedoverthewebinSummer
1999, Summer 2000, Summer 2001, Fall 2002, and Fall 2005 as part of the UIUC computer science
department’sonlinemaster’sprogram. Lecturenoteswerepostedtothecoursewebsiteafewdays(on
average)aftereachlecture. Homeworks,exams,andsolutionswerealsodistributedovertheweb.
Most (but not all) of the exercises at the end of each lecture note have been used at least once in
ahomeworkassignment,discussionsection,orexam. Youcanalsofindanear-completecollectionof
homeworks and exams from past semesters of my class online at http://www.cs.illinois.edu/~jeffe/
teaching/algorithms/. Alargefractionoftheseexerciseswerecontributedbysomeamazingteaching
assistants:
AdityaRamani,AlinaEne,AmirNayyeri,AshaSeetharam,BenMoseley,BrianEnsink,Chris
Neihengen,DanBullok,DanCranston,DavidMorrison,JohnathonFischer,EktaManaktala,
Erin Wolf Chambers, Igor Gammer, Gio Kao, Kevin Milans, Kevin Small, Kyle Fox, Lan
Chen, Michael Bond, Mitch Harris, Nick Hurlburt, Nitish Korula, Rachit Agarwal, Reza
Zamani-Nasab,RishiTalreja,RobMcCann,ShripadThite,andYasuFurakawa.
(cid:63)
Stars indicate more challenging problems; many of these appeared on qualifying exams for the
(cid:198)
algorithmsPhDstudentsatUIUC.Asmallnumberofreally hardproblemsaremarkedwitha larger
star;oneortwoopenproblemsareindicatedby(cid:198)enormousstars.
Pleasedonotaskmeforsolutionstotheexercises. Ifyou’reastudent,seeingthesolutionwillrob
you of the experience of solving the problem yourself, which is the only way to learn the material. If
you’reaninstructor,youshouldn’tassignproblemsthatyoucan’tsolveyourself! (Idonotalwaysfollow
myownadvice;someoftheseproblemshaveseriousbugs.)
Acknowledgments
Thelecturenotesandexercisesdrawheavilyonthecreativity,wisdom,andexperienceofthousandsof
algorithmsstudents,teachers,andresearchers. Inparticular,Iamimmenselygratefultothealmost1400
Illinoisstudentswhohaveusedthesenotesasaprimaryreference,offereduseful(ifsometimespainful)
criticism,andsufferedthroughsometrulyawfulfirstdrafts. I’malsogratefulforthecontributionsand
feedbackfromteachingassistants,alllistedabove.
Naturally,thesenotesoweagreatdealtothepeoplewhotaughtmethisalgorithmsstuffinthefirst
place: BobBixbyandMichaelPerlmanatRice;DavidEppstein,DanHirshberg,andGeorgeLuekeratUC
Irvine;andAbhiramRanade,DickKarp,ManuelBlum,MikeLuby,andRaimundSeidelatUCBerkeley.
I’vealsobeenhelpedtremendouslybymanydiscussionswithfacultycolleaguesatUIUC—EdgarRamos,
HerbertEdelsbrunner,JasonZych,LennyPitt,MaheshViswanathan,MargaretFleck,Shang-HuaTeng,
Steve LaValle, and especially Chandra Chekuri, Ed Reingold, and Sariel Har-Peled. I stole the first
iterationoftheoverallcoursestructure,andtheideatowriteupmyownlecturenotes,fromHerbert
Edelsbrunner.
Finally,“Johnny’s”multi-coloredcrayonhomeworkwasfoundundertheTAofficedooramongthe
otherFall2000Homework1submissions. ThesquareKufirenditionofthename“al-Khwa¯rizm¯ı”onthe
backofthecoverpageismine.
Additional References
Istronglyencouragemystudents(andotherreaders)nottorestrictthemselvestoasingletextual
reference. Authorsandreadersbringtheirownperspectivestothematerial;noinstructor‘clicks’with
every student, or even every very strong student. Finding the author that most effectively gets their
intuitionintoyourheadtakesomeeffort,butthateffortpaysoffhandsomelyinthelongrun.
Thefollowingreferenceshavebeenparticularlyvaluablesourcesofinspiration,intuition,examples,
andproblems. Thislistisincomplete!
