If you find the book helpful, please purchase a copy to support the authors! 11 Adnan Aziz is a professor at the Department of Electrical and Computer Engineering at The University of Texas at Austin , where he conducts re- search and teaches classes in applied algorithms. He received his PhD from The University of California at Berkeley; his undergraduate degree is from IIT Kanpur. He has worked at Google, Qua 1c omm , IBM, and sev- eral software startups. 叭Th en 且 ot designing algorithms, he plays withhis children, Laila, Imran , and Omar. Amit Prakash is a Member of the Technical Staff at Google, where he works primarily on machine learning problems that arise in the context of online advertising. Prior to that he worked at Microsoft in the web search te 缸 n. He received his PhD from TheUniversity ofTexas at Austin; his undergraduate degree is from IIT Kanpur. 叭Th en he is not improving the quality of ads , he indulges in his passions for puzzles , movies, travel, and adventures with his wife. All rights reserved. No part of this publicatio 丑 may be reproduced, stored in a retrieval system, or transmitted, in any form , or by any means , electronic, mechanical, photocopying, recording, or otherwise, without the prior consent of the authors. This book was typeset by the authors using Lesley L 缸 npor t' s 匹趴 document preparatio 丑 system and Peter Wilson's Memoir class. The cover design was done using In kscape. MacOSaiX was used to cre 伽 ate the front cover image; it approximates Shela Nye's portrait of Alan Turing using a collection of public domain images of famous computer scientists and mathematicians. 古 le graphic on the back cover was cre- ated by Nidhi Rohatgi. The companion website for the book includes a list of known errors for each version of the boo k. If you come across a technical error, please write to us and we will cheerfully send you $0.42. Please refer to the website for details. Ver 咀 on 1. 0.0 (September I, 2010) L 叩 ebsite:http://algorithmsforinterviews.com ISBN: 1453792996 EAN-13: 9781453792995 To my father! Ishrat Aziz! for giving me my l~ 作 long love of learning AdnanAziz To my parents! Manju Shree and Arun Prakash! the most loving parents I can imagine Amit Prakash If you find the book helpful, please purchase a copy to support the authors! Table of Contents Prologue·1 Problem Solving 丁 'echniques . 5 I Problems 13 1 Searching. 14 2 Sorting. 23 3 Meta 四 algorithms. 29 4 Algorithms on Graphs· 41 5 Algorithms on 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 If you find the book helpful, please purchase a copy to support the authors! Let's begin with the picture on the front cover. You may have observed that the portra 让 of Alan Turing is constructed from a number of pictures ("tiles") of great computer scientists and mathematicians. Suppose you were asked in an interview to design a program that takes an 垃 nage and a collection of s x s-sized tiles and produce a mosaic from the tiles that resembles the image. A good way to begin may be to partition the image into s x s-sized squares, compute the average color of each such image square, and then find the tile that is closest to it in the color space. Here distance in color space can be L2-norm over Red- Green-Blue (RGB) intensities for the color. As you look more carefully at the problem , you might conclude that it would be better to match each tile with an image square that has a similar structure. One way could be to perform a coarse pixelization (2 x 2 or 3 x 3) of each 挝 lage square and finding the tile that is "closest" to the image square under a distance V1 TABLE OF CONTENTS III Solutions 109 l Searching· 110 2 Sorting' 123 3 Meta-algorithms· 130 4 Algorithms on Graphs· 144 5 Algorithms on Strings· 156 6 Intractability· 160 7 Parallel Computing· 167 Prologue 8 Design Problems· 174 9 Discrete Mathematics· 186 10 Probability· 194 11 Programming· 206 Index of Problems· 212 工 ι~N 时R \T~ 于C.Pt N .s O R " 2::Tit13;?32L;t23 山附吨吵 77 γ au ARf / / SOMEW 且已民 E // 10 0\ I INYεNTE j) \坠主) A NEWMo):> E. L I) / o~ Cot 吨 Pυ 1' A ， ION / 五夺 (!?Cn MY A. t 喝 oS 一一 lit干气" iau c. H G. VE R.'f U /" P Pl CK E'T oN \~ THE INT€ 民 NeT / Figure 1. Evolution of a computer scientist If you find the book helpful, please purchase a copy to support the authors! 