Question

Write a linear-time sorting algorithm that sorts a permutation of integers 1 through \(n\), inclusive.

Short answer

Use a variant of the Counting sort algorithm that requires no additional space. Iterate over the elements, for each element, find where it should be placed in the sorted version, and swap it with the element present there. Keep swapping until the element is in the correct place, then move to the next one.

Step-by-step solution

Understand the nature of input

Note that the given input is a permutation of numbers from 1 to \(n\), not just any random sequence of integers. It simply means that each element from 1 to \(n\) is present exactly once, only that their order is shuffled.

Decide on sorting algorithm

There are many sorting algorithms, but not all offer linear time complexity. However, given the nature of our input, one that offers linear time complexity is the Counting Sort, but it usually requires additional space. Fortunately, for this exercise, a variant of counting sort can be employed that does not need any additional space.

Describe the sorting algorithm

Step through each element in the array. Let the current element be \(arr[i]\), find the index that this element is supposed to be placed in the sorted version of the array, i.e, index \(arr[i]-1\). Then swap \arr[i] with \arr[arr[i]-1]. Repeat this process for the same \arr[i] until \arr[i] is at its right place. Repeat this for the next \arr[i+1], \arr[i+2]..., \arr[n] till the entire array is sorted.

Write Algorithm

1. For i = 0 to i = n-1; loop\n 2. While (arr[i] != i+1); loop\n a. Swap \arr[i] and \arr[\arr[i] - 1]

Key concepts

Counting Sort

Counting Sort is a highly efficient sorting algorithm that works best when you have a known range of integer values. It operates on the principle of counting the occurrences of each unique element in the input and then using this count to determine the position of each element in a sorted array.

Instead of comparing elements against one another, Counting Sort creates a count array (or a frequency array) to tally how many times each value appears. Then, it calculates the starting index for each value based on the cumulative sum of counts. Finally, it walks through the input array, placing each element in its correct sorted position as per the count array. This method works especially well when sorting a permutation of numbers from 1 through \(n\), since the range is tight and each element appears exactly once, as seen in the exercise.

The main advantage of Counting Sort is its linear-time complexity, \(O(n + k)\), where \(n\) is the number of elements in the array and \(k\) is the range of input values. However, because it requires additional space proportional to the range of data, its space complexity can be a disadvantage in scenarios with large ranges.

Algorithm Complexity

Algorithm complexity is a measure of the efficiency of an algorithm in terms of time taken and space used as a function of the input size. It's categorized into two types: time complexity and space complexity.

Time complexity is a function that gives the number of basic operations done by an algorithm for an input of size \(n\), often noted as \(O(n)\), \(O(n^2)\), etc., where \(O\) denotes the order of the function. In the case of linear-time algorithms like the one used in the exercise for sorting permutations, their time complexity is \(O(n)\), meaning the time it takes to complete the task grows proportionally with the size of the input.

Space complexity, on the other hand, refers to the amount of memory space required by the algorithm to process data as a function of the input size. A sorting algorithm with space complexity \(O(n)\) means that the amount of memory required grows linearly with the size of the input array. Understanding both complexities is crucial for selecting the most appropriate sorting algorithm for a given problem.

Array Permutations

An array permutation refers to an arrangement of all the elements of the array in a particular sequence or order. In mathematical terms, a permutation of a set is any arrangement of its members into a sequence, where each member of the set appears exactly once. This concept is central to the algorithmic exercise, understanding that the input represents a scrambled order of integers from 1 to \(n\).

In sorting problems, knowing that an array is a permutation can significantly simplify the process. For instance, when sorting a permutation of integers 1 through \(n\), we already know what the sorted array should look like—it would be an ascending sequence from 1 to \(n\). Given this information, we can optimize our sorting method to leverage this insight, which is exactly what the Counting Sort variant does by repeatedly swapping elements to their correct positions without the need for additional space for counting.