Step by Step solution about How to resolve the algorithm Sorting Algorithms/Circle Sort step by step in the C# programming language

Overview:

The provided C# program implements a "Circle Sort" algorithm to sort an input array of integers in ascending order. The algorithm is a variation of the Merge Sort algorithm, optimized for circular arrays, where the last element wraps around to the beginning.

CircleSort Method:

• Accepts an array of integers as input.
• Repeatedly calls the CircleSortR method to sort the array until no swaps are needed.
• Returns the sorted array.

CircleSortR Method (Recursive):

• Accepts the input array, start index, end index, and number of swaps.
• Recursively divides the array into two halves and sorts them separately.
• Merges the sorted halves and counts the number of swaps during merging.
• Returns the updated number of swaps.

Implementation Details:

• The CircleSortR method uses a divide-and-conquer approach to sort the array in a circular fashion.
• It finds the middle index mid for each recursive call.
• The first while loop sorts the left half of the array (from low to low + mid) and the right half (from low + mid + 1 to high).
• If the element at the merge point (arr[lo]) is greater than the element to its right (arr[hi + 1]), it swaps them.
• The numSwaps variable keeps track of the number of swaps performed during each recursive call.
• After sorting the left and right halves, the method recursively calls itself on those halves to complete the merging process.

Example Usage:

In the Main method, three test cases are used to demonstrate the sorting algorithm:

• { 6, 7, 8, 9, 2, 5, 3, 4, 1 }
• { 2, 14, 4, 6, 8, 1, 3, 5, 7, 11, 0, 13, 12, -1 }
• { 2, 3, 3, 5, 5, 1, 1, 7, 7, 6, 6, 4, 4, 0, 0 }

The sorted arrays are printed to the console.

Time Complexity:

• The algorithm resembles the Merge Sort, which has a time complexity of O(n log n) in the average and worst cases.

Advantages:

• Optimized for circular arrays, where the end wraps around to the beginning.
• Relatively simple and efficient implementation.

Disadvantages:

• Not as efficient as other sorting algorithms (e.g., Quick Sort, Heap Sort) for very large arrays.

using System;
using System.Linq;

namespace CircleSort
{
    internal class Program
    {
        public static int[] CircleSort(int[] array)
        {
            if (array.Length > 0)
                while (CircleSortR(array, 0, array.Length - 1, 0) != 0)
                    continue;
            return array;
        }

        private static int CircleSortR(int[] arr, int lo, int hi, int numSwaps)
        {
            if (lo == hi)
                return numSwaps;

            var high = hi;
            var low = lo;
            var mid = (hi - lo) / 2;

            while (lo < hi)
            {
                if (arr[lo] > arr[hi])
                {
                    (arr[lo], arr[hi]) = (arr[hi], arr[lo]);
                    numSwaps++;
                }
                lo++;
                hi--;
            }

            if (lo == hi && arr[lo] > arr[hi + 1])
            {
                (arr[lo], arr[hi + 1]) = (arr[hi + 1], arr[lo]);
                numSwaps++;
            }

            numSwaps = CircleSortR(arr, low, low + mid, numSwaps);
            numSwaps = CircleSortR(arr, low + mid + 1, high, numSwaps);

            return numSwaps;
        }

        private static void Main(string[] args)
        {
            var sortedArray = CircleSort(new int[] { 6, 7, 8, 9, 2, 5, 3, 4, 1 });
            sortedArray.ToList().ForEach(i => Console.Write(i.ToString() + " "));
            Console.WriteLine();
            var sortedArray2 = CircleSort(new int[] { 2, 14, 4, 6, 8, 1, 3, 5, 7, 11, 0, 13, 12, -1 });
            sortedArray2.ToList().ForEach(i => Console.Write(i.ToString() + " "));
            Console.WriteLine();
            var sortedArray3 = CircleSort(new int[] { 2, 3, 3, 5, 5, 1, 1, 7, 7, 6, 6, 4, 4, 0, 0 });
            sortedArray3.ToList().ForEach(i => Console.Write(i.ToString() + " "));
            Console.ReadKey();
        }
    }
}