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Chapter 7: Problem 39

Write a linear-time sorting algorithm that sorts a list of values of a given ordinal type

Short Answer

Expert verified
The Counting Sort algorithm is a suitable solution for this problem. It works by counting the occurrence of each value in the original input, then placing the values in a new output array based on these counts. It achieves linear time complexity in cases where the range of input values is relatively small.

Step by step solution

01

- Determine the range of the input values

To perform counting sort, the first thing to know is the range of the data that needs to be sorted. In other words, find the minimum and maximum values from the list.
02

- Initialize the Count array and Output array

Create a 'count' array with size equal to the range of input values. Initialize each value in this array to 0. The count array is used to store the count of each individual input value. Also create an 'output' array of the same size as the input data.
03

- Count the occurrence of each value

Iterate through the original data, increase the count of the corresponding index in the count array by 1 for each value in the input data.
04

- Calculate the cumulative count

Update the count array to reflect the actual position of each element in the output array. Iterate through the count array, such that the value at each index i will be the cumulative sum of the values at the previous indices.
05

- Place the elements in the Output array

Iterate through the original data again. For each element, go to the corresponding index in the Count array, decrement that value by 1, and then place the element at that index of the Output array.
06

- Copy the Output array to the original data

Finally, copy the sorted elements from the Output array back to the original data array. Now the original data array is sorted in increasing order.

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Most popular questions from this chapter

Show that there are 2 , nodes with depth \(j\) for \(j

Implement the Insertion Sort algorithm (Algorithm 7.1), run it on your system, and study its best-case, average-case, and worst-case time complexities using several problem instances.

Give two instances for which the Mergesort algorithm is the most appropriate choice.

Modify Heapsort so that it stops after it finds the \(k\) largest keys in nonincreasing order. Analyze your algorithm, and show the results using order notation.

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.
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