Quiz 1 Solution. Figure 1: grade distribution. Problem Points Grade Initials. Total 80

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1 Introduction to Algorithms October 17, 21 Massachusetts Institute of Technology 6.46J/1.41J Singapore-MIT Alliance SMA553 Professors Erik Demaine, Lee Wee Sun, and Charles E. Leiserson Quiz 1 Solution Quiz 1 Solution Figure 1: grade distribution Problem Points Grade Initials Total Name: MIT students: Circle the name of your recitation instructor: Bob Chong George Jennifer Rachel

2 /. 6.46J/1.41J/SMA553 Quiz 1 Solution Name 2 Problem 1. Match-up [16 points] Fill in the following table by specifying, for each algorithm, the letter of the recurrence for its running time and the numeral of the solution to that recurrence. Points will be deducted for wrong answers, so do not guess unless you are reasonably sure. Algorithm Recurrence Solution Merge sort Binary search Randomized quicksort (expected) Randomized quicksort Strassen s algorithm Selection Randomized selection Randomized selection (expected) d 6 b 2 g 6 h 5 a 3 c 1 h 5 e 1 Letter a b c d e f g h Recurrence! "$#$ % & '( )*+,- & ' :.(;=< / >19? 79 A@B Numeral Solution C C DFE C HGJIKL C C C MDFE

3 /V V/ O N 6.46J/1.41J/SMA553 Quiz 1 Solution Name 3 Problem 2. Merging several sorted lists [19 points] Professor Fiorina uses the following algorithm for merging sorted lists, each having elements. She takes the first list and merges it with the second list using a linear-time algorithm for merging two sorted lists, such as the merging algorithm used in merge sort. Then, she merges the resulting list of ( elements with the third list, merges the list of N elements that results with the fourth list, and so forth, until she ends up with a single sorted list of all elements. (a) Analyze the worst-case running time of the professor s algorithm in terms of and. Merging the first two lists,each of elements, takes ( time. Merging the resulting ( elements with the third list of elements takes N time, and so on. Thus for a total of list, we have: O7PRQ ( 1 O' TSSS 1 7W X 'Y@B Z9 9 U

4 6.46J/1.41J/SMA553 Quiz 1 Solution Name 4 (b) Briefly describe an algorithm for merging sorted lists, each of length, whose MDFE\9 worst-case running time is [. Briefly justify the running time of your algo- MD]E\9 rithm. (If you cannot achieve [, do the best you can for partial credit.) There are several possible answers (only one is required). One method is to repeatedly pair up the lists, and merge each pair. This method can also be seen as a tail component of the execution merge sort, where the analysis is clear. Finally, a conceptually simple method is to store a min priority queue of the minimum elements of each of the lists. At each step, we output the extracted minimum of the priority queue, and using sattelite data determine from which of the lists it came, and insert the next element from that list into the priority queue.

5 6.46J/1.41J/SMA553 Quiz 1 Solution Name 5 (c) Argue that there are 4^ _ 9` different ways to interleave lists, each of length, into a single list of length. For the first list, number of ways of choosing elements from elements 4^ R@a. L^b 2. possible permutation of the elements in the list, only 1 permuta- term in Note that of the. tion is valid because the elements in the list is already sorted. Hence the. the denominator. For the second list, number of ways of choosing elements from the remaining A@ elements... R@ L^ A@ L^b L^ And so on. Therefore, total number of ways of forming the lists, each of size, from elements is: c^ A@. A@ed ; :f. L^ R@gd ;=< f. A@. c^ h i` ]. A@. ] L. L^ SSS R@gd ;=< f. ]. A@. This is exactly the number of different ways of interleaving the list, into the single list of elements ].

6 m m j Q 6.46J/1.41J/SMA553 Quiz 1 Solution Name 6 (d) Prove a lower bound of j MDFEk9 algorithm for merging sorted lists each of length. Q.l 4^ l..) on the worst-case running time of any comparison (Hint: Use the fact that Consider the decision tree used to make the comparisons. The worst-case running time corresponds to the height of the tree:. O7PRQ DFE 4^ _ 9` DFE\c^@%D]E= h ` DFE\c^@%D]E= h $no) DFE\c^@%MD]E DFE\c^@%MD]ECpMD]Ek MDFE\A@%MDFECQr@sutbvUpMD]Ek MDFEw&@sMDFE\Q MD]Ek9

7 [ 6.46J/1.41J/SMA553 Quiz 1 Solution Name 7 Problem 3. True or False, and Justify [45 points] Circle T or F for each of the following statements, and briefly explain why. The more content you provide in your justification, the higher your grade, but be brief. Your justification is worth more points than your true-or-false designation. (a) T F A constant? m exists such that for any m?, there is an array of elements such that insertion sort runs faster than merge sort on that input. true. If the elements are already in sorted order, the running time for insertion sort is tbv), whereas that of merge sort is [. (b) T F Consider the algorithm from the textbook for building a max-heap: BUILD-MAX-HEAP xm 1 heap-sizey xz { xrz lengthy xz! ~ downto 2 for Oc{ } lengthy 3 do x ho MAX-HEAPIFY On an array of elements, this code runs in MDFE\ time in the worst case, because there are calls to MAX-HEAPIFY, and the worst-case time for any call is D]EC. false. MDFEC is not a tight bound. In fact, Build-Max-Heap runs in time, as proven during recitation.

8 [ l m z z m 6.46J/1.41J/SMA553 Quiz 1 Solution Name (c) T F Given a number and a positive integer, the value n MD]E can be computed in [ time by repeated squaring. (Assume that multiplying two numbers takes constant time.) d n f (d) T F true. Letting = n, we have: n n (ƒ nl ƒ MD]E Note: it is important to understand that MD]E = [ An adversary can force randomized quicksort to run in j as input an already sorted or reverse-sorted array of size.. time by providing false. The running time is not dependent on the input that the adversary provides. This is because radomized quicksort will randomly choose a pivot to partition, independent of the input. (e) T F For any integer-valued random variable ˆ Š *UŒ m m * +@p, we have yj zž true Ĉ Š ˆ Š m Œ m Œ l yj yj ˆ Š ä*uœ +@%ˆ Š +@% yj z Œ

9 6.46J/1.41J/SMA553 Quiz 1 Solution Name 9 (f) T F Bucket sort is a suitable auxiliary sort for radix sort. true. An auxilliary sort for radix sort must be stable. Bucket sort is stable and hence suitable for use as an auxilliary sort for radix sort. P (g) T F Consider two implementations of a hash table with slots storing keys, where P. Let P > be the expected time for an unsuccessful search in the table if collisions are resolved by chaining, using the assumption of simple uniform hashing. Let P h be the expected time for an unsuccessful search in the table if collisions are resolved by open addressing, using the assumption of uniform hashing. Then, we have P > Cä P h h. false. P > P > :k P +@. š In the worst case, in the chaining version, we need to search through all the values in one particular slot. On the other hand, in the open addressing version, we need to search through every slot. Hence P > M ä P h h.

10 [ <. l P 6.46J/1.41J/SMA553 Quiz 1 Solution Name 1 (h) T F Let œ be a class of universal hash functions for a hash table of size P ž. Then, if we use a random Ÿggœ to hash keys into the table, the expected number of collisions is at most $. true. Number of ways to choose 2 keys out of keys is. d.(;=< f Therefore, the expected number of collisions, (i) T F Given a set ŠU < > y number of collisions z R@ > XŽ Ž ŽX. > LŒ R@ of points in the plane, the points closest to the origin '*U2*- (using normal Euclidean distance) can be found in time in the worst case. true. Finding the distance of every point from the origin takes [. We can then use Select to get the Ÿ point. That again takes [. Finally, partitioning around the Ÿ point requies [. Each of the steps takes [. Therefore, total time is [ ž