• Alfred V. Aho, John E. Hopcroft, and Jeffrey D. Ullman. The Design and Analysis of Computer
Algorithms. Addison-Wesley,1974. (IusedthistextbookasanundergradatRice,andagainasa
mastersstudentatUCIrvine.)
• ThomasCormen,CharlesLeiserson,RonRivest,andCliffStein. IntroductiontoAlgorithms,third
edition. MITPress/McGraw-Hill,2009. (Thesecondeditionwasmyrecommendedtextbookuntil
2005. IalsousedthefirsteditionasateachingassistantatBerkeley.)
• Sanjoy Dasgupta, Christos H. Papadimitriou, and Umesh V. Vazirani. Algorithms. McGraw-Hill,
2006. (Thisisthecurrentrecommendedtextbookformyundergraduateclasses.)
• JeffEdmonds. HowtoThinkaboutAlgorithms. CambridgeUniversityPress,2008.
• Michael R. Garey and David S. Johnson. Computers and Intractability: A Guide to the Theory of
NP-Completeness. W.H.Freeman,1979.
• MichaelT.GoodrichandRobertoTamassia. AlgorithmDesign: Foundations,Analysis,andInternet
Examples. JohnWiley&Sons,2002.
• Jon Kleinberg and Éva Tardos. Algorithm Design. Addison-Wesley, 2005. (This is the current
recommendedtextbookformygraduatealgorithmsclasses.)
• DonaldKnuth. TheArtofComputerProgramming,volumes1–3. Addison-Wesley,1997. (Myparents
gavemetheseforChristmaswhenIwas14. Ididn’tactuallyreadthemuntilmuchlater.)
• UdiManber. IntroductiontoAlgorithms: ACreativeApproach. Addison-Wesley,1989. (Iusedthis
textbookasateachingassistantatBerkeley.)
• RajeevMotwaniandPrabhakarRaghavan. RandomizedAlgorithms. CambridgeUniversityPress,
1995.
• IanParberry. ProblemsonAlgorithms. Prentice-Hall,1995. (Thisbookisoutofprint,butitcanbe
downloadedfor‘free’fromhttp://www.eng.unt.edu/ian/books/free/license.html.)
• AlexanderSchrijver. CombinatorialOptimization: PolyhedraandEfficiency. Springer,2003.
• RobertSedgewick. Algorithms. Addison-Wesley,1988. (Thisbookanditssequelshavebyfarthe
bestalgorithmillustrationsI’veseenanywhere.)
• RobertEndreTarjan. DataStructuresandNetworkAlgorithms. SIAM,1983.
• RobertJ.Vanderbei. LinearProgramming: FoundationsandExtensions. Springer,2001.
• ClassnotesfrommyownalgorithmsclassesatBerkeley,especiallythosetaughtbyDickKarpand
RaimundSeidel.
• Lecturenotes,slides,homeworks,exams,andvideolecturespostedbyinnumerablecolleagues
aroundtheworld.
• TheSourceofAllKnowledge(Google)andTheSourceofAllLies(Wikipedia).
Prerequisites
For the most part, these notes assume the reader has mastered the material covered in the first
twoyearsofastrongundergraduatecomputersciencecurriculum,andhastheintellectualmaturityto
recognizeandrepairanyremaininggapsintheirmastery. (Masteryisnotthesamethingas‘exposure’or
‘agoodgrade’;thisiswhyIstarteverysemesterwithHomeworkZero.) Specificprerequisitesinclude:
• Proof techniques: direct proof, indirect proof, proof by contradiction, combinatorial proof, and
induction(includingits“strong”,“structural”,and“recursive”forms). Lecture0requiresinduction,
andwheneverLecture n−1requiresinduction,sodoesLecture n.
• Discretemathematics: High-schoolalgebra,naivesettheory,Booleanalgebra,first-orderpredicate
logic,sets,functions,relations,modulararithmetic,recursivedefinitions,trees(asabstractobjects,
notdatastructures),graphs.
• Elementarydiscreteprobability: uniformvsnon-uniformprobabilitydistributions,expectation,
linearityofexpectation,independence.
• Iterativeprogrammingconcepts: variables,conditionals,iteration,subroutines,indirection(ad-
dresses/pointers/references),recursion. Programmingexperienceinanylanguagethatsupports
pointersandrecursionisaplus.
