Question

Write a \(\Theta(n)\) algorithm that sorts \(n\) distinct integers, ranging in size between 1 and \(k n\) inclusive, where \(k\) is a constant positive integer. (Hint: Use a kn-element array.)

Short answer

The algorithm that fulfils the requirements is basically the counting sort algorithm. Here we initialize a kn-element array to hold the count of each number. After recording the frequencies of numbers in the input, we calculate cumulative frequencies. Then, we traverse the input from end to start, and for each element, we use its count as its sorted position. After placing all elements in the sorted sequence, we copy it back to the original array.

Step-by-step solution

Initialize Counting Array

Allocate an array \(Count[kn+1]\). This array will be used to keep track of the count of each number in the input. Initialize all the counts in \(Count[kn+1]\) to 0.

Calculate Frequencies

For each number \(x\) in the input array, increment \(Count[x]\). After this step, \(Count[x]\) will hold the frequency of \(x\) in the input.

Calculate Cumulative Frequencies

For each index \(i\) from 1 to \(kn\), set \(Count[i] = Count[i] + Count[i-1]\). After this step, \(Count[x]\) will hold the count of numbers less than or equal to \(x\) in the input.

Rearrange The Input

Allocate a temporary array \(Temp[n]\). Starting from the last element of the input, use \(Count[input[i]]\) as the position of \(input[i]\) in \(Temp\) and decrement \(Count[input[i]]\) by 1. After this step, \(Temp\) will hold the sorted sequence of the input.

Copy The Sorted Sequence

Finally, copy the sorted sequence from \(Temp\) back to the input array.

Key concepts

Counting Sort

Counting sort is a non-comparative sorting algorithm that operates by counting the number of objects that possess a distinct key value. It's particularly efficient when the range of values to sort (\between 1 and k) is not significantly greater than the number of objects to sort (n).

Let's walk through the algorithm. First, count how many times each distinct value appears in the input array. Next, create a cumulative frequency array to determine the position of each integer in the sorted array. Finally, sort the values by placing them into a new array based on their determined positions. Unlike comparison-based algorithms like quicksort or mergesort, counting sort can achieve linear time complexity, which is very efficient. It's important to note that counting sort is best suited for sorting integers. For this exercise, where the integers are distinct and range between 1 and kn, counting sort works exceptionally well, adhering to the \(\Theta(n)\) time complexity guideline.

Algorithm Complexity

Algorithm complexity is a measure of the computational resources required by an algorithm to complete its task, typically time and space. Time complexity deals with the amount of time an algorithm takes as the input size grows, usually expressed using Big O notation. To meet the \(\Theta(n)\) time complexity, we need an algorithm that can sort in linear time relative to the input.

The counting sort algorithm accomplishes this by using additional space to store counts and accumulate frequencies, thus avoiding the need for nested iterations typical of \(O(n^2)\) algorithms, like bubble sort or insertion sort. In essence, counting sort exchanges extra space for time, which is why its space complexity is \(O(nk)\). Moreover, since we assume that \(k\) is a constant, the space complexity can also be seen as linear.

Sorting Distinct Integers

Sorting a set of distinct integers can be optimized since you know each number will appear at most once in the sorted array. This is an excellent fit for counting sort because it simplifies the algorithm—you don't need to manage collisions or track multiple occurrences of the same value.

The assumption of distinct integers allows us to prepare a counting array of size \(kn+1\), anticipating that the highest possible integer does not exceed \(kn\). It simplifies the tracking process and aids in achieving the desired linear \(\Theta(n)\) complexity. This efficiency arises from knowing in advance that what we are sorting is not just any integers but a unique set distinguished by certain constraints (range and uniqueness).

Cumulative Frequency

Cumulative frequency is a pivotal concept in counting sort, representing the number of data points that are less than or equal to a certain value. Within the context of counting sort, once the initial counting of elements is complete, cumulative frequencies are computed to determine the exact position of each integer in the final sorted array.

After populating the counting array with the frequencies of each distinct number, we sequentially add each count to the sum of the previous counts. This process transforms the count array into a cumulative frequency array. It effectively maps each input integer to its correct position in the sorted array, by determining how many elements are smaller or equal. When the cumulative frequency is coupled with a temporary array to hold sorted values, you can achieve sorting in linear time, while still preserving the order and uniqueness of the original elements.