Elements of Programming Interviews in Python

109 Pages·2017·0.63 MB·English

Elements
of
Programming
Interviews in Python
The Insiders’ Guide
AdnanAziz
Tsung-HsienLee
AmitPrakash
Thisdocumentisasamplingofourbook,ElementsofProgramming
Interviews in Python (EPI). Its purpose is to provide examples of
EPI’sorganization,content,style,topics,andquality. Thesampler
focusessolelyonproblems;inparticular,itdoesnotincludethree
chaptersonthenontechnicalaspectsofinterviewing. We’dloveto
hearfrom you—we’reespecially interestedinyour suggestionsas
to where the exposition can be improved, as well as any insights
intointerviewingtrendsyoumayhave.
YoucanbuyEPIPythonatAmazon.com(paperback).
http://ElementsOfProgrammingInterviews.com
AdnanAzizisaResearchScientistatFacebook. Previously,hewasaprofessorattheDepartment
ofElectricalandComputerEngineeringatTheUniversityofTexasatAustin,whereheconducted
research and taught classes in applied algorithms. He received his Ph.D. from The University of
California at Berkeley; his undergraduate degree is from Indian Institutes of Technology Kanpur.
He has worked at Google, Qualcomm, IBM, and several software startups. When not designing
algorithms,heplayswithhischildren,Laila,Imran,andOmar.
Tsung-Hsien Lee is a Senior Software Engineer at Uber. Previously, he worked as a Software
Engineer at Google and as Software Engineer Intern at Facebook. He received both his M.S. and
undergraduate degrees from National Tsing Hua University. He has a passion for designing and
implementingalgorithms. Helikestoapplyalgorithmstoeveryaspectofhislife. Hetakesspecial
prideinhelpingtoorganizeGoogleCodeJam2014and2015.
AmitPrakashisaco-founderandCTOofThoughtSpot,aSiliconValleystartup. Previously,hewasa
MemberoftheTechnicalStaffatGoogle,whereheworkedprimarilyonmachinelearningproblems
thatariseinthecontextofonlineadvertising. BeforethatheworkedatMicrosoftinthewebsearch
team. HereceivedhisPh.D.fromTheUniversityofTexasatAustin; hisundergraduatedegreeis
fromIndianInstitutesofTechnologyKanpur. Whenheisnotimprovingbusinessintelligence,he
indulgesinhispassionforpuzzles,movies,travel,andadventureswithNidhiandAanya.
ElementsofProgrammingInterviews: TheInsiders’Guide
byAdnanAziz,Tsung-HsienLee,andAmitPrakash
Copyright©2017AdnanAziz,Tsung-HsienLee,andAmitPrakash. Allrightsreserved.
Nopartofthispublicationmaybereproduced,storedinaretrievalsystem,ortransmitted,inany
form,orbyanymeans,electronic,mechanical,photocopying,recording,orotherwise,withoutthe
priorconsentoftheauthors.
The views and opinions expressed in this work are those of the authors and do not necessarily
reflecttheofficialpolicyorpositionoftheiremployers.
WetypesetthisbookusingLATEXandtheMemoirclass. WeusedTikZtodrawfigures. AllanYtac
createdthecover,basedonadesignbriefweprovided.
The companion website for the book includes contact information and a list of known errors for
eachversionofthebook. Ifyoucomeacrossanerrororanimprovement,pleaseletusknow.
Website: http://elementsofprogramminginterviews.com
DistributedundertheAttribution-NonCommercial-NoDerivs3.0License
Table of Contents
I DataStructuresandAlgorithms 1
1 PrimitiveTypes 2
1.1 Computingtheparityofaword . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
2 Arrays 7
2.1 TheDutchnationalflagproblem . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
3 Strings 13
3.1 Interconvertstringsandintegers . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
3.2 Baseconversion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
4 LinkedLists 17
4.1 Testforcyclicity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
5 StacksandQueues 21
5.1 ImplementastackwithmaxAPI . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
5.2 Computebinarytreenodesinorderofincreasingdepth . . . . . . . . . . . . . . . 26
6 BinaryTrees 28
6.1 Testifabinarytreeisheight-balanced . . . . . . . . . . . . . . . . . . . . . . . . . 30
7 Heaps 33
7.1 Mergesortedfiles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
8 Searching 37
8.1 Searchasortedarrayforfirstoccurrenceofk . . . . . . . . . . . . . . . . . . . . . 39
9 HashTables 42
9.1 Isananonymousletterconstructible? . . . . . . . . . . . . . . . . . . . . . . . . . 46
10 Sorting 48
10.1 Computetheintersectionoftwosortedarrays . . . . . . . . . . . . . . . . . . . . 50
10.2 Renderacalendar . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
11 BinarySearchTrees 54
i
11.1 TestifabinarytreesatisfiestheBSTproperty . . . . . . . . . . . . . . . . . . . . . 56
12 Recursion 59
12.1 Generatethepowerset . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
13 DynamicProgramming 63
13.1 Countthenumberofwaystotraversea2Darray . . . . . . . . . . . . . . . . . . . 65
14 GreedyAlgorithmsandInvariants 69
14.1 The3-sumproblem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
15 Graphs 73
15.1 PaintaBooleanmatrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76
16 ParallelComputing 78
16.1 ImplementaTimerclass . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80
II DomainSpecificProblems 81
17 DesignProblems 82
17.1 Designasystemfordetectingcopyrightinfringement . . . . . . . . . . . . . . . . 83
18 LanguageQuestions 85
18.1 Closure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85
18.2 Shallowanddeepcopy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85
19 Object-OrientedDesign 87
19.1 CreationalPatterns . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87
20 CommonTools 89
20.1 Merginginaversioncontrolsystem . . . . . . . . . . . . . . . . . . . . . . . . . . 89
IIITheHonorsClass 93
21 HonorsClass 94
21.1 Computethegreatestcommondivisor . . . . . . . . . . . . . . . . . . . . . . . 95
21.2 Computethemaximumproductofallentriesbutone . . . . . . . . . . . . . . 96
IV NotationandIndex 99
Notation 100
IndexofTerms 102
Part I
Data Structures and Algorithms
hapter
C
1
Primitive Types
Representationistheessenceofprogramming.
—“TheMythicalManMonth,”
F.P.Brooks,1975
Aprogramupdatesvariablesinmemoryaccordingtoitsinstructions. Variablescomeintypes—a
typeisaclassificationofdatathatspellsoutpossiblevaluesforthattypeandtheoperationsthat
canbeperformedonit. Atypecanbeprovidedbythelanguageordefinedbytheprogrammer. In
Pythoneverythingisanobject—thisincludesBooleans,integers,characters,etc.
Primitivetypesbootcamp
Writingaprogramtocountthenumberofbitsthataresetto1inanintegerisagoodwaytoget
uptospeedwithprimitivetypes. Thefollowingprogramtestsbitsone-at-a-timestartingwiththe
least-significantbit. Itillustratesshiftingandmasking;italsoshowshowtoavoidhard-codingthe
sizeoftheintegerword.
def count_bits(x):
num_bits = 0
while x:
num_bits += x & 1
x >>= 1
return num_bits
SinceweperformO(1)computationperbit,thetimecomplexityisO(n),wherenisthenumberof
bitsneededtorepresenttheinteger,e.g.,4bitsareneededtorepresent12,whichis(1100) inbinary.
2
ThetechniquesinSolution1.1onthefacingpage,whichisconcernedwithcountingthenumberof
bitsmodulo2,i.e.,theparity,canbeusedtoimprovetheperformanceoftheprogramgivenabove.
Knowyourprimitivetypes
Python has a number of build-in types: numerics (e.g., integer), sequences (e.g., list), mappings
(e.g.,dict),aswellasclasses,instancesandexceptions. Allinstancesofthesetypesareobjects.
Integers in Python3 are unbounded—the maximum integer representable is a function of the
available memory. The constant sys.maxsize can be used to find the word-size; specifically, it’s
the maximum value integer that can be stored in the word, e.g., 2**63 - 1 on a 64-bit machine.
Boundsonfloatsarespecifiedinsys.float_info.
• Beveryfamiliarwiththebit-wiseoperatorssuchas6&4,1|2,8>>1,-16>>>2,1<<10,˜0,15ˆx.
Negative numbers are treated as their 2’s complement value. (There is no concept of an
unsignedshiftinPython,sinceintegershaveinfiniteprecision.)
2
Beverycomfortablewiththebitwiseoperators,particularlyXOR.
Understandhowtousemasksandcreatetheminanmachineindependentway.
Knowfastwaysto clearthelowermostsetbit (andbyextension, setthelowermost0, getits
index,etc.)
Understandsignednessanditsimplicationstoshifting.
Considerusingacachetoaccelerateoperationsbyusingittobrute-forcesmallinputs.
Be awarethat commutativity and associativity can beused toperform operations in parallel
andreorderoperations.
Table1.1:TopTipsforPrimitiveTypes
• The key methods for numeric types are abs(-34.5), math.ceil(2.17), math.floor(3.14),
min(x,-4), max(3.14, y), pow(2.71, 3.14) (alternately, 2.71 ** 3.14), and
math.sqrt(225).
• Knowhowtointerconvertintegersandstrings,e.g.,str(42), int(’42’),floatsandstrings,
e.g.,str(3.14), float(’3.14’).
• Unlike integers, floats are not infinite precision, and it’s convenient to refer to infinity as
float(’inf’)andfloat(’-inf’). Thesevaluesarecomparabletointegers,andcanbeused
tocreatepsuedomax-intandpseudomin-int.
• When comparing floating point values consider using math.isclose()—it is robust, e.g.,
whencomparingverylargevalues,andcanhandlebothabsoluteandrelativedifferences.
• The key methods in random are random.randrange(28), random.randint(8,16),
random.random(),random.shuffle(A),andrandom.choice(A).
1.1 Computingtheparityofaword
The parity of a binary word is 1 if the number of 1s in the word is odd; otherwise, it is 0. For
example, the parity of 1011 is 1, and the parity of 10001000 is 0. Parity checks are used to detect
singlebiterrorsindatastorageandcommunication. Itisfairlystraightforwardtowritecodethat
computestheparityofasingle64-bitword.
Howwouldyoucomputetheparityofaverylargenumberof64-bitwords?
Hint:Usealookuptable,butdon’tuse264entries!
Solution: Thebrute-forcealgorithmiterativelyteststhevalueofeachbitwhiletrackingthenumber
of 1sseen sofar. Since weonly care ifthe numberof 1s iseven orodd, wecan store thenumber
modulo2.
def parity(x):
result = 0
while x:
result ^= x & 1
3
x >>= 1
return result
ThetimecomplexityisO(n),wherenisthewordsize.
Wewillnowdescribeseveralalgorithmsforparitycomputationthataresuperiortothebrute-
forcealgorithm.
The first improvement is based on erasing the lowest set bit in a word in a single opera-
tion, thereby improving performance in the best- and average-cases. Specifically, the expres-
sion y = x&~(x − 1), where & is the bitwise-AND operator and ~ is the bitwise complement
operator. The variable y is 1 at exactly the lowest bit of x that is 1; all other bits in y are
0. For example, if x = (00101100) , then x − 1 = (00101011) , ~(x − 1) = (11010100) , and
2 2 2
y = (00101100) &(11010100) = (00000100) . Thiscalculationisrobust—itiscorrectforunsigned
2 2 2
andtwo’s-complementrepresentations. Consequently,thisbitmayberemovedfromxbycomput-
ingx⊕y,where⊕isthebitwise-XORfunction. ThetimecomplexityisO(s),wheresisthenumber
ofbitssetto1inx.
def parity(x):
result = 0
while x:
result ^= 1
x &= x - 1 # Drops the lowest set bit of x.
return result
Letkbethenumberofbitssetto1inaparticularword. (Forexample,for10001010,k = 3.) Then
timecomplexityofthealgorithmaboveisO(k).
Thefactthatx&~(x−1)isolatesthelowestbitthatis1inxisimportantenoughthatyoushould
memorizeit. However,itisalsofairlyeasytoderive. First,supposexisnot0,i.e.,ithashasabit
thatisone. Subtractingonefromxchangestherightmostbittozeroandsetsallthelowerbitsto
one(ifyouaddonenow,yougettheoriginalvalueback). Theeffectistomaskouttherightmost
one. Thereforex&~(x−1)hasasinglebitsettoone,namely,therightmost1inx. Nowsupposex
is0. Subtractingonefromxunderflows,resultinginawordinwhichallbitsaresettoone. Again,
x&~(x−1)is0.
Asimilarderivationshowsthatx&(x−1)replacesthelowestbitthatis1with0. Forexample,
ifx=(00101100) ,thenx−1=(00101011) ,sox&(x−1)=(00101100) &(00101011) =(00101000) .
2 2 2 2 2
Thisfactcanalsobeveryuseful.
Wenowconsideraqualitativelydifferentapproach. Theproblemstatementreferstocomputing
theparityforaverylargenumberofwords. Whenyouhavetoperformalargenumberofparity
computations,and,moregenerally,anykindofbitfiddlingcomputations,twokeystoperformance
areprocessingmultiplebitsatatimeandcachingresultsinanarray-basedlookuptable.
Firstwedemonstratecaching. Clearly,wecannotcachetheparityofevery64-bitinteger—we
would need 264 bits of storage, which is of the order of ten trillion exabytes. However, when
computing the parity of a collection of bits, it does not matter how we group those bits, i.e., the
computationisassociative. Therefore,wecancomputetheparityofa64-bitintegerbygroupingits
bitsintofournonoverlapping16bitsubwords, computingtheparityofeachsubwords, andthen
computingtheparity ofthesefour subresults. We choose16since216 = 65536is relativelysmall,
4
whichmakesitfeasibletocachetheparityofall16-bitwordsusinganarray. Furthermore,since16
evenlydivides64,thecodeissimplerthanifwewere,forexample,touse10bitsubwords.
We illustrate the approach with a lookup table for 2-bit words. The cache is (cid:104)0,1,1,0(cid:105)—these
aretheparitiesof(00),(01),(10),(11), respectively. Tocomputetheparityof(11001010)wewould
computetheparitiesof(11),(00),(10),(10). Bytablelookupweseetheseare0,0,1,1,respectively,
sothefinalresultistheparityof0,0,1,1whichis0.
Tolookuptheparityofthefirsttwobitsin(11101010),werightshiftby6,toget(00000011),and
usethisasanindexintothecache. Tolookuptheparityofthenexttwobits,i.e.,(10),werightshift
by4,toget(10)inthetwoleast-significantbitplaces. Therightshiftdoesnotremovetheleading
(11)—it results in (00001110). We cannot index the cache with this, it leads to an out-of-bounds
access. Togetthelasttwobitsaftertherightshiftby4,webitwise-AND(00001110)with(00000011)
(this is the “mask” used to extract the last 2 bits). The result is (00000010). Similar masking is
neededforthetwoother2-bitlookups.
def parity(x):
MASK_SIZE = 16
BIT_MASK = 0xFFFF
return (PRECOMPUTED_PARITY[x >> (3 * MASK_SIZE)] ^
PRECOMPUTED_PARITY[(x >> (2 * MASK_SIZE)) & BIT_MASK] ^
PRECOMPUTED_PARITY[(x >> MASK_SIZE) & BIT_MASK] ^
PRECOMPUTED_PARITY[x & BIT_MASK])
Thetimecomplexityisafunctionofthesizeofthekeysusedtoindexthelookuptable. LetLbe
the width of the words for which we cache the results, and n the word size. Since there are n/L
terms, the timecomplexity is O(n/L), assuming word-leveloperations, suchas shifting, take O(1)
time. (Thisdoesnotincludethetimeforinitializationofthelookuptable.)
WecanimproveontheO(n)worst-casetimecomplexityofthepreviousalgorithmsbyexploiting
somesimplepropertiesofparity. Specifically,theXORoftwobitsisdefinedtobe0ifbothbitsare
0 or both bits are 1; otherwise it is 1. XOR has the property of being associative, i.e., it does not
matterhowwegroupbits, aswellascommutative, i.e., theorderinwhichweperformtheXORs
doesnotchangetheresult. TheXORofagroupofbitsisitsparity. Wecanexploitthisfacttouse
theCPU’sword-levelXORinstructiontoprocessmultiplebitsatatime.
For example, the parity of (cid:104)b ,b ,...,b ,b ,b ,b (cid:105) equals the parity of the XOR of
63 62 3 2 1 0
(cid:104)b ,b ,...,b (cid:105)and(cid:104)b ,b ,...,b (cid:105). TheXORofthesetwo32-bitvaluescanbecomputedwitha
63 62 32 31 30 0
single shift and a single 32-bit XOR instruction. We repeat the same operation on 32-, 16-, 8-, 4-,
2-,and1-bitoperandstogetthefinalresult. Notethattheleadingbitsarenotmeaningful,andwe
havetoexplicitlyextracttheresultfromtheleast-significantbit.
Weillustratetheapproachwithan8-bitword. Theparityof(11010111)isthesameastheparity
of(1101)XORedwith(0111),i.e.,of(1010). Thisinturnisthesameastheparityof(10)XORedwith
(10), i.e., of (00). The final result is the XOR of (0) with (0), i.e., 0. Note that the first XOR yields
(11011010),andonlythelast4bitsarerelevantgoingforward. ThesecondXORyields(11101100),
andonlythelast2bitsarerelevant. ThethirdXORyields(10011010). Thelastbitistheresult,and
toextractitwehavetobitwise-ANDwith(00000001).
def parity(x):
x ^= x >> 32
x ^= x >> 16
5
x ^= x >> 8
x ^= x >> 4
x ^= x >> 2
x ^= x >> 1
return x & 0x1
ThetimecomplexityisO(logn),wherenisthewordsize.
Note that we can combine caching with word-level operations, e.g., by doing a lookup in the
XOR-basedapproachoncewegetto16bits.
Theactualruntimesdependontheinputdata,e.g.,therefinementofthebrute-forcealgorithm
is very fast on sparse inputs. However, for random inputs, the refinement of the brute-force is
roughly 20% faster than the brute-force algorithm. The table-based approach is four times faster
still,andusingassociativityreducesruntimebyanotherfactoroftwo.
Variant:Writeexpressionsthatusebitwiseoperators,equalitychecks,andBooleanoperatorstodo
thefollowinginO(1)time.
• Rightpropagatetherightmostsetbitinx,e.g.,turns(01010000) to(01011111) .
2 2
• Computexmoduloapoweroftwo,e.g.,returns13for77mod64.
• Testifxisapowerof2,i.e.,evaluatestotrueforx=1,2,4,8,...,falseforallothervalues.
6