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Exercise 2. Present an O(n) algorithm that sorts n positive integer numbers a1, a2,..., an which are known to be bounded by n² – 1 (so 0 ≤ ai ≤ n² − 1, for every i = 1,...,n. Use the idea of Radix Sort (discussed in class and presented in Section 8.3 in the textbook). Note that in order to obtain O(n) you have to do Radix Sort by writing the numbers in a suitable base. Recall that the runtime of Radix Sort is O(d(n+k)), where d is the number of digits, and k is the base, so that the number of digits in the base is also k. The idea is to represent each number in a base k chosen so that each number requires only 2 "digits," so d = 2. Explain what is the base that you choose and how the digits of each number are calculated, in other words how you convert from base 10 to the base. Note that you cannot use the base 10 representation, because n² - 1 (which is the largest possible value) requires log10 (n²-1) digits in base 10, which is obviously not constant and therefore you would not obtain an O(n)-time algorithm. Illustrate your algorithm by showing on paper similar to Fig. 8.3, page 198 in the textbook (make sure you indicate clearly the columns) how the algorithm sorts the following sequence of 12 positive integers: 45, 98, 3, 82, 132, 71, 72, 143, 91, 28, 7, 45. In this example n = 12, because there are 12 positive numbers in the sequence bounded by 143 122 - 1.

Exercise 2. Present an O(n) algorithm that sorts n positive integer numbers a1, a2,..., an which are known to be bounded by n² – 1 (so 0 ≤ ai ≤ n² − 1, for every i = 1,...,n. Use the idea of Radix Sort (discussed in class and presented in Section 8.3 in the textbook). Note that in order to obtain O(n) you have to do Radix Sort by writing the numbers in a suitable base. Recall that the runtime of Radix Sort is O(d(n+k)), where d is the number of digits, and k is the base, so that the number of digits in the base is also k. The idea is to represent each number in a base k chosen so that each number requires only 2 "digits," so d = 2. Explain what is the base that you choose and how the digits of each number are calculated, in other words how you convert from base 10 to the base. Note that you cannot use the base 10 representation, because n² - 1 (which is the largest possible value) requires log10 (n²-1) digits in base 10, which is obviously not constant and therefore you would not obtain an O(n)-time algorithm. Illustrate your algorithm by showing on paper similar to Fig. 8.3, page 198 in the textbook (make sure you indicate clearly the columns) how the algorithm sorts the following sequence of 12 positive integers: 45, 98, 3, 82, 132, 71, 72, 143, 91, 28, 7, 45. In this example n = 12, because there are 12 positive numbers in the sequence bounded by 143 122 - 1.

C++ for Engineers and Scientists
C++ for Engineers and Scientists
4th Edition
ISBN:9781133187844
Author:Bronson, Gary J.
Publisher:Bronson, Gary J.
Chapter4: Selection Structures
Section: Chapter Questions
Problem 14PP
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= Exercise 2. Present an O(n) algorithm that sorts n positive integer numbers a1, a2, • An which are known to be bounded by n² - 1 (so 0 ≤ ai ≤ n² − 1, for every i 1,..., n. Use the idea of Radix Sort (discussed in class and presented in Section 8.3 in the textbook). Note that in order to obtain O(n) you have to do Radix Sort by writing the numbers in a suitable base. Recall that the runtime of Radix Sort is O(d(n+k)), where d is the number of digits, and k is the base, so that the number of digits in the base is also k. The idea is to represent each number in a base k chosen so that each number requires only 2 "digits," so d = 2. Explain what is the base that you choose and how the digits of each number are calculated, in other words how you convert from base 10 to the base. Note that you cannot use the base 10 representation, because n² – 1 (which is the largest possible value) requires log₁0 (n² - 1) digits in base 10, which is obviously not constant and therefore you would not obtain an O(n)-time algorithm. Illustrate your algorithm by showing on paper similar to Fig. 8.3, page 198 in the textbook (make sure you indicate clearly the columns) how the algorithm sorts the following sequence of 12 positive integers: 45, 98, 3, 82, 132, 71, 72, 143, 91, 28, 7, 45. In this example n = 12, because there are 12 positive numbers in the sequence bounded by 143 122 1. -
Transcribed Image Text:= Exercise 2. Present an O(n) algorithm that sorts n positive integer numbers a1, a2, • An which are known to be bounded by n² - 1 (so 0 ≤ ai ≤ n² − 1, for every i 1,..., n. Use the idea of Radix Sort (discussed in class and presented in Section 8.3 in the textbook). Note that in order to obtain O(n) you have to do Radix Sort by writing the numbers in a suitable base. Recall that the runtime of Radix Sort is O(d(n+k)), where d is the number of digits, and k is the base, so that the number of digits in the base is also k. The idea is to represent each number in a base k chosen so that each number requires only 2 "digits," so d = 2. Explain what is the base that you choose and how the digits of each number are calculated, in other words how you convert from base 10 to the base. Note that you cannot use the base 10 representation, because n² – 1 (which is the largest possible value) requires log₁0 (n² - 1) digits in base 10, which is obviously not constant and therefore you would not obtain an O(n)-time algorithm. Illustrate your algorithm by showing on paper similar to Fig. 8.3, page 198 in the textbook (make sure you indicate clearly the columns) how the algorithm sorts the following sequence of 12 positive integers: 45, 98, 3, 82, 132, 71, 72, 143, 91, 28, 7, 45. In this example n = 12, because there are 12 positive numbers in the sequence bounded by 143 122 1. -
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