Algorithms for Interviews

222 Pages·2016·22.04 MB·English

by  Aziz and Prakash

A l g o r i t h m s
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Adnan Aziz
Amit Prakash
Algorithms
For
Interviews
AProblem SolvingApproach
AdrianAziz
AmitPrakash
algorithmsforinterviews.com
AdnanAzizisaprofessorattheDepartmentofElectricalandComputer
EngineeringatTheUniversityofTexas atAustin,whereheconductsre
search and teaches classes inapplied algorithms. Hereceived hisPhD
fromTheUniversityofCaliforniaatBerkeley;hisundergraduatedegree
isfromIITKanpur. HehasworkedatGoogle,Qualcomm,IBM,andsev
eralsoftwarestartups. Whennotdesigningalgorithms,heplayswithhis
children,Laila,Imran,and Omar.
Amit Prakash is aMember of the Technical Staff atGoogle, where he
works primarilyonmachine learningproblems thatarise inthecontext
ofonline advertising. Priortothatheworked atMicrosoft in theweb
searchteam. HereceivedhisPhDfromTheUniversityofTexasatAustin;
hisundergraduatedegreeisfromIITKanpur. Whenheisnotimproving
thequalityofads,heindulgesinhispassionsforpuzzles,movies,travel,
andadventureswithhiswife.
All rights reserved. No part of this publication may be reproduced,
storedinaretrievalsystem,ortransmitted,inanyform,orbyanymeans^
electronic, mechanical, photocopying, recording, orotherwise, without
the priorconsentoftheauthors.
This book was typeset by the authors using Lesley Lamport's
ETEXdocument preparation systemand Peter Wilson's Memoir class.
Thecoverdesignwasdone usingInkscape. MacOSaiXwasusedtocre
ate the front cover image; itapproximates Shela Nye's portrait ofAlan
Turing using acollection ofpublic domain images offamous computer
scientists andmathematicians. The graphic onthebackcover wascre
atedby NidhiRohatgi.
Thecompanionwebsiteforthebookincludesalistofknownerrorsfor
each version of the book. If you come across atechnical error, please
writeto us and we willcheerfully sendyou $0.42. Please referto the
websitefor details.
Version1.0.0(September1,2010)
Website:http://algorithmsforinterviews.com
ISBN: 1453792996
EAN-13: 9781453792995
To myfather, IshratAziz,forgiving me my lifelong love of
learning
Adrian Aziz
To my parents,Manju ShreeandArun Prakash, the most loving
parents Ican imagine
AmitPrakash
Table of Contents
Prologue •1
Problem Solving Techniques •5
I Problems 13
1 Searching•14
2 Sorting •23
3 Meta-algorithms •29
4 Algorithms on Graphs•41
5 Algorithmson Strings •52
6 Intractability- 56
7 Parallel Computing-62
8 Design Problems •67
9 Discrete Mathematics •73
10 Probability •80
11 Programming- 88
II The Interview 99
12 Strategies For A Great Interview•100
13 Conducting An Interview •105
VI TABLEOFCONTENTS
III Solutions 109
1 Searching•110
2 Sorting. 123
3 Meta-algorithms-130
4 Algorithms on Graphs•144
5 Algorithms onStrings•156
6 Intractability-160
7 Parallel Computing•167
8 Design Problems •174
9 Discrete Mathematics •186
10 Probability-194
11 Programming•206
Index of Problems•212
Prologue
Let'sbeginwith the picture onthe front cover. Youmayhave observed
thattheportraitofAlanTuringisconstructedfrom anumberofpictures
("tiles") ofgreatcomputerscientistsandmathematicians.
Suppose you were asked in an interview to design aprogram that
takesanimageandacollectionofsxs-sizedtilesandproduceamosaic
from thetiles thatresembles theimage. Agoodwaytobeginmaybeto
partition the image into 5xs-sized squares, compute the average color
of each such image square, and then find the tile that isclosest toitin
the color space. Here distance in color spacecanbe L2-norm over Red-
Green-Blue (RGB)intensitiesfor thecolor. Asyoulookmorecarefullyat
the problem, you might conclude that itwouldbebetter to match each
tile with an image square that has asimilar structure. One way could
betoperformacoarsepixelization (2 x2or3x3) ofeachimagesquare
andfindingthetilethatis"closest"totheimagesquareunderadistance
X £AN WRITE I CAN SORT I INVENTED
rtot-ttPucvnoN coj>ethatsays A N6W MOfcfcL
CAN BE POKE *H£UOUlORLt>M OFCOtnPUTAT»0N
AS A SERIES OP
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TO MY AU»OS
TOUCH £VERY
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Figure1.Evolutionofacomputerscientist
function defined over all pixel colors (for example, L2-norm over RGB
values for each pixel). Depending on how you represent the tiles, you
endupwiththeproblemoffinding theclosestpointfrom asetofpoints
in a/c-dimensionalspace.
Ifthere are mtiles and the image is partitioned into nsquares, then
abrute-forceapproachwouldhave 0(rri'n) timecomplexity. Youcould
improve on this by first indexing the tiles using an appropriate search
tree. Amore detailed discussion onthis approach ispresented inProb
lem8.1 anditssolution.
Ifina45-60minuteinterview,youcanworkthroughtheaboveideas,
write some pseudocode for your algorithm, and analyze its complex
ity, youwouldhavehad afairly successfulinterview. Inparticular, you
would have demonstrated to yourinterviewer thatyoupossess several
keyskills:
- Theabilitytorigorouslyformulate real-worldproblems.
- Theskillstosolveproblemsanddesignalgorithms.
- Thetools togofromanalgorithmtoaworkingprogram.
- Theanalyticaltechniquesrequiredtodeterminethecomputational
complexityofyoursolution.
BookOverview
Algorithmsfor Interviews (AFI) aims tohelp engineersinterviewingfor
software developmentpositions. Theprimaryfocus ofAFIis algorithm
design. Theentirebookispresentedthroughproblemsinterspersedwith
discussions. The problems cover key concepts and are well-motivated,
challenging,and fun tosolve.
We do not emphasize platforms and programming languages since
they differ across jobs, and can be acquired fairly easily. Interviews at
mostlargesoftwarecompaniesfocus moreonalgorithms,problemsolv
ing, and design skills than on specific domain knowledge. Also, plat
forms andprogramminglanguages canchangequicklyas requirements
changebutthequalitiesmentionedabovewillalwaysbefundamentalto
anysuccessfulsoftwareendeavor.
The questions we present should all be solvable within a one hour
interview and inmany cases, take substantially less time. Aquestion
maytake more orless time tocomplete, depending ontheamount of
codingthatis askedfor.
Oursolutionsvaryintermsofdetail—forsomeproblemswepresent
detailed implementations in Java/C++/Python; for others, we simply
sketch solutions. Some use fairly technical machinery, e.g., max-flow,
randomized analysis, etc. Youwillencountersuchproblems onlyifyou
PROLOGUE 3
claimspecializedknowledge, e.g., graphalgorithms, complexitytheory,
etc.
Interviewing is about more than being able to design algorithms
quickly. You alsoneed to knowhow to presentyourself, how to askfor
helpwhenyouare stuck,howto comeacross asbeingexcitedaboutthe
company, and knowingwhatyoucando for them. We discuss thenon
technical aspects of interviewing inChapter 12. You can practice with
friends orbyyourself; ineithercase,besuretotimeyourself. Interview
atas many places as you can without ittaking away from your job or
classes. The experience will help you and you may discover you like
companiesthatyoudidnotknowmuchabout.
Although an interviewer may occasionally ask a question directly
from AFI, you should not base your preparation on memorizing solu
tions from AFI. We sincerelyhope thatreading thisbookwillbeenjoy
able and improveyour algorithmdesignskills. Theend goalis to make
youabetterengineeraswellasbetterpreparedfor softwareinterviews.
Level and Prerequisites
MostofAFIrequiresitsreaderstohavebasicfamiliaritywithalgorithms
taught in atypical undergraduate-level algorithms class. The chapters
on meta-algorithms, graphs, and intractability use more advanced ma
chineryandmayrequireadditionalreview.
Eachchapterbeginswithareviewofkeyconcepts. Thisreviewisnot
meanttobecomprehensiveandifyouarenotfamiliarwiththematerial,
you should first study the correspondingchapterinan algorithms text
book. Thereare dozens ofsuchtexts andourpreferenceisto masterone
ortwo goodbooksratherthansuperficiallysamplemany. We likeAlgo
rithms byDasgupta, Papadimitriou, and Vazirani because itissuccinct
andbeautifullywritten;IntroductiontoAlgorithmsbyCormen,Leiserson,
Rivest,andSteinismoredetailedandservesasagoodreference.
Sinceourfocus isonproblemsthatcanbesolvedinaninterviewrel
atively completely, there are many elegant algorithm design problems
whichwedonotinclude. Similarly,wedonothaveanystraightforward
review-type problems; you may want tobrushuponthese using intro
ductoryprogrammingand data-structurestexts.
Thefield ofalgorithmsisvastandtherearemanyspecializedtopics,
such ascomputational geometry, numerical analysis, logic algorithms,
etc. Unlessyouclaimknowledgeofsuchtopics,itishighlyunlikelythat
you willbeasked aquestionwhichrequires esoteric knowledge. While
aninterviewproblemmayseemspecializedatfirstglance,itisinvariably
thecasethatthebasic algorithms described inthisbookaresufficientto
solveit.
Acknowledgments
The problems in this book come from diverse sources—our own expe
riences, colleagues, friends, papers,books, Internetbulletinboards, etc.
ToparaphrasePaulHalmosfromhiswonderfulbookProblemsforMath
ematicians, Youngand Old: "Idonotgivecredits—whodiscoveredwhat?
Who was first? Whose solutionis thebest? Itwould notbe fair to give
creditinsomecasesandnotinothers.Nooneknowswhodiscoveredthe
theoremthatbearsPythagoras'nameanditdoesnotmatter. Thebeauty
ofthe subjectspeaks foritselfand sobeit."
Onepersonwhosehelpandsupporthasimprovedthequalityofthis
book and made itfun to read is our cartoonist, editor, and proofreader,
Nidhi Rohatgi. Several of our friends and students gave feedback on
thisbook—wewould especiallylike tothankIanVarley, whowrote so
lutionstoseveralproblems,andSenthilChellappan,GayatriRamachan-
dran,andAlperSenforproofreadingseveralchapters.
Webothwanttothankallthepeoplewhohavebeenasourceofen
lightenmentandinspirationtousthroughtheyears.
I,AdnanAziz, wouldlike tothankteachers, friends, and students
from IIT Kanpur, UC Berkeley, and UT Austin. Iwould especially like
to thank my friends Vineet Gupta and Vigyan Singhal, and my teach
ersRobertSolovay, RobertBrayton,Richard Karp,Raimund Seidel,and
Somenath Biswas for introducingme to the joys of algorithms. My co
author,AmitPrakash,hasbeenawonderfulcollaborator—thisbookisa
testament tohisintellect,creativity,andenthusiasm.
I, Amit Prakash, have myco-author and mentor, AdnanAziz, to
thankthemostforthisbook. Toagreatextent,myproblemsolvingskills
have been shaped by Adnan. Therehave been occasionsin lifewhen
Iwould nothave made through without hishelp. Heisalso thebest
possiblecollaboratorIcanthinkofforanyintellectualendeavor.
Over theyears, Ihave been fortunate tohave great teachers at IIT
KanpurandUTAustin. IwouldespeciallyliketothankProfessorsScott
Nettles, VijayaRamachandran, and Gustavo de Veciana. I would also
like tothank myfriends andcolleagues atGoogle, Microsoft, andUT
Austin for all the stimulating conversations and problem solving ses
sions. Lastlyandmostimportantly,Iwanttothankmyfamilywhohave
beenaconstantsourceofsupport,excitement,andjoyfor allmylifeand
especiallyduringtheprocessofwritingthisbook.
AdnanAziz
adnanOalgorithmsforinterviews.com
AmitPrakash
amitOalgorithmsforinterviews.com