2 PROLOGUE 3 function defined over all pixel colors (for example , L2-norm over RGB alues for each pixel). Depending 0 日 how you represent the tiles, you eI1d up with the problem of findirlg the closest point koma set of pohts in a k-dimensional space. If there are m tiles aRd the image is partitiORed into nsquaresr then a brute-force approach would have O(m· η) time complexity. You could improve 0 口 this by first indexhg the tiles ushg aIIappropriate search tree.Amore detailed disωsion on this approach is presented in Prob- lem 8.1 and its solution. If h a E-60miRUte hterviewy you can work thTough the above ideasr write some pseudocode for your algorithm , and analyze its complex- iiF youwo 讪 d have had a fairly successful ir 阳忧 w. In 丑叼 1p 归 a 盯 M 削 r 时叫 ticu 吐 1 址阳 lar 艾盯: would have d 由 em 丑 10 ∞ I 丑 1st 位 ra 挝 ted tωo your in 口 1t 怡 erv 飞 vi 坦 ewe 盯 rt 也 ha 挝 t you possess several key skills: _ The ability to rigorously formulate real-world problems. 一Th e skills to solve problems and design algorithms. 一 The tools to go from an algorithm to a working program. 一 The malytical techMques required to determhe the computatioml complexity of your solution. Book Overview Alσorithms for In terviews (AFI) aims to help engineers interviewing for SOLar-e developnmtpmωns. The 严 in 叫 T foω 仙 FI is algorithm design. The entire book is prese 口 ted through problems interspersed with discussions. 白 1e problems cover key comepts md are well-motivatedr challenging, and fun to solve. We do not emphasize platforms and programmi 哆 languages since they differ across jobsy md cm be acquired fairly emly.II1terviews at st large software compmies focus more on algorithmsr problem solv- iLaJdesign skills than O 丑 specific domain knowledge. Also, pI 斗 fobs aM progmm1hgla 吨 mges cm chmge quickly as requirements chmge but the qualities mmtiORed above will always be hmdameI1tal to anv successful software endeavor. JTM questiom we pment should allbe solvable withh a om hour iew and in rna 叮 cases ， take s 由阳巾 lly less time. A question may take more or less time to completeF depmdhg OIIthe amOUIIt of oding that is asked for. 。品 soldomvaryhtems ofdetail m-for some pdlemswe prese 口 t detailed implementations in Java/C 十十 IPytho 刊 for othersr we siTPly sketch solutions. Some use fairly technical machinery , e.g., max- t1 ow , raI1domized malysisy etc.You will enComter such problems only if you claim specialized knowledge , e.g., graph algorithms, complexity theory , etc. In terviewing is about more than being able to design algorithms quickly. You also need to know how to present yourself, how to ask for help when you are stuck , how to come across as being excited about the company , and knowing what you can do for them. We discuss the non- technical aspects of interviewing in Chapter 12. You can practice with friends or by yourself; in either case, be sure to time yourself. In terview at as many places as you can without it taking away from your job or classes. The experience will help you and you may discover you like companies that you did not know much abou t. Although an interviewer may occasionally ask a question directly from AFI, you should not base your preparation on memor 恒 ing solu 用 tions from AFI. We sincerely hope that reading this book will be enjoy- able and improve your algorithm design skills. The end goal is to make you a better engineer as well as better prepared for software interviews. Level and Prerequisites Most of AFI requires its readers to have basic familiarity with algorithms taught in a typical undergr 叫 uate-Ievel algorithms class. 古 le chapters O 口 meta-algorithms ， gr 叩 hs ， and intractability use more advanced ma- chinery and may require additional review. Each chapter begins with a review of key concepts. This review is not meant to be comprehensive and if you are not familiar with the material , you should first study the corresponding chapter in an algorithms text- boo k. There are dozens of such texts and our preference is to master one or two good books rather than super 血 cially sample many. We like Algo- rithms by Dasgupta , Papadirnitriou , and Vazirani because it is succinct and beautifully written; Introduction to Algorithms by Cormen , Leiserson, Ri vest , and Stein is more detailed and serves as a good reference. Since our focus is on problems that can be solved in an interview rel- atively