• Fundamentaldatastructures: arrays,linkedlists,binarysearchtrees,atleastonebalancedsearch
tree(suchasAVLtrees,red-blacktrees,B-trees,skiplists,splaytrees,ortreaps),binaryheaps.
• Fundamental abstract data types: dictionaries, stacks, queues, priority queues; the difference
betweenthislistandthepreviouslist.
• Fundamentalalgorithms: elementaryarithmetic,linearsearch,binarysearch,sorting(selection,
insertion,merge-,heap-,quick-,radix,anythingbutbubble-),pre-/post-/inordertreetraversal.
• Basic algorithm analysis: Asymptotic notation (o, O, Θ, Ω, ω), translating loops into sums and
recursivecallsintorecurrences,evaluatingsimplesumsandrecurrences.
• Mathematical maturity: facility with abstraction, formal (especially recursive) definitions, and
(especiallyinductive)proofs;followingmathematicalarguments;recognizingsyntactic,semantic,
and/orlogicalnonsense;writingtheformerratherthanthelatter.
Someofthisprerequisitematerialiscoveredbrieflyinthesenotes,butmoreasareminderthana
goodintroduction.
Caveat Lector!
Withfewexceptions,eachofthesenotescontainsfartoomuchmaterialtocoverinasinglelecture.
In a typical 75-minute lecture, I tend to cover 4 to 5 pages of material—a bit more if I’m lecturing to
graduate students than to undergraduates. Your mileage may vary! (Arguably, that means that as I
continuetoaddmaterial,thelabel“lecturenotes”becomeslessandlessaccurate.)
Despite several rounds of revision, these notes still contain lots of mistakes, errors, bugs, gaffes,
omissions, snafus, kludges, typos, mathos, grammaros, thinkos, brain farts, nonsense, garbage, cruft,
junk,andoutrightlies,all of which are entirely Steve Skiena’s fault. Ireviseandupdatethesenotes
everytimeIteachthecourse,sopleaseletmeknowifyoufindabug. (Steveisunlikelytocare.)
Whenever I teach the algorithms class, I award extra credit points to the first student to post an
explanation and correction of any error in the lecture notes to the course newsgroup. Obviously, the
number of extra credit points depends on the severity of the error and the quality of the correction.
If I’m not teaching the course, encourage your instructor to set up a similar extra-credit scheme, and
forwardthebugreportstoSteveme!
Ofcourse,anyotherfeedbackisalsowelcome!
Enjoy!
—Jeff
It is traditional for the author to magnanimously accept the blame for whatever
deficienciesremain. Idon’t. Anyerrors,deficiencies,orproblemsinthisbookare
somebodyelse’sfault,butIwouldappreciateknowingaboutthemsoastodetermine
whoistoblame.
—StevenS.Skiena,TheAlgorithmDesignManual(1997)
(cid:13)c Copyright2011JeffErickson. ReleasedunderaCreativeCommonsAttribution-NonCommercial-ShareAlike3.0License(http://creativecommons.org/licenses/by-nc-sa/3.0/).
Freedistributionisstronglyencouraged;commercialdistributionisexpresslyforbidden. Seehttp://www.cs.uiuc.edu/~jeffe/teaching/algorithms/forthemostrecentrevision.
Algorithms Lecture 0: Introduction [F10]
Weshouldexplain,beforeproceeding,thatitisnotourobjecttoconsiderthisprogramwith
referencetotheactualarrangementofthedataontheVariablesoftheengine,butsimply
asanabstractquestionofthenatureandnumberoftheoperationsrequiredtobeperfomed
duringitscompletesolution.
—AdaAugustaByronKing,CountessofLovelace,translator’snotesforLuigiF.Menabrea,
“SketchoftheAnalyticalEngineinventedbyCharlesBabbage,Esq.”(1843)
Youarerighttodemandthatanartistengagehisworkconsciously, butyouconfusetwo
differentthings:solvingtheproblemandcorrectlyposingthequestion.
—AntonChekhov,inalettertoA.S.Suvorin(October27,1888)
Themorewereduceourselvestomachinesinthelowerthings,
themoreforceweshallsetfreetouseinthehigher.
—AnnaC.Brackett,TheTechniqueofRest(1892)
Themomentamanbeginstotalkabouttechnique
that’sproofthatheisfreshoutofideas.
