Database System Concepts
Database System Concepts
7th Edition
ISBN: 9780078022159
Author: Abraham Silberschatz Professor, Henry F. Korth, S. Sudarshan
Publisher: McGraw-Hill Education
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Fig. 8.3 included

**Exercise 2.** Present an \(O(n)\) algorithm that sorts \(n\) positive integer numbers \(a_1, a_2, \ldots, a_n\) which are known to be bounded by \(n^2 - 1\) (so \(0 \leq a_i \leq n^2 - 1\), for every \(i = 1, \ldots, n\). 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 \(n^2 - 1\) (which is the largest possible value) requires \(\log_{10}(n^2 - 1)\) 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 = \(12^2 - 1\).
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Transcribed Image Text:**Exercise 2.** Present an \(O(n)\) algorithm that sorts \(n\) positive integer numbers \(a_1, a_2, \ldots, a_n\) which are known to be bounded by \(n^2 - 1\) (so \(0 \leq a_i \leq n^2 - 1\), for every \(i = 1, \ldots, n\). 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 \(n^2 - 1\) (which is the largest possible value) requires \(\log_{10}(n^2 - 1)\) 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 = \(12^2 - 1\).
The image displays a series of number columns that undergo transformations through two processes. Here's a description of the process as if it were part of an educational piece explaining sorting techniques: ### Initial List: - 329 - 457 - 657 - 839 - 436 - 720 - 355 ### Step 1: The numbers are rearranged into new columns through a sorting transformation: - Column 1: 720, 355, 436, 457, 657, 329, 839 ### Step 2: The numbers are further rearranged: - Column 2: 329, 436, 839, 355, 457, 657, 720 ### Final Sorted List: The final sorting arranges the numbers in ascending order: - 329 - 355 - 436 - 457 - 657 - 720 - 839 These transformations demonstrate the steps of a sorting algorithm, showcasing how an initially unordered list of numbers is processed into a sorted sequence through successive restructuring.
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Transcribed Image Text:The image displays a series of number columns that undergo transformations through two processes. Here's a description of the process as if it were part of an educational piece explaining sorting techniques: ### Initial List: - 329 - 457 - 657 - 839 - 436 - 720 - 355 ### Step 1: The numbers are rearranged into new columns through a sorting transformation: - Column 1: 720, 355, 436, 457, 657, 329, 839 ### Step 2: The numbers are further rearranged: - Column 2: 329, 436, 839, 355, 457, 657, 720 ### Final Sorted List: The final sorting arranges the numbers in ascending order: - 329 - 355 - 436 - 457 - 657 - 720 - 839 These transformations demonstrate the steps of a sorting algorithm, showcasing how an initially unordered list of numbers is processed into a sorted sequence through successive restructuring.
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