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Chapter 7: Problem 44

In the process of rebuilding the master list, the Radix Sort Algorithm (Algorithm 7.6 ) wastes a lot of time examining empty sublists when the number of piles (radix) is large. Is it possible to check only the sublists that are not empty?

Short Answer

Expert verified
No, it is not possible to check only the non-empty sublists in the Radix Sort Algorithm. Doing so would compromise the stability of the sorting method by breaking the order of equal values in the final output.

Step by step solution

01

Understanding Radix Sort

The Radix Sort is a non-comparative sorting algorithm. It sorts numbers by processing individual digits. 'n' numbers are sorted in 'd' digits, therefore the complexity of the Radix Sort algorithm is O(n*d). Each integer is processed in a specific way, digit by digit starting from least significant digit to most significant. It uses counting sort as a subroutine to sort.
02

Examining the role of sublists/piles

Within the Radix Sort, a pile or sublist is created for each unique digit. If the radix is large, there will be more piles. It is in these piles where the algorithm can potentially waste time, as it will examine these even if they are empty.
03

Consider bypassing empty piles

In theory, it seems that if a pile is empty, it could be bypassed which could reduce the time complexity. However, because Radix Sort is a stable sort, it maintains the relative order of records with equal values. Therefore, every sublist or pile has to be examined to maintain the stability of the sort.
04

Final Verdict

While bypassing empty piles might sound like a good idea to speed up the process, doing so would compromise the stability of the Radix Sort algorithm. It is not feasible to check only the non-empty sublists in Radix Sort Algorithm because it will break the order of equal values in the final output.

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Most popular questions from this chapter

Implement the Quicksort algorithm using different strategies for choosing a pivot item, run it on your system, and study its best-case, average-case, and worst-case performances for different strategies using several problem instances.

Show that a heap with \(n\) nodes has \(\lceil n / 2\rceil\) leaves.

An algorithm called Shell Sort is inspired by Insertion Sort's ability to take advantage of the order of the elements in the list. In Shell Sort, the entire list is divided into noncontinuous sublists whose elements are a distance \(h\) apart for some number \(h\). Each sublist is then sorted using Insertion Sort. During the next pass, the value of \(h\) is reduced, increasing the size of each sublist. Usually the value of each \(h\) is chosen to be relatively prime to its previous value. The final pass uses the value 1 for \(h\) to sort the list.. Write an algorithm for Shell Sort, study its performance, and compare the result with the per. formance of Insertion Sort.

Another way to sort a list by exchanging out-of-order keys is called Bubble Sort. Bubble Sort scans adjacent pairs of records and exchanges those found to have out-of-order keys, After the first time through the list, the record with the largest key (or the smallest key) is moved to its proper position. This process is done repeatedly on the remaining, unsorted part of the list until the list is completely sorted. Write the Bubble Sort algorithm. Analyze your algorithm, and show the results using order notation. Compare the performance of the Bubble Sort algorithm against those of Insertion Sort, Exchange Sort, and Selection Sort.

Write a version of mergesort3 (Algorithm 7.3), and a corresponding version of merge3, that reverses the rolis of two arrays \(S\) and \(U\) in each pass through the repeat loop.
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