—RaymondChandler
0 Introduction
0.1 What is an algorithm?
An algorithm is an explicit, precise, unambiguous, mechanically-executable sequence of elementary
instructions. Forexample,hereisanalgorithmforsingingthatannoyingsong‘99BottlesofBeeronthe
Wall’,forarbitraryvaluesof99:
BOTTLESOFBEER(n):
Fori←ndownto1
Sing“i bottlesofbeeronthewall,i bottlesofbeer,”
Sing“Takeonedown,passitaround,i−1bottlesofbeeronthewall.”
Sing“Nobottlesofbeeronthewall,nobottlesofbeer,”
Sing“Gotothestore,buysomemore,nbottlesofbeeronthewall.”
Theword‘algorithm’doesnotderive,asalgorithmophobicclassicistsmightguess,fromtheGreek
root algos (ἄλγος), meaning ‘pain’. Rather, it is a corruption of the name of the 9th century Persian
mathematician Abu¯ ’Abd Alla¯h Muh.ammad ibn Mu¯sa¯ al-Khwa¯rizm¯ı.1 Al-Khwa¯rizm¯ı is perhaps best
knownasthewriterofthetreatiseAl-Kita¯bal-mukhtas.arf¯ıh¯ısa¯bal-g˘abrwa’l-muqa¯bala2,fromwhichthe
modernwordalgebraderives. Inanothertreatise,al-Khwa¯rizm¯ıpopularizedthemoderndecimalsystem
forwritingandmanipulatingnumbers—inparticular,theuseofasmallcircleor.sifrtorepresentamissing
quantity—which had originated in India several centuries earlier. This system later became known
in Europe as algorism. Thanks to the efforts of the medieval Italian mathematician Leonardo of Pisa,
betterknownasFibonacci,algorismbegantoreplacetheabacusasthepreferredsystemofcommercial
calculation3 inEuropeinthelate12thcentury,althoughcyphersbecametrulyubiquitousinWestern
Europe only after the French revolution 600 years later. The more modern word algorithm is a false
1‘Mohammad,fatherofAdbdulla,sonofMoses,theKwa¯rizmian’. Kwa¯rizmisanancientcity,nowcalledKhiva,inthe
KhorezmProvinceofUzbekistan.
2‘TheCompendiousBookonCalculationbyCompletionandBalancing’.
3fromtheLatinwordcalculus,meaningliterally‘smallrock’,referringtothestonesonacountingboard,orabacus
©Copyright2010JeffErickson. ReleasedunderaCreativeCommonsAttribution-NonCommercial-ShareAlike3.0License(http://creativecommons.org/licenses/by-nc-sa/3.0/).
Freedistributionisstronglyencouraged;commercialdistributionisexpresslyforbidden. Seehttp://www.cs.uiuc.edu/~jeffe/teaching/algorithms/forthemostrecentrevision.
1
Algorithms Lecture 0: Introduction [F10]
cognatewiththeGreekwordarithmos(ἀριθμός),meaning‘number’(andperhapstheaforementioned
άλγος). Thus,untilveryrecently,thewordalgorithmreferredexclusivelytopencil-and-papermethods
fornumericalcalculations. Peopletrainedinthereliableexecutionofthesemethodswerecalled—you
guessedit—computers.
0.2 A Few Simple Examples
Multiplication by compass and straightedge
Althoughtheyhaveonlybeenanobjectofformalstudyforafewdecades,algorithmshavebeenwithus
sincethedawnofcivilization,forcenturiesbeforeAl-Khwa¯rizm¯ıandFibonaccipopularizedthecypher.
