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# how to generate pdf in mvc 4 using itextsharp : How to change pdf to powerpoint on Library software class asp.net winforms wpf ajax DataStructures9-part1828

72
Chapter2 AlgorithmAnalysis
(1) sum m = = 0;
for( i i = = 0; ; i i < n; ++i i )
++sum;
(2) sum m = = 0;
for( i i = = 0; ; i i < n; ++i i )
for( j = 0; j j < < n; ; ++j j )
++sum;
(3) sum m = = 0;
for( i i = = 0; ; i i < n; ++i i )
for( j = 0; j j < < n * n; ++j j )
++sum;
(4) sum m = = 0;
for( i i = = 0; ; i i < n; ++i i )
for( j = 0; j j < < i; ; ++j j )
++sum;
(5) sum m = = 0;
for( i i = = 0; ; i i < n; ++i i )
for( j = 0; j j < < i * i; ++j j )
for( k k = = 0; k < j; ++k k )
++sum;
(6) sum m = = 0;
for( i i = = 1; ; i i < n; ++i i )
for( j = 1; j j < < i * i; ++j j )
if( j j % i i == = 0 0 )
for( k = 0; k k < < j; ++k )
++sum;
2.8
SupposeyouneedtogeneratearandompermutationoftheﬁrstNintegers.For
example,{4,3,1,5,2}and{3,1,4,2,5}arelegalpermutations,but{5,4,1,2,
1}isnot,becauseonenumber(1)isduplicatedandanother(3)ismissing.This
routineisoftenusedinsimulationofalgorithms.Weassumetheexistenceofa
randomnumbergenerator,
r
,withmethod
randInt(i,j)
,thatgeneratesintegers
between
i
and
j
withequalprobability.Herearethreealgorithms:
1. Fillthearray
a
from
a[0]
to
a[N-1]
asfollows:Toﬁll
a[i]
,generaterandom
a[0]
,
a[1]
,...,
a[i-1]
.
2. Sameasalgorithm(1),butkeep p anextraarray called the
used
array.When
arandomnumber,
ran
,isﬁrstputinthearray
a
,set
used[ran] = true
.This
meansthatwhenﬁlling
a[i]
witharandomnumber,youcantestinonestep
i
stepsintheﬁrstalgorithm.
3. Fillthearraysuchthat
a[i] = = i+1
.Then
for( i i = 1; i < < n; ; ++i )
swap( a[ i i ], a[ [ randInt( ( 0, i i ) ) ] );
a. Provethatallthreealgorithmsgenerateonlylegalpermutationsandthatall
permutationsareequallylikely.
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Exercises
73
b.Giveasaccurate(Big-Oh)ananalysisasyoucanoftheexpectedrunningtimeof
eachalgorithm.
c. Write(separate)programstoexecuteeachalgorithm10times,togetagood
average.Runprogram(1)for= 250,500,1,000,2,000;program(2)for
= 25,000, , 50,000,100,000, 200,000, 400,000,800,000; and d program
(3) for = 100,000, , 200,000,400,000, 800,000, 1,600,000, 3,200,000,
6,400,000.
d.Compareyouranalysiswiththeactualrunningtimes.
e. Whatistheworst-caserunningtimeofeachalgorithm?
2.9
CompletethetableinFigure2.2withestimatesfortherunningtimesthatwere
toolongtosimulate.Interpolatetherunningtimesforthesealgorithmsandesti-
matethetimerequiredtocomputethemaximumsubsequencesumof1million
2.10 Determine,forthetypicalalgorithmsthatyouusetoperformcalculationsbyhand,
therunningtimetodothefollowing:
b.MultiplytwoN-digitintegers.
c. DividetwoN-digitintegers.
2.11 Analgorithmtakes0.5msforinputsize100.Howlongwillittakeforinputsize
500iftherunningtimeisthefollowing(assumelow-ordertermsarenegligible)?
a. linear
b.O(NlogN)
d.cubic
2.12 Analgorithmtakes0.5msforinputsize100.Howlargeaproblemcanbesolvedin
1miniftherunningtimeisthefollowing(assumelow-ordertermsarenegligible)?
a. linear
b.O(NlogN)
d.cubic
2.13 Howmuchtimeisrequiredtocomputef(x)=
N
i=0
a
i
x
i
:
a. Usingasimpleroutinetoperformexponentiation?
b.UsingtheroutineinSection2.4.4?
2.14 Consider the following algorithm(known as Horner’s rule) toevaluate f(x) =
N
i=0
a
i
x
i
:
poly = = 0;
for( i i = = n; i >= 0; --i )
poly = x * * poly y + a[i];
a. Showhowthestepsareperformedbythisalgorithmforx=3,f(x)=4x
4
+
8x
3
+x+2.
b.Explainwhythisalgorithmworks.
c. Whatistherunningtimeofthisalgorithm?
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74
Chapter2 AlgorithmAnalysis
2.15 GiveanefﬁcientalgorithmtodetermineifthereexistsanintegerisuchthatA
i
=i
inanarrayofintegersA
1
<A
2
<A
3
< ···< A
N
.Whatistherunningtimeof
youralgorithm?
2.16 Writeanalternative
gcd
algorithmbasedonthefollowingobservations(arrangeso
thata>b):
a. gcd(a,b)=2gcd(a/2,b/2)ifaandbarebotheven.
b.gcd(a,b)=gcd(a/2,b)ifaisevenandbisodd.
c. gcd(a,b)=gcd(a,b/2)ifaisoddandbiseven.
d.gcd(a,b)=gcd((a+b)/2,(ab)/2)ifaandbarebothodd.
2.17 Giveefﬁcientalgorithms(alongwithrunningtimeanalyses)to
a. Findtheminimumsubsequencesum.
b.Findtheminimumpositivesubsequencesum.
c. Findthemaximumsubsequenceproduct.
2.18 Animportantprobleminnumericalanalysisistoﬁndasolutiontotheequation
f(X)=0forsomearbitraryf.Ifthefunctioniscontinuousandhastwopointslow
andhighsuchthatf(low)andf(high)haveoppositesigns,thenarootmustexist
betweenlowandhighandcanbefoundbyabinarysearch.Writeafunctionthat
takesasparametersf,low,andhighandsolvesforazero.Whatmustyoudoto
ensuretermination?
2.19 Themaximumcontiguoussubsequencesumalgorithmsinthetextdonotgiveany
indicationoftheactualsequence.Modifythemsothattheyreturninasingleobject
thevalueofthemaximumsubsequenceandtheindicesoftheactualsequence.
2.20 a. Writeaprogramtodetermineifapositiveinteger,N,isprime.
b.IntermsofN,whatistheworst-caserunningtimeofyourprogram?(Youshould
beabletodothisinO(
N).)
c. LetBequalthenumberofbitsinthebinaryrepresentationofN.Whatisthe
valueofB?
d.IntermsofB,whatistheworst-caserunningtimeofyourprogram?
e. Compare e the running g times s to determine if a 20-bit number and d a a 40-bit
numberareprime.
f. IsitmorereasonabletogivetherunningtimeintermsofNorB?Why?
2.21 TheSieveofEratosthenesisamethodusedtocomputeallprimeslessthanN.We
beginbymakingatableofintegers2toN.Weﬁndthesmallestinteger,i,thatis
notcrossedout,printi,andcrossouti,2i,3i,....Wheni>
N,thealgorithm
terminates.Whatistherunningtimeofthisalgorithm?
2.22 ShowthatX
62
canbecomputedwithonlyeightmultiplications.
2.23 Writethefastexponentiationroutinewithoutrecursion.
2.24 Giveaprecisecountonthenumberofmultiplicationsusedbythefastexponenti-
ationroutine.(Hint:ConsiderthebinaryrepresentationofN.)
2.25 ProgramsAandBareanalyzedandfoundtohaveworst-caserunningtimesno
greaterthan150Nlog
2
NandN
2
possible:
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Exercises
75
a. Whichprogramhasthebetterguaranteeontherunningtimeforlargevaluesof
N(N>10,000)?
b.Whichprogramhasthebetterguaranteeontherunningtimeforsmallvaluesof
N(N<100)?
c. WhichprogramwillrunfasteronaverageforN=1,000?
d.Is itpossiblethat programwillrunfasterthanprogramAonallpossible
inputs?
2.26 Amajorityelementinanarray,A,ofsizeNisanelementthatappearsmorethan
N/2times(thus,thereisatmostone).Forexample,thearray
3,3,4,2,4,4,2,4,4
hasamajorityelement(4),whereasthearray
3,3,4,2,4,4,2,4
doesnot.Ifthereisnomajorityelement,yourprogramshouldindicatethis.Here
First,acandidatemajorityelementisfound(thisistheharderpart).Thiscandidateis
theonlyelementthatcouldpossiblybethemajorityelement.Thesecondstepdetermines
ifthiscandidateisactuallythemajority.Thisisjustasequentialsearchthroughthearray.
Toﬁndacandidateinthearray,A,formasecondarray,B.ThencompareA
1
andA
2
.
3
andA
4
.
candidateforA(why?).
a. Howdoestherecursionterminate?
b.HowisthecasewhereNisoddhandled?
c. Whatistherunningtimeofthealgorithm?
d.Howcanweavoidusinganextraarray,B?
e. Writeaprogramtocomputethemajorityelement.
ualrowisincreasingfromlefttoright.Eachindividualcolumnisincreasingfrom
toptobottom.GiveanO(N)worst-casealgorithmthatdecidesifanumberXisin
thematrix.
2.28 Designefﬁcientalgorithmsthattakeanarrayofpositivenumbers
a
,anddetermine:
a. themaximumvalueof
a[j]+a[i]
,with
j
i
.
b.themaximumvalueof
a[j]-a[i]
,with
j
i
.
c. themaximumvalueof
a[j]
*
a[i]
,with
j
i
.
d.themaximumvalueof
a[j]/a[i]
,with
j
i
.
2.29
Whyisitimportanttoassumethatintegersinourcomputermodelhaveaﬁxed
size?
2.30 Considerthewordpuzzleproblemonpage2.Supposeweﬁxthesizeofthelongest
wordtobe10characters.
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76
Chapter2 AlgorithmAnalysis
a. IntermsofRandC,whicharethenumberofrowsandcolumnsinthepuzzle,
and W,which is the number of words, , what arethe running g times ofthe
algorithmsdescribedinChapter1?
b.Supposethewordlistispresorted.Showhowtousebinarysearchtoobtainan
algorithmwithsigniﬁcantlybetterrunningtime.
2.31 Supposethatline 15in thebinary search routinehad thestatement
low = mid
low = = mid d + 1
.Wouldtheroutinestillwork?
2.32 Implementthebinarysearchsothatonlyonetwo-waycomparisonisperformedin
eachiteration.
2.33 Supposethatlines15and16inalgorithm3(Fig.2.7)arereplacedby
15
int maxLeftSum m = = maxSumRec( a, left, center - - 1 1 );
16
int maxRightSum m = maxSumRec( a, center, right );
Wouldtheroutinestillwork?
2.34 The inner r loop p of f the cubic maximum subsequence sum algorithm performs
N(N+1)/2iterations.ThelinearversionperformsNiterations.Whatpatternis
evident?Canyougiveacombinatoricexplanationofthisphenomenon?
References
partseries[5],[6],and[7].Analysisofthegcdalgorithmappearsin[6].Anotherearlytext
onthesubjectis[1].
Thereisstillnouniformagreementonthematter,especiallywhenitcomestousing().
usedinsomecornerstoexpressalowerbound,when()iscalledfor.
Themaximumsubsequencesumproblemisfrom[3].Theseriesofbooks[2],[3],and
[4]showhowtooptimizeprogramsforspeed.
1.A.V.Aho,J.E.Hopcroft,andJ.D.Ullman,TheDesignandAnalysisofComputerAlgorithms,
2.J.L.Bentley,WritingEfﬁcientPrograms,PrenticeHall,EnglewoodCliffs,N.J.,1982.
5.D. E. Knuth,TheArtofComputer Programming,Vol1:FundamentalAlgorithms,3ded.,
6.D.E.Knuth, TheArtofComputer Programming,Vol2:SeminumericalAlgorithms,3ded.,
8.D.E.Knuth,“BigOmicronandBigOmegaandBigTheta,”ACMSIGACTNews,8(1976),
18–23.
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C H A P P T E R
3
Lists,Stacks,andQueues
Thischapterdiscussesthreeofthemostsimpleandbasicdatastructures.Virtuallyevery
signiﬁcantprogramwilluseatleastoneofthesestructuresexplicitly,andastackisalways
implicitlyusedinaprogram,whetherornotyoudeclareone.Amongthehighlightsofthis
chapter,wewill...
Showhowtoefﬁcientlyperformoperationsonlists.
Inthischapter,weprovidecodethatimplementsasigniﬁcantsubsetoftwolibrary
classes:
vector
and
list
.
tionofhowthesetofoperationsisimplemented.Objectssuchaslists,sets,andgraphs,
datatypes.Integers,reals,andbooleanshaveoperationsassociatedwiththem,andsodo
Alternatively,wemightonlywantthetwooperationsunionandﬁnd,whichwoulddeﬁnea
TheC
++
classallowsfortheimplementation ofADTs,with appropriatehiding of
implementationdetails.Thus,any otherpartoftheprogramthatneeds toperforman
implementationdetailsneedtobechanged,itshouldbeeasytodosobymerelychanging
completelytransparenttotherestoftheprogram.
uptotheprogramdesigner.Thethreedatastructuresthatwewillstudyinthischapterare
77
78
Chapter3 Lists,Stacks,andQueues
iftheyaredonecorrectly,theprogramsthatusethemwillnotnecessarilyneedtoknow
whichimplementationwasused.
WewilldealwithagenerallistoftheformA
0
,A
1
,A
2
,...,A
N−1
.Wesaythatthesizeof
thislistisN.Wewillcallthespeciallistofsize0anemptylist.
Foranylistexcepttheemptylist,wesaythatA
i
follows(orsucceeds)A
i−1
(<N)
andthatA
i−1
precedesA
i
(i>0).TheﬁrstelementofthelistisA
0
,andthelastelement
isA
N−1
.WewillnotdeﬁnethepredecessorofA
0
orthesuccessorofA
N−1
.Theposition
ofelementA
i
inalistisi.Throughoutthisdiscussion,wewillassume,tosimplifymatters,
thattheelementsinthelistareintegers,butingeneral,arbitrarilycomplexelementsare
allowed(andeasilyhandledbyaclasstemplate).
Associatedwiththese“deﬁnitions”isasetofoperationsthatwewouldliketoperform
printList
and
makeEmpty
,whichdothe
obviousthings;
find
,whichreturnsthepositionoftheﬁrstoccurrenceofanitem;
insert
and
remove
,whichgenerallyinsertandremovesomeelementfromsomepositioninthe
list;and
findKth
,whichreturnstheelementinsomeposition(speciﬁedasanargument).
Ifthelistis34,12,52,16,12,then
find(52)
mightreturn2;
insert(x,2)
mightmakethe
listinto34,12,
x
,52,16,12(ifweinsertintothepositiongiven);and
remove(52)
might
turnthatlistinto34,12,
x
,16,12.
Ofcourse,theinterpretationofwhatisappropriateforafunctionisentirelyupto
the programmer,as is the handling of special cases (for example,what does
find(1)
next
and
previous
,whichwould
takeaposition asargument andreturn thepositionofthesuccessorand predecessor,
respectively.
3.2.1 SimpleArrayImplementationofLists
Alltheseinstructionscanbeimplementedjustbyusinganarray.Althougharraysarecre-
atedwithaﬁxedcapacity,the
vector
class,whichinternallystoresanarray,allowsthearray
togrowbydoublingitscapacitywhenneeded.Thissolvesthemostseriousproblemwith
usinganarray—namely,thathistorically,touseanarray,anestimateofthemaximumsize
ofthelistwasrequired.Thisestimateisnolongerneeded.
Anarrayimplementationallows
printList
tobecarriedoutinlineartime,andthe
findKth
operationtakesconstanttime,which isasgood as canbeexpected.However,
insertionanddeletionarepotentiallyexpensive,dependingonwheretheinsertionsand
deletionsoccur.Intheworstcase,insertingintoposition0(inotherwords,atthefront
ofthelist)requirespushingtheentirearraydownonespottomakeroom,anddeleting
theﬁrstelementrequiresshiftingalltheelementsinthelistuponespot,sotheworst
casefortheseoperationsisO(N).Onaverage,halfofthelistneedstobemovedforeither
operation,solineartimeisstillrequired.Ontheotherhand,ifalltheoperationsoccurat
takeO(1)time.
79
Therearemany situationswherethelist isbuiltup byinsertionsatthehigh end,
andthenonlyarrayaccesses(i.e.,
findKth
operations)occur.Insuchacase,thearrayis
asuitableimplementation.However,ifinsertionsanddeletionsoccurthroughoutthelist
and,inparticular,atthefrontofthelist,thenthearrayisnotagoodoption.Thenext
Inordertoavoidthelinearcostofinsertionanddeletion,weneedtoensurethatthelist
isnotstoredcontiguously,sinceotherwiseentirepartsofthelistwillneedtobemoved.
callthisthe
next
next
nullptr
.
Toexecute
printList()
or
find(x)
,wemerelystartattheﬁrstnodein thelistand
thentraversethelistbyfollowingthe
next
inthearrayimplementation;although,theconstantislikelytobelargerthanifanarray
implementation wereused.The
findKth
operation isnolongerquiteasefﬁcientasan
arrayimplementation;
findKth(i)
takesO(i)timeandworksbytraversingdownthelistin
theobviousmanner.Inpractice,thisboundispessimistic,becausefrequentlythecallsto
findKth
areinsortedorder(byi).Asanexample,
findKth(2)
,
findKth(3)
,
findKth(4)
,and
findKth(6)
canallbeexecutedinonescandownthelist.
The
remove
methodcanbeexecutedinone
next
pointerchange.Figure3.2showsthe
resultofdeletingthethirdelementintheoriginallist.
The
insert
methodrequiresobtaininganewnodefromthesystembyusinga
new
call
andthenexecutingtwo
next
pointermaneuvers.ThegeneralideaisshowninFigure3.3.
Thedashedlinerepresentstheoldpointer.
A
0
A
1
A
2
A
3
A
4
A
0
A
1
A
2
A
3
A
4
80
Chapter3 Lists,Stacks,andQueues
A
0
A
1
A
2
A
3
A
4
X
first
last
a
b
c
d
constant-time,aslong as we maintain alink tothe last node. . Thus, atypicallinked
nullptr
,andthenupdatethe
linkthat maintainsthe lastnode.In theclassic linked list,whereeach nodestoresa
next-to-lastnode.
3.3
vector
and
list
intheSTL
TheC
++
languageincludes,initslibrary,animplementationofcommondatastructures.
ThispartofthelanguageispopularlyknownastheStandardTemplateLibrary(STL).
othersin Chapters4and5.In general,thesedatastructuresarecalledcollectionsor
containers.