Let P be a mergesort program to sort numbers in ascendinng order on a unknown data structure which take O(n^2) time to find the mid element, rest property is unknown. Then recurrence relation for the same is ? code example

Example 1: merge sort c#

using System;
using System.Collections.Generic;
using System.Linq;
using System.Text;

namespace Merge_sort
{    
    class Program
    {
        static void Main(string[] args)
        {
            List<int> unsorted = new List<int>();
            List<int> sorted;

            Random random = new Random();

            Console.WriteLine("Original array elements:" );
            for(int i = 0; i< 10;i++){
                unsorted.Add(random.Next(0,100));
                Console.Write(unsorted[i]+" ");
            }
            Console.WriteLine();

            sorted = MergeSort(unsorted);

            Console.WriteLine("Sorted array elements: ");
            foreach (int x in sorted)
            {
                Console.Write(x+" ");
            }
			Console.Write("\n");
        }
		

        private static List<int> MergeSort(List<int> unsorted)
        {
            if (unsorted.Count <= 1)
                return unsorted;

            List<int> left = new List<int>();
            List<int> right = new List<int>();

            int middle = unsorted.Count / 2;
            for (int i = 0; i < middle;i++)  //Dividing the unsorted list
            {
                left.Add(unsorted[i]);
            }
            for (int i = middle; i < unsorted.Count; i++)
            {
                right.Add(unsorted[i]);
            }

            left = MergeSort(left);
            right = MergeSort(right);
            return Merge(left, right);
        }

        private static List<int> Merge(List<int> left, List<int> right)
        {
            List<int> result = new List<int>();

            while(left.Count > 0 || right.Count>0)
            {
                if (left.Count > 0 && right.Count > 0)
                {
                    if (left.First() <= right.First())  //Comparing First two elements to see which is smaller
                    {
                        result.Add(left.First());
                        left.Remove(left.First());      //Rest of the list minus the first element
                    }
                    else
                    {
                        result.Add(right.First());
                        right.Remove(right.First());
                    }
                }
                else if(left.Count>0)
                {
                    result.Add(left.First());
                    left.Remove(left.First());
                }
                else if (right.Count > 0)
                {
                    result.Add(right.First());

                    right.Remove(right.First());    
                }    
            }
            return result;
        }
    }
}

Example 2: merge sort

// @see https://www.youtube.com/watch?v=es2T6KY45cA&vl=en
// @see https://www.cs.usfca.edu/~galles/visualization/ComparisonSort.html

function merge(list, start, midpoint, end) {
    const left = list.slice(start, midpoint);
    const right = list.slice(midpoint, end);
    for (let topLeft = 0, topRight = 0, i = start; i < end; i += 1) {
        if (topLeft >= left.length) {
            list[i] = right[topRight++];
        } else if (topRight >= right.length) {
            list[i] = left[topLeft++];
        } else if (left[topLeft] < right[topRight]) {
            list[i] = left[topLeft++];
        } else {
            list[i] = right[topRight++];
        }
    }
}

function mergesort(list, start = 0, end = undefined) {
    if (end === undefined) {
        end = list.length;
    }
    if (end - start > 1) {
        const midpoint = ((end + start) / 2) >> 0;
        mergesort(list, start, midpoint);
        mergesort(list, midpoint, end);
        merge(list, start, midpoint, end);
    }
    return list;
}

mergesort([4, 7, 2, 6, 4, 1, 8, 3]);