Question

Write a linear-time sorting algorithm that sorts a permutation of integers 1 through \(n,\) inclusive. (Hint: Use an \(n\) -element array.)

Short answer

The sorting algorithm leverages the nature of the problem - the input array is a permutation of numbers from 1 to \(n\). It iterates through the array, and for each number \(i\), it swaps it with the number at the \(i - 1\) position in the array. This process results in each number being in its correct position in the array, effectively sorting the array.

Step-by-step solution

Understanding the problem

The key to solving this problem is understanding that if we have a permutation of integers from 1 to \(n\), each integer \(i\) can be placed at the index \(i - 1\). This is because array indices start from 0, while our integers start from 1. Hence, we need to place each integer \(i\) at index \(i - 1\). Doing so will sort the array in ascending order.

Implementing the algorithm

In the case of a permutation of \(n\) numbers, we can make a linear pass through the array. Every time we encounter a number \(x\), we swap it with the number that currently resides at position \(x - 1\). We continue doing this until the number \(x\) is at the position \(x - 1\). We repeat this process for each number in the array.

Iterating over the array

We start iterating over the array from the first element (index 0). For each element at position \(i\), if it's not equal to \(i + 1\) (since our array starts at 1, not at 0), we need to move it to its correct position. We accordingly swap this element with the element at its correct position. We repeat this action until the array is sorted.

Validating the solution

After running our algorithm, we can validate if the solution is correct by performing a simple check: iterate over the array and for every index \(i\) the value should equal \(i + 1\). If this condition holds true for all \(i\) from 0 to \(n - 1\), then the array is correctly sorted.

Key concepts

Algorithm Complexity

Understanding the complexity of an algorithm is crucial because it provides insight into how well the algorithm scales with the size of the input data. In computer science, algorithm complexity is categorized as time complexity and space complexity, which refer to the amount of time and storage an algorithm requires, respectively.

The goal is often to find the most efficient algorithm - one that can handle large amounts of data without requiring a proportional increase in processing time or memory.

In the case of the linear-time sorting algorithm for sorting a permutation of integers, the time complexity is linear, denoted as 'O(n)', which means that the time taken to sort the numbers increases in direct proportion to the number of elements. This efficiency is possible because the algorithm takes advantage of the given structure, that is a permutation of integers from 1 to n, hence each element can be mapped directly to a unique position.

Array Indexing

Array indexing is a fundamental concept when it comes to working with arrays, a core data structure used in programming to store collections of elements. Indexing allows access to each element in the array using its position within the array.

The important thing to remember is that arrays are zero-indexed in most programming languages, which means the first element is accessed with an index of 0.

In our specific exercise, the array needs to sort a sequence by leveraging array indexing, placing each number in the position corresponding to its value minus one. This takes advantage of array indexing to perform efficient in-place sorting, and it's vital for the students to understand this concept to properly implement linear-time sorting algorithms.

Sorting Permutation of Integers

Sorting a permutation of integers refers to arranging a shuffled sequence of distinct numbers into a specific order, usually in ascending (increasing) or descending (decreasing) value.

Because we know the sequence contains all integers from 1 to n without repeats, the sorting problem has a unique structure that allows for an optimized solution. Each integer already 'knows' its target position - the integer 3, for example, belongs at index 2. This allows for a more efficient sorting method compared to generic sorting algorithms like quicksort or mergesort, which don't leverage such specific knowledge.

This characteristic makes it possible to design a linear-time algorithm that completes the sorting process in 'O(n)' time, which is a major improvement over the 'O(n log n)' time complexity that is the best possible for comparison-based sorting algorithms on arbitrary lists of numbers. Hence, understanding this property of the sequence is key to developing the most efficient sorting solution.