Merge Sort Algorithm

Merge sort works as follows:

1. Divide the unsorted list into n sublists, each containing one element. A list of one element is sorted by definition.
2. Repeatedly merge sorted sublists (called "runs") to produce longer runs until there is only one run remaining. This is the sorted list.

   

   Pseudocode

   This algorithm makes use of a variable length temporary array, which is most easily represented in C++ using the vector class from the standard library. When implementing the algorithm, include the following code at the top of the source file:

   #include <vector>
   
   using std::vector;
   

   
   

   procedure merge_sort(array : list of sortable items, start : first element of list, 
   end : last element of list)
       if start < end
           mid ← (start + end) / 2
       
           merge_sort(array, start, mid)
           merge_sort(array, mid + 1, end)
       
           merge(array, start, mid, end)
       end if
   end procedure

   
   

   procedure merge(array : list of items to merge, start : first element of first sublist, mid : