EmilSekerinski,McMasterUniversity,WinterTerm16/17COMPSCI1MD3IntroductiontoProgramming

Liketuples,listsareorderedcollections.Elementsdon'thavetobeofthesametype.Elementscanoccurmultipletimes

Operation Notation

emptylist,listwithE,listwithE,F,… [],[E],[E, F],[E, F, G],…

listLindexedatI L[I]

suffixofLstartingatI L[I:]

prefixofLuptoI L[:I]

sliceofLstartingatIuptoJ L[I:J]

listLconcatenatedwithM L+M

lengthofL len(L)

EmemberofL,notmemberofL E in L,E not in L

LrepeatedItimes L*I,I*L

occurrencesofEinL L.count(E)

firstindexofEinL L.index(E)

Unliketuplesandstrings,listsaremutable>>> a = [1, 2, 3]; a[1] = 4; a>>> a = [1, 2, 3]; a[1:] = [5, 6]; a

Anumberoffunctionsupdatelistsinsitu Operation Notation

appendelementEtolistL L.append(E)

extendlistLwithlistM L.extend(M)

insertEatindexI L.insert(I, E)

deleteelementatindexI L.remove(I)

returnelementatindexIanddeleteit L.pop(I)

reverselistL L.reverse()

However,mutabilitymayleadtoaliasing,whichcanbeasourceoferrors>>> a = [1, 2, 3]; b = a; a; b>>> a = a + [4]; a; b

>>> a = [1, 2, 3]; b = a; a; b>>> a[1] = 5; a; b

a

b [1,2,3]

b

a

[1,2,3]

[1,2,3,4]

a

[1,2,3]b

a

[1,5,3]b

>>> a = [1, 2, 3]; b = [7, a, 9]; a; b>>> a.append(4); a; b

a [1,2,3]

b [7, ,9]

a [1,2,3,4]

b [7, ,9]

a = [1, 2, 3]

b = a

c = [4, 5, b]

a[0] = c[1]

d = c[2]

e = d[0]

a

[1,2,3]

a

b [1,2,3]

a

b [1,2,3][4,5, ]

c

a

b [5,2,3][4,5, ]

c

e == 5

a

b [5,2,3][4,5, ]

cd

Setcomprehensioninmathematics:{2i|i∈[0,10)}={1,2,4,8,16,32,64,128,256,512}

InPython,thegeneralformoflistcomprehensionis

[E for X in F if B]withvariableX,expressionsE,F,andBooleanexpressionB.Similartoforloops,Xtakesvaluesinthespecifiedorder>>> [2**i for i in range(10)]>>> [i for i in range(20) if i%2 == 0]>>> [i%10 for i in range(20)]>>> [i%3 == 0 for i in range(20)]

Allcandidateprimenumbersarewrittendown,startingfrom2,sayuptobutnotincludingn=20:2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19Allmultiplesof2areremoved,asthesearenotprimes:2 3 5 7 9 11 13 15 17 19Thenextunvisitednumberis3.Allmultiplesof3areremoved:2 3 5 7 11 13 17 19Thisprocesscontinueswithunvisitednumbers.Akeyobservationiswhenthiscanstop:thereisnopointinremovingmultiplesof5withfactorslessthan5,like2*5,3*5asthesehavebeeneliminatedalready;thereisnopointinremovingmultiplesof5withfactorsof5orgreater,asthesearegreaterthan20:onlymultipliesoffactorsupto√nneedtobeeliminated!

SifttheTwosandsifttheThrees,TheSieveofEratosthenes.Whenthemultiplessublime,ThenumbersthatremainarePrime.EratosthenesofCyrene,Greekmathematician,circa200bc

WeuseaBooleanlistoflengthntokeepprospectiveprimes.Whenmultiplesofiareeliminated,thefirstonetobeconsideredisi*i,asthoselesshavealreadybeeneliminated.Inrange(a, b, c),parametercspecifiesthestep:def eratosthenes1(n): a = n*[True] for i in range(2, binarySqrt(n)+1): if a[i]: for j in range(i*i, n, i): a[j] = False return [i for i in range(2, n) if a[i]]

>>> eratosthenes1(20)

Slicingandmanyotheroperatorsworkuniformlyonallsequencetypes:strings,tuples,listsIfIisleftout,0istaken;ifJisleftout,len(S)istaken;ifKisleftout,1istaken>>> 'AaBbCcDd'[1:4:2]>>> 'AaBbCcDd'[1::2]>>> 'AaBbCcDd'[:4:2]>>> 'AaBbCcDd'[::2]

Slicingalwaysreturnsanewlist,alsoforS[:]!

Operation Notation

sliceofSfromItoJstepK S[I:J:K]

¡  '0123456'[0:6:1]¡  '0123456'[0:6:2]¡  '0123456'[0:6:3]¡  '0123456'[0:6:4]¡  '0123456'[0:6:5]¡  '0123456'[0:6:6]

¡  '0123456'[0:5:2]¡  '0123456'[0:5:3]¡  '0123456'[0:5:4]¡  '0123456'[0:5:5]¡  '0123456'[0:5:6]

Ingeneral,S[I:J:K]has(J-I-1)//K+1elements

def eratosthenes1(n): a = n*[True] for i in range(2, binarySqrt(n)+1): if a[i]: for j in range(i*i, n, i): a[j] = False return [i for i in range(2, n) if a[i]]Replacingtheinnerforloopwithslicingdef eratosthenes2(n): a = n*[True] for i in range(2, binarySqrt(n)+1): if a[i]: a[i*i: n: i] = \

((n-i*i-1)//i+1)*[False] return [i for i in range(2, n) if a[i]]

add,discard,poparein-situoperations,modifyingtheset

Operation Notation

emptyset set()

setwithE,setwithE,F,… {E}, {E, F}, {E, F, G}

union(∪)ofSandT S|T

intersection(∩)ofSandT S&T

difference(–)ofSandT S-T

cardinalityofS len(S)

EelementofS,notelementofS E in S,E not in S

addEtoS S.add(E)

removeEfromS S.discard(E)

returnandremovearbitraryelement S.pop()

def uniqueElements1(l): s = set() for e in l: s = s|{e} return len(s)

>>> uniqueElements1('abracadabra')>>> uniqueElements1(['Ho', 'Ho', 'Ho'])

Howmuchfasterisin-situinsertion?def uniqueElements2(l): s = set() for e in l: s.add(e) return len(s)

ForlistsL,M,N: ForsetsS,T,U:L+[]=L=[]+L S|set()=S=set()|S (unit)(L+M)+N=L+(M+N) (S|T)|U=S|(T|U) (associativity)

S|T=T|S (commutativity)S|S=S (idempotence)

Tuplesarelikelists,butimmutable,thusnoaliasing,andfaster?def weekday(n): return ['Mon', 'Tue', … , 'Sat', 'Sun'][n]

def weekday(n): return ('Mon', 'Tue', … , 'Sat', 'Sun')[n]

¡  {1, 2} | {2, 3}¡  {1, 2} | {2, 1}¡  {1, 2} & {2, 3}¡  {1, 2} - {2, 3}¡  {2, 3} - {1, 2}¡  set() - {1, 2}¡  s = {2}; s.add(3).add(4); s¡  s = {2}; s.add(3).discard(2); s¡  s = {1, 2, 3}; s.pop()

Lists,strings,tuplesareindexedbyintegers.Dictionariescanbeindexedbyarbitrarytypes.>>> d = {'Jan': 1, 'Feb': 2, 'Mar': 3}

Operation Notation

emptydictionary {}

dictionarymappingkeyEtovalueF,… {E: F}, {E: F, G: H}

lookingupkeyEinD D[E]

keyEinD,keyEnotinD E in S,E not in S

deletingkeyEanditsvaluefromD del D[E]

sizeofD len(D)

Ahistogramdeterminesthe(absolute)frequencyofitemsinacollection.def histogram(s): d = {} for c in s: if c not in d: d[c] = 1 else: d[c] = d[c] + 1 return d

>>> histogram('DOORBELL')>>> histogram([85, 92, 83, 85, 92, 90])

Giventhedenominationsofacurrency,whatistheleastnumberofcoinsneededforagivenamount?Assumethedenominationsaregivenasalistinincreasingorder.def change(a, d): i, r, c = len(d), a, {} while i > 0: i = i - 1 c[d[i]], r = r // d[i], r % d[i] return c

>>> change(57, [1, 5, 10, 25, 100])Thisisanexampleofagreedyalgorithm:pickingthemostcoinsofthehighestdenominationmaynotbeoptimal.When?

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