completely, there are many elegant algorithm design problems which we do not include. Similarly, we do not have any straightforward review-type problems; you may want to brush up ∞ these using intro- ductory programming and data-structures texts. The field of algorithms is vast and there are many specialized topics, such as computational geometry , numerical analysis, logic algori 仕 lms ， etc. Unless you claim knowledge of such topics, it is highly unlikely that you will be asked a question which requires esoteric knowledge. While an interview problem may seem specialized at first glance, it is invariably the case that the basic algorithms described in this book are sufficient to solve i t. If you find the book helpful, please purchase a copy to support the authors! 4 Acknowledgments The problems in this book come from diverse sources-our own expe- riences , colleagues, friends, papers, books, In ternet bulletin boards, etc. To paraphrase Paul Halmos from his wo 口 derful book Problems for Math- ematicians , Young and Old: "I do not give credits-who discovered what? Who was first? Whose solution is the best? It would not be fair to give credit in some cases 缸 1d not in others. No one knows who discovered the theorem that bears Pythagoras' name and it does not matter. The beauty of the subject speaks for itself and so be it." One person whose help and support has improved the quality of this book and made it fun to read is our cartoonist, editor, and proofreader, Nidhi Rohatgi. Several of our friends and students gave feedback on this book-we would especially like to thank Ian Varley, who wrote so- lutions to several problems , and Senthil Chellappan, Gayatri Ramachan- dran , and Alper Sen for proofreading several chapters. We both want to thank all the people who have been a source of en- lightenment and inspiratio 口 to us through the years. 1/ Adnan Aziz, would like to thank teachers, friends, and students from IIT Kanpur , UC Berkeley, and UT Austin. I would especially like to thank my friends Vineet Gupta and Vigyan Singhal, and my teach- ers Robert Solovay , Robert Brayton, Richard Karp/ Raimund Seidel, and Somenath Biswas for introducing me to the joys of algorithms. My co- author , Amit Prakash , has been a wonderful collaborator-this book is a testament to his intellect , creativity, and enthusiasm. 1/ Amit Prakash, have my co-author and mentor , Adnan Aziz, to thank the most for this boo k. To a great extent, my problem solving skills have been shaped by Adnan. There have been occasions in life when I would 口 ot have made 吐 Hough without his help. He is also the best possible collaborator I can think of for any intellectual endeavor. Over the years , I have been fortunate to have great teachers at IIT Kanpur and UT Austin. I would especially like to thank Professors Scott Nettles, Vijaya Ramachandran, and Gustavo de Veciana. I would also like to thank my friends and colleagues at Google, Microsoft, and UT Austin for all the stimulating conversations and problem solving ses- sions. Lastly and most importantl )T, I want to thank my family who have been a constant source of support , exciteme 时/ and joy for all my life and especially during the process of writing this boo k. ADNAN AZIZ ad 丑臼 l@a工 gorithmsforinterviews.com AMIT PRAKASH amit@algorithmsforinterviews.com Problem Solving Techniques It 's not that I/m so smartt it's just that I stay with problems longer. A. Einstein. -Developing problem solving skills is1ikekamizlg to p1ay a m · instrument 一】-a book or a teacher cm poht you h the right directiOIL but O 干 ly your hayd workwill take you where you want to g0·Like a m- / youn 创 tohowunde 蝴吨 concepts but theory is no substitute for practice;for th1s reasonr AFI consists primarily ofproblems Great problem sokers have ski11s that carmot be captured by a set of rules.Stilly whm faced with a cEIdleI1ging algoyithm desigIIprob1em it is rlpfu1Mwe a sma11setdgm 向阳 iples that may be applicable we eIImerate a c01lection of such prhciples h Table 1.ofteIL you may have to use more than one of these techdques. We will I1ow look at some concrete examples of how these techRiques an be applied. DIVIDE-AND-CONQUER AND GENERALIZATION A triomho is formed by joining three unit-sized squares in m L-shape. Amu 也 ted ches 由 oar 斗 (he 丑 cefor 也 8 x 8 Mboard) is made up of 64 unit- sized squares arzmged m m 8 × 8squarey miI111s the topleft squaye-sup- 2:12oua 盯 r 把.