Here is an algorithm, popularized (but almost certainly not discovered) by Euclid about 2500 years
ago,formultiplyingordividingnumbersusingarulerandcompass. TheGreekgeometersrepresented
numbersusinglinesegmentsoftheappropriatelength. Inthepseudo-codebelow,CIRCLE(p,q)represents
thecirclecenteredatapoint p andpassingthroughanotherpointq. Hopefullytheotherinstructions
areobvious.4
Z
〈〈Constructthelineperpendicularto(cid:96)andpassingthrough P.〉〉
RIGHTANGLE((cid:96),P):
ChooseapointA∈(cid:96)
A,B←INTERSECT(CIRCLE(P,A),(cid:96))
C,D←INTERSECT(CIRCLE(A,B),CIRCLE(B,A))
returnLINE(C,D)
〈〈Constructapoint Z suchthat|AZ|=|AC||AD|/|AB|.〉〉
C
MULTIPLYORDIVIDE(A,B,C,D):
α←RIGHTANGLE(LINE(A,C),A) D
E←INTERSECT(CIRCLE(A,B),α)
F ←INTERSECT(CIRCLE(A,D),α)
β ←RIGHTANGLE(LINE(E,C),F) B
γ←RIGHTANGLE(β,F)
returnINTERSECT(γ,LINE(A,C)) b(cid:13)
F a(cid:13)
A E
g(cid:13)
Multiplyingordividingusingacompassandstraightedge.
This algorithm breaks down the difficult task of multiplication into a series of simple primitive
operations: drawingalinebetweentwopoints,drawingacirclewithagivencenterandboundarypoint,
and so on. These primitive steps are quite non-trivial to execute on a modern digital computer, but
thisalgorithmwasn’tdesignedforadigitalcomputer;itwasdesignedforthePlatonicIdealClassical
Greek Mathematician, wielding the Platonic Ideal Compass and the Platonic Ideal Straightedge. In
thisexample,Euclidfirstdefinesanewprimitiveoperation,constructingarightangle,by(asmodern
programmerswouldputit)writingasubroutine.
4Euclidandhisstudentsalmostcertainlydrewtheirconstructionsonanἄβαξ,atablecoveredindustorsand(orperhaps
verysmallrocks).Overthenextseveralcenturies,theGreekabaxevolvedintothemedievalEuropeanabacus.
2
Algorithms Lecture 0: Introduction [F10]
Multiplication by duplation and mediation
Here is an even older algorithm for multiplying large numbers, sometimes called (Russian) peasant
multiplication. AvariantofthismethodwascopiedintotheRhindpapyrusbytheEgyptianscribeAhmes
around 1650 BC, from a document he claimed was (then) about 350 years old. This was the most
commonmethodofcalculationbyEuropeansbeforeFibonacci’sintroductionofArabicnumerals;itwas
stilltaughtinelementaryschoolsinEasternEuropeinthelate20thcentury. Thisalgorithmwasalso
commonly used by early digital computers that did not implement integer multiplication directly in
hardware.
x y prod
PEASANTMULTIPLY(x,y):
0
prod←0
123 +456 =456
while x >0
61 +912 =1368
if x isodd
30 1824
prod←prod+y
15 +3648 =5016
x ←(cid:98)x/2(cid:99)
7 +7296 =12312
y ← y+y
3 +14592 =26904
return p
1 +29184 =56088
The peasant multiplication algorithm breaks the difficult task of general multiplication into four
simpler operations: (1) determining parity (even or odd), (2) addition, (3) duplation (doubling a
number),and(4)mediation(halvinganumber,roundingdown).5 Ofcourseafullspecificationofthis
algorithmrequiresdescribinghowtoperformthosefour‘primitive’operations. Peasantmultiplication
requires(aconstantfactor!)morepaperworktoexecutebyhand,butthenecessaryoperationsareeasier
(forhumans)torememberthanthe10×10multiplicationtablerequiredbytheAmericangradeschool
algorithm.6
Thecorrectnessofpeasantmultiplicationfollowsfromthefollowingrecursiveidentity,whichholds
foranynon-negativeintegers x and y:

0 if x =0

x· y = (cid:98)x/2(cid:99)·(y+ y) if x iseven

(cid:98)x/2(cid:99)·(y+ y)+ y if x isodd
Congressional Apportionment
Hereisanothergoodexampleofanalgorithmthatcomesfromoutsidetheworldofcomputing. ArticleI,
Section2oftheUnitedStatesConstitutionrequiresthat
RepresentativesanddirectTaxesshallbeapportionedamongtheseveralStateswhichmaybeincluded
withinthisUnion,accordingtotheirrespectiveNumbers.... TheNumberofRepresentativesshall
notexceedoneforeverythirtyThousand,buteachStateshallhaveatLeastoneRepresentative....
