Question

Write a linear-time algorithm that sorts \(n\) distinct integers ranging from 1 to 500 , inclusive. (Hint: Use a 500 -element array.)

Short answer

The problem is solved using the counting sort algorithm. First, create a counting array of 500 elements all initialized to 0. Then, count the occurrence of all integers in the given array updating the corresponding index in the counting array. Finally, iterate over the counting array and add the index number to the sorted output array as many times as the value from the counting array dictates.

Step-by-step solution

Prepare the Counting Array

First, it's necessary to initialize an array of 500 elements, named counting array. Since indexing starts from 0, the array would have indices running from 0 to 499. All elements in the counting array are initially set to 0.

Count the Occurrences

Next, counting the occurrences of each integer in the given array. For each number \(i\) in the given array, increment the value at the \(i-1^{th}\) index of the counting array by 1.

Construct the Sorted Array

Now, the sorted array can be constructed from the counting array. Starting from the first element of the counting array, and proceeding to the last, for each counting array index \(i\) that has a non-zero value, append \(i+1\) to the output (sorted) array as many times as specified by the value at that index in the counting array.

Key concepts

Counting Sort

Regarded as one of the most intuitive sorting algorithms, Counting Sort operates on a simple principle—counting the occurrences of each unique element within a list and using that information to place them in the correct order.

Imagine you have a handful of red, blue, and green marbles that you want to sort by color. Counting Sort would involve tallying up how many marbles there are of each color, and then arranging the marbles in groups based on these tallies.

For the exercise at hand, dealing with integers ranging from 1 to 500, Counting Sort is especially effective. The approach involves three main steps:

• Initialize the Counting Array: This begins with setting up an array, with a length equal to the highest integer in the range—500 in this case. Each index corresponds to an integer you are sorting, and initially, all counts are set to zero.
• Count Occurrences: Traverse through the input array, and for each integer encountered, increment the count at the corresponding index in the Counting Array.
• Construct Sorted Output: Finally, build the sorted array by appending each integer to a new array the number of times indicated by its count in the Counting Array.

Counting Sort excels not only because it's easy to understand but also due to its efficiency. It's particularly beneficial when sorting a set of items with a known, limited range, as it can render an impressively fast sorting time.

Algorithm Complexity

When assessing an algorithm's efficiency, we delve into Algorithm Complexity, which includes both time complexity—how long an algorithm takes to run, as a function of the input size—and space complexity—how much extra memory it requires.

Typically, time complexity is expressed in 'Big O' notation, like O(n), O(n^2), or O(log n), to provide a high-level idea of the algorithm's performance as the input size grows. For instance, O(n) indicates that the running time increases linearly with the input size. Similarly, space complexity gives us insight into scalability regarding memory consumption.

In the context of the given exercise, Counting Sort has a time complexity of O(n+k), where 'n' is the number of elements to sort, and 'k' is the range of input numbers (here, k=500). Because of the fixed range of inputs, it's possible to sort in linear time, O(n), which is incredibly efficient. As for space complexity, Counting Sort requires additional memory to house the counting array, so it has O(k) space complexity—again, k being the range of the input integers. The key takeaway is that Counting Sort becomes less efficient as the range 'k' increases, especially if 'k' becomes larger than 'n'.

Data Structures

Data Structures are an essential aspect of computer science, providing a means to manage and organize data efficiently. In the context of our sorting exercise, arrays are the chosen data structure, utilized in two significant ways:

• Input Array: The original unsorted collection of integers, where each element represents a particular value within the specified range (1 to 500 in this exercise).
• Counting Array: A specially crafted array that captures the number of times each integer appears in the input array. Instead of storing the raw data (the integers), it stores the meta-data—the counts of each integer.

By exploiting the properties of these data structures, Counting Sort uses the Counting Array to track frequencies, followed by constructing a new, sorted array. The choice of using an array is integral to the algorithm's linear-time performance because arrays allow for quick access (in constant time, O(1)) to read and write operations by index. This rapid index-based access permits Counting Sort to work its magic efficiently. The nuanced application of data structures, like creating a Counting Array in this instance, exemplifies how a proper understanding of data structures can lead to powerful algorithmic solutions.