它 eas 咏 ωk om 口 lin 丑 10ωst 白 ha 挝 t covers the 8 x 8 Mb o 侃 ard. (Since there are 63 squares h the 8 × 8Mboard and-we have ntriomhosr a valid phcement canmt have overlapping triommos or trioIIlinos which extend out of the 8 × 8 Mboard.) Divide-aI1d-COIlquer is a good strategyto attackthis problem-k1stead of the 8 × 8Lfboardr1etFs consider m n × nLfboard-A2 × 2Mboard c-n be covered w 圳 triomir 叫阳比 of tl 盯 ame exact shape. You 阳 m41 二 Z ; i 己 2? 吃 :3:2:::; 江 i 工::?俨俨 hat 挝 ta 创阳 om 红 min 丑 linO 口∞ 1 旧盯 O 叩 pI 阳丑 m 阳 nen 臼 m 叫 n S 臼 sing can be used to c ∞ omput 怡 e a plac 臼 em 工丑 len 丑 lt for an r ηZ 叶十 Ixη 十 1 If you find the book helpful, please purchase a copy to support the authors! 6 PROBLEM SOLVING TECHNIQUES 7 Technique Description Divide-and- Can you divide the problem into two or more conquer smaller independent sUbproblems and solve the original problem using solutions to the subproblems? Recursion , dynamic If you have access to solutions for smaller in- programmmg stances of a givenproblem, can you easily con- struct a solution to the problem? Case analysis Can you split the input/ execution into a num- ber of cases and solve each case in isolation? Generalization Is there a problem that subsumes your prob- lem and is easier to solve? Data-structures Is there a data-structure that directly maps to the given problem? It erative refinement Most problems can be solved using a brute- force approach. Can you formalize such a so- lution and improve up ∞ it? Small examples Can you find a solution to small concrete in- stances of the problem and then build a so- lution that can be generalized to arbitrary 让卜 stances? Reduction Can you use a problem with a known solution as a subroutine? Graph modeling Can you describe your problem using a graph and solve it using an existing algorithm? Write an equatio 丑 Can you express relatio 口 ships in your problem in the form of equations (or inequalities)? Auxiliary elements Can you add some new element to your prob- lem to get closer to a solution? Variation Can you solve a slightly differentproblem and map its solution to your problem? Parallelism Can you decompose your problem into sub- problems that can be solved independently on different machines? Caching Can you store some of your computation and look it up later to save work? Symmetry Is there symmetry in the input space or solu- tion space that car 飞 be explo 让 ed? Table 1. Commo 日 problem solving techniques. Mboard-However you wmquickly see tht tkislhe of reasonhg does not lead you anywhere. Another hypothesis is that if a placement exists for an n xη Mb oard ， then one also exists for a 2n x 2η Mb oard. This does wor k: take 4 n x n Mboards and arrange them 以 orma 严 x2ηsquareimuchawdthi three of theMboards have theEIIUSSIng square set towards the center a 时 one Mb oard has its missi 吨叩 are outward to coin 创 e with the missing cornerof a 2ηx2η 孔1b oard. The gap in the ce 口 ter can be covered with a trio 吨。 and ， by hypothesis, we ~a~ cover the 4η × η Mb oards with triominos as well. Her 丑 nee aplacαemen 时 1 吐 te 以 χi 妇 st 怡 sforany ηt 也 ha 挝 ti 扫 s a power O 们 f 归 2. In 丑叩 pa 缸削 I 挝 ωO ∞ I 丑川 1 used inη 川 t 白 he pm f cm be dimetly coded to fh d th e ac tu aiC OL e-::L as well. Obs 盯 e 阳恤 P 伊阳 ro 伪 blemd 由 err 红 mo ella 臼 sg 伊 ene 咀 eraliz 泣 za 挝 tion (from 8 x 8 tω02 俨饥 x2 俨 η). RECURSION AND DYNAMIC PROGRAMMING Suppose you were to desig1aI1algorithm that takes az111npareIIthesized e 创 S 臼蚓 S 剖 i ∞町 C ∞ O 叫 1 干卢 1 让 t 牛归 gμad 创 d 出创 i 让挝 t 赳 i 妇阳 Oαna 叫 mu 均 lica 甜 ti ∞ O 句 perato 创创 r 吼. thepa 盯 ren 丑 lt 白 he 臼 Sl 曰 za 拍 tio ∞ I 丑 1 t 出:h a 挝 tm 工丑 laxi 垃 m 丑 li 坦 ze 臼 s the 飞 value 淀 e of the expressio ∞ n. For mpley the expression5-3·4+6yields my of the followiIIgva111es: -25 = 5 一 (3 ， (4 + 6)) -13 = 5 一 ((3 . 4) + 6) 20 = (5 - 3) . (4 + 6) -1 = (5 一 (3 . 4)) 十 6 14 = ((5 - 3) . 4) + 6 S 」;芷: 2C 甲吟叫 S 由如 M 均中抄 C ∞ om 叫 pu 盹 t 怡 e 由 the p 归 a 盯盯削 r 跄 ren 时.它吱 m 创叫 e 缸 m 创 I 丑时毗 1 吐巾 t 也 thesizatio ∞ I 口山 1 aX1 牛中叫 I 虹 m 削 n 旧 1 让 ize 臼 sits 怡 sva 叫 alue 叩 1 凡 e 今， i 扰 t is easy to ider 哟 T the optimum top level parenthesization • pareRtheSIze on each side of the operators azld determ mt whi 中 operator r 虹叫 I e 仅 cur 岛 Sl 凹飞 ve c ∞ ompu 时 ta 挝甜 t 拄 io ∞ n of the rnaχi 垃 m 丑 III 曰 zln 口 19 pa 盯 ren 口时 t 白 he 臼 si 垃 za 剖 tim 丑 1 for u 由 be 今产 pre 蚓 O ∞丑 m S leads to repeated calls with idmtical argume 时 s. Dy- programming avoids these repeated computations;refer to Prob- lem 3.11for a detailed exposition-I CASE ANALYSIS Y 沟 ou are gi 扣 Vmemaset S ofE distincthtegm mdaCPUthathas aspecial mstruetiOIL SORt-Ethat cm sort5htegers h OIIe cycle.Your task is to identi 命 the 3largest integers h S ushg SORt-5to compaye and sort subsets of afurthermorer you must miIIimize the number of calls to SORT5. If you find the book helpful, please purchase a copy to support the authors! 8 PROBLEM SOLVING TECHNIQUES 9 If all we had to compute was the largest integer in the set, the opti- mum approach would be to form 5 di 司 oint subsets 8 1 ,…, 8 5 of 8/ sort each subset , and then sort {max8 1 ,…,max 8 5 }. This takes 6 calls to SORT5 but leaves ambiguity about the second and third largest integers. It may seem like many calls to SORT5 are still needed. However if you do a careful case analysis and eliminate all x ε8 for which there are at least 3 integers in 8 larger than X/ 0 口 ly 5 integers remain and hence just one more call to SORT5 is needed to compute the result. Details are given in the solutionto Problem 2 .5. FIND A GOOD DATA STRUCTURE Suppose you are given a set of files, each containing stock quote infor- mation. Each line contains starts with a timest 缸丑 p. The files are indi 飞 rid ually sorted by this value. You are to design an algorithm that combines these quotes into a single file R containing these quotes , sorted by the 出 nestamps. This problem can be solved by a multistage merge process, but there is a trivial solution using a min-heap data structure, where quotes are ordered by timestamp. First build the min-heap with the first quote from eachfile; then iteratively extract the minimum entry e from the min-heap/ write 让 to R, and add in the next entry in the file corresponding to e. Details are given in Problem 2.10. ITERATIVE REFINEMENT OF BRUTE-FORCE SOLUTION Consider the problem of string search (cf Problem 5.1): given two strings s (search string) and T (text), find all occurrences of s in T. Since scan occur at any offset in T, the brute-force solution is to test for a match at every offse t. This algorithm is perfectly correct; its time complexity is O(η. m)/ where n and m are the lengths of sand T. After trying some examples , you may see that there are several ways in which to ir 口 prove the time complexity of the brute-force algorithm. For example , if the character T[i] is not present in s you can suitably ad- vance the matching. Furthermore , this skipping works betterif we match the search string from its end and work backwards. These refinements willmake the algorithm very fast (linear-time) on random text and search strings; however , the worst case complex 让 y remains O(η ·m). You can make the additional obser 飞 ration that a partial match of s which does not result in a full match implies other offsets which cannot lead to full matches. For ex 缸 nple ， if s 二 α bd α beabe and iff starting back- wards , we have a partial match up toα be α be that does 口 ot result in a full match , we know that the next possible matching offset has to be at least 3 positions ahead (where we can match the secondα be from the partial match). By puttiI1g together these refinemeI1ts you will have arrived at the famous Boyer-Moore string search algorithm-its worstmcase time C m- plexityis oh+m)(whichis thebestpossible ffomatheoreticalperspecm tivek it is also one of the fastest strhg search dFrithms h practice. SMALL EXAMPLES Problems that seem difficult to solve in the abstract, can become much mo 时 ractable when you examine small concrete instances. For instan 二 co 且 sider tl 时 ollowi 口 g problem: there a 时 00 clo 时 doors alo 吨 a … ri dorr numbered from1t0500.A persOI1walks through the corridor and opens each door.AIIother person walks through tke corridor and closes every alternate door. Continuing 如 this m 缸 me 乙 the i-th person c aI1d toggles the position of every t-th door starthgfrom door t.y; to determine exactly how many doors are opmafter the 500-th persOII has walked through the corridor. It is very difficult to solve this problemushg abstract variables.How- ever if you try the problem for ljp374710?md20doorsr it takes mder ammte to see that the doors that remah opmare l?479716.··F regard- less of the totalI1umber of doors.The pattern is obvious-the doors that re 中 am op 中 are 中 ose numbered by perfect squares. Once you make ths cOImeetlOIL1tls easy tOEZ?ve it for the generalcase-HeIIce the totd number of open doors is l y500 J= 22. Refer~to Problem 9A for a detailed solution. REDUCTION Conside 气 the probkm of fiMing if om st 血 g is a rotation of the other, e.g., "car" 缸 1d Harc"are rotatiORs of each other A I1aturd approach may be to rotate the first strhgby everypomible offset aM ttmcomar4 wi 中 the second st 出 g. This algorithm would have quadratic ti 斗 com plexity. You may I1otice that this problem is quite s 扛 nilar to string search which cm be domh1inear-tmer albeit mhg a somewhat complex alm gorithm.So it would be I1aturd to try to reduce this problem to string search.IndeedrifwecomatemtethesecondstringwithitselfaMsearcE for the first stying h tke resulting string, we will find a match iff the two original strhgs are rotatiOI1s of each other.This reduction yields a linear- time algorithm for our problem;details are giveR iRProbkII15.4. Usually you try to reduce your proble~ to an easier problem. But sometmesr you need to reduce a problem bmWI1to be difficult to your giveI1problem to show that your problem is difficult.Such probkms are described in Chapter 6. If you find the book helpful, please purchase a copy to support the authors! 10 PROBLEM SOLVING TECHNIQUES 11 GRAPH MODELING Drawing pictures is a great way to brainstorm for a potential solution. 日 the relati(;nships in a given problem can be represented using a graph , auite often the problem can be reduced to a well-known graph problem. 至 or example, suppose you are given a set of barter rates between com- modities and you are supposed to find out if 缸 1 arbitrage exists, i. e. , there is a way by which you can start with αunits of some commodity C and perform a series of barters which results in having more thanαunits of C. We can model the problem with a graph where commodities corre- spond to vertices, barters correspond t? edges , .~~ the :d?e ,: eight ~s s~t to the logarithm of the barter rate. If we can find a cycle in the graph with a positke weightrwe wouldhave fOUI1d such a series of exchnges. Such a cycle can be solved using the Bellma 扣 Ford algorithm (c f. Prob- lem 4.19). After some (or a lot) of tr 划 -and-error ， you may begin to wonder if a such a configuration exists. Prov hard. However if you think of the 8 x 8 square board as a chessboard, you will observe that the removed comers are of the same color. Hence the board consists of either 30 white squares and 32 black squares or vice versa. Since a domino will always cover two adjacent squares, anyar- rangement of dominoes must cover the same number ofblack and white squares. Hence no such configuration exists. The or 地 inal problem did not talk about the colors of the squares. Adding these colors to the squares makes it easy to prove impossibility, illustrating the strategy of adding auxiliary elements. VARIATION WRITEANEQUATION Some problems can be solved by expressing them in the language of mathematics. For example , suppose you were aske~ to write an algo- rithm that computed binomial coefficien 怡， G) =硕兰布 The problem with computing the binomial coefficient directly from the definition is that the factorial function grows very quickly and can overflow an integer variable. If we use floating point represe 口 tations for numbers , we lose precision and the problem of overflow does n?~ go away. These proble~s potentially exist even if the final value .of G) i~ small. One c~ try to factor the numerator and denominator and try and cancel out commo 日 terms but factorization is itself a hard problem. The binomial coefficients satisfy the addition formula: Suppose we were asked to design an algorithm which takes as input an undirected graph and produces as output a black or white coloring of the vertices such that for every vertex, at least half of its neighbors differ in color from 让. We could try to solve this problem by assigning arbitrary colors to vertices and then flipping colors wherever constraints are not me t. How- ever this approach does not converge 0 口 all examples. It turns out we can define a slightly different problem whose solution will yield the coloring we are looking for. Define an edge to be diverse if its ends have different colors. It is easy to verify that a color assignment that maximizes the number of diverse edges also satisfies the constraint of the original problem. The number of diverse edges can be maχ 迦 lized greedily flipping the colors of vertices that would lead to a higher num- ber of diverse edges; details are give 口 in Problem 4.11. PARALLELISM //IlI\ + -k nk In the context of interview questio 口 s ， parallelism is useful when dealing with scale, i.e., when the problem is so large that it is 红卫 possible to solve it on a single machine or it would take a very long time. The key insight you need to display is how to decompose the problem such that (1.) each subproblem can be solved relatively independently and (2.) constructing the solution to the or 培 inal problem from solutions to the subproblems is not expensive in terms of CPU time , main memory , and network usage. Consider the problem of sorting a petascale integer array. If we know the distribution of the numbers , the best approach would be to define equal-sized ranges of integers and send one range to one machine for sorting. The sorted numbers would just need to be concatenated in the correct order. If the distribution is 卫 ot known then we can send equal- sized arbitrarysubsets to each machine and then merge the sorted results This identity leads to a straightforward recursion for computing (~) which avoids the problems mentioned above. Dynamic programming has to be used to achieve good time complexity-details are in Prob 阳 lem 9.1. AUXILIARY ELEMENTS Consider an 8x 8 square board in which two squares 0 且 diagonally oppo- site corners are removed. You are given a set of thirty-one 2 x 1 dominoes and are asked to cover the board with them. If you find the book helpful, please purchase a copy to support the authors! 12 using a min-heap. For details on petascale sorting, please refer to Prob- lem2.2. CACHING Caching is a great tool whenever there is a possibility of repeating com- putations. For example , the central idea behind dynamic programming is caching results from intermediate computations. Cachingbecomes ex- tremely useful in another setting where requests come to a service in an online fashion and a small number of requests take up a significant amount of compute power. Workloads on web services exhibit this prop- erty; Problem 7.1 describes one such problem. SYM 如 1ETRY While symmetry is a simple concept it can be used to solve very difficult problems , sometimes in less than intuitive ways. Consider a 2-player g 缸 ne in which players alternately take bites from a chocolate bar. The chocolate bar is an ηx m rectangle; a bite must remove a square and all squares above and to the right in the chocolate bar. The first pI 句 rer to eat the lower leftmost square loses (think of it as being poisoned). Suppose we are asked whether we would prefer to play first or sec- ond. One approach is to make the obser 飞 ration that the game is sym- metrical for Player 1 and Player 2/ except for their starting state. If we assume that there is no winning strategy for Player 1/ then there must be a way for Player 2 to win if Player 1 bites the top right square in his first move. Whatever move Player 2 makes after that can always be made by Player 1 as his f 让 st move. Hence Player 1 can always win. For a detailed discussion , refer to the Problem 9.13. CONCLUSION In addition to developing intuition for which technique may apply to which problem , it is also important to know when your technique is not wor 烛 19 and quickly move to your next best guess. In an interview set- ting/ even if you do not end up solving the problem entirely, you will get credit for applying these tecm 问 ues in a systematic way and clearly communicating your approach to the problem. We cover nontechnical aspects of problem solving in Chapter 12. Part I Problems If you find the book helpful, please purchase a copy to support the authors! [...]... right subchild 1 16 SEARCH BST FOR A KEY Searching for a key in a BST is very similar to searching in a sorted array Recursion is more natural but for performance, a while-loop is preferred Problem 1.16: Given a BST T， 丘rst write a recursive function that searches for key K , then write an iterative function 1.17 SEARCH BST FOR x> k BSTs offer more than the abili可 to search for a key-they can be used to... time lookups for the min element.) Searching for arbitrary keys has O(η) time complexity-a 町thi吨 that can be done with a heap can be done with a balanced BST with the same complexity but with possibly some space and time overhead 2.10 MERGING SORTED ARRAYS You are given 500 files , each containing stock quote information for an SP500 company Each line contains an update of the following form: 1232111... amount of chmge that remahs to h made If we try this for 48 pe口ce， we get 30 , 12 , 6 However the optimum would be 24 , 24 In its most general form, the coin changing problem is NPMrd(4.ChapteJ6)but for some coimgesr the greedy fl写frit』mm optimum-e-E-r if tt1e denomiRatiom are ofthe form {lAT27俨} Ad hoc guments can be applied to show that 让 is also optimum for VS coins iiZgeneralproblem ca由e solved in pseudopolynomial... algorithm is 如nd缸nental to the performance of many applications and there are several elega时 algorithms proposed for it, each with its own tradeoffs As a result, there is no one perfect answer to 让 If someone asks you this questio口 in an interview, the best way to approach this problem would be to work through one good algorithm in detail and discuss the breadth of other algorithms for solving this problem... tharIthe original proMem .For exampley COI1sider the following problem: give口 an array of integers A of length 凡 find the interval indices a and b such that 2:~=α A[i] is maximized Letrs try to solve this problem assumiRg we have the s0111tiORfor the subarray A[l ， 饥- 1] In this case, even if we knew the largest sum subarray for array A[l ， η-I]， it does not help us solve the problem for A[l ， η] Now, consider... Rehashing is expensive (e (η+ m) time) but if it is performed infrequently (for example, if performed every time the load increases by 2x) , its amortized cost is low Compared to binary search trees (discussed on Page 20), inserting and deleting in a hash table is more efficient (assuming the load is constant) One disadvantage of hashing is the need for a good hash function but this is rarely an issue... classmates in a graduate course on information theory at MIT were given the choice of a term paper or a final exam For the term pape乙 Huffman's professor, Robert M Fano, had given the problem of finding an algorithm for assigning binary codes to symbols such that a given set of symbols can be represented in the smallest number of bits Huf缸lan worked on the problem for months, developing a number of approaches... assign the shorter strings to more probable characters and the longer strings to less probable characters 3.12 SCHEDULING TUTORS ￥'ou are responsible for scheduling tutors for the day at a tutoring com- 卢町 For 二ach day, you have received a number of r叩臼ts for tutors Each wquest has a specified start time md each lessORis thirty miI111tes 10吨 YLu have more tutors ttm reqlmts.Each tutor cm start work at... 征伍 ime client 俑阳创ore t四 For example, if the lookups are s凹ed 削 1 i t ake st 耐.它吐urn 怡 臼 怡 吐 i口 order of 让lcreasing i , then the client making the i-th query has to wait 2:~=1 tj milliseconds Problem 3.13: Design an efficient algorithm for computing 缸loptimum order for processing the queries If you find the book helpful, please purchase a copy to support the authors! Greedy Algorithms HUFFMAN CODING... system that customers use to trigger certain tasks He wants to minimize the average amount of time it takes for a customer to perform tasks If a menu item is at the i-th positio凡 it takes i units of time for the user to reach there (linear scan) and it takes c units of time to click on it MEL ALGORITHMS Each menu item can have multiple levels of sub-menus and a sub-menu can be reached by clicking on . of support , exciteme 时/ and joy for all my life and especially during the process of writing this boo k. ADNAN AZIZ ad 丑臼 l@a工 gorithmsforinterviews.com AMIT PRAKASH amit@algorithmsforinterviews.com Problem Solving Techniques It 's not. cheerfully send you $0.42. Please refer to the website for details. Ver 咀 on 1. 0.0 (September I, 2010) L 叩 ebsite:http://algorithmsforinterviews.com ISBN: 1453792996 EAN-13: 9781453792995 To my father! Ishrat Aziz! for giving me my l~ 作 long love of learning AdnanAziz To my parents! Manju Shree and Arun Prakash! the most loving parents I can imagine Amit