SincetherearealimitednumberofseatsavailableintheHouseofRepresentatives,exactproportional
representation is impossible without either shared or fractional representatives, neither of which are
5TheversionofthisalgorithmactuallyusedinancientEgyptdoesnotusemediationorparity,butitdoesusecomparisons.
To avoid halving, the algorithm pre-computes two tables by repeated doubling: one containing all the powers of 2 not
exceeding x,theothercontainingthesamepowersof2multipliedby y. Thepowersof2thatsumto x arethenfoundby
greedysubtraction,andthecorrespondingentriesintheothertableareaddedtogethertoformtheproduct.
6AmericanschoolkidslearnavariantofthelatticemultiplicationalgorithmdevelopedbyIndianmathematiciansand
describedbyFibonacciinLiberAbaci.Thetwoalgorithmsareequivalentiftheinputnumbersarerepresentedinbinary.
3
Algorithms Lecture 0: Introduction [F10]
legal. Asaresult,severaldifferentapportionmentalgorithmshavebeenproposedandusedtoround
thefractionalsolution fairly. The algorithmactuallyusedtoday, calledtheHuntington-Hillmethodor
themethodofequalproportions,wasfirstsuggestedbyCensusBureaustatisticianJosephHillin1911,
refinedbyHarvardmathematicianEdwardHuntingtonin1920,adoptedintoFederallaw(2U.S.C.§§2a
and2b)in1941,andsurvivedaSupremeCourtchallengein1992.7 TheinputarrayPop[1..n]stores
the populations of the n states, and R is the total number of representatives. Currently, n = 50 and
R=435. TheoutputarrayRep[1..n]storesthenumberofrepresentativesassignedtoeachstate.
APPORTIONCONGRESS(Pop[1..n],R):
PQ←NEWPRIORITYQUEUE
fori←1ton
Rep[i]←1
(cid:112)
INSERT(cid:128)PQ,i,Pop[i]/ 2(cid:138)
R←R−1
whileR>0
s←EXTRACTMAX(PQ)
Rep[s]←Rep[s]+1
(cid:16) (cid:46)(cid:112) (cid:17)
INSERT PQ,s,Pop[s] Rep[s](Rep[s]+1)
R←R−1
returnRep[1..n]
Thispseudocodedescriptionassumesthatyouknowhowtoimplementapriorityqueuethatsupports
theoperationsNEWPRIORITYQUEUE,INSERT,andEXTRACTMAX. (Theactuallawdoesn’tassumethat,of
course.) The output of the algorithm, and therefore its correctness, does not depend at all on how
the priority queue is implemented. The Census Bureau uses an unsorted array, stored in a column of
an Excel spreadsheet; you should have learned a more efficient solution in your undergraduate data
structuresclass.
A bad example
Consider“Martin’salgorithm”:8
BECOMEAMILLIONAIREANDNEVERPAYTAXES:
Getamilliondollars.
Don’tpaytaxes.
Ifyougetcaught,
Say“Iforgot.”
Prettysimple,exceptforthatfirststep;it’sadoozy. AgroupofbillionaireCEOsmightconsiderthis
analgorithm,sinceforthemthefirststepisbothunambiguousandtrivial,butfortherestofuspoor
slobs,Martin’sprocedureistoovaguetobeconsideredanactualalgorithm. Ontheotherhand,thisisa
perfectexampleofareduction—itreducestheproblemofbeingamillionaireandneverpayingtaxesto
the‘easier’problemofacquiringamilliondollars. We’llseereductionsoverandoveragaininthisclass.
7Overrulinganearlierrulingbyafederaldistrictcourt,theSupremeCourtunanimouslyheldthatanyapportionment
method adopted in good faith by Congress is constitutional (United States Department of Commerce v. Montana). The
current congressional apportionment algorithm is described in gruesome detail at the U.S. Census Department web site
http://www.census.gov/population/www/censusdata/apportionment/computing.html. Agoodhistoryoftheapportionment
problemcanbefoundathttp://www.thirty-thousand.org/pages/Apportionment.htm. AreportbytheCongressionalResearch
Servicedescribingvariousapportionmentmethodsisavailableathttp://www.rules.house.gov/archives/RL31074.pdf.
8S.Martin,“YouCanBeAMillionaire”,SaturdayNightLive,January21,1978. AppearsonComedyIsNotPretty,Warner
Bros.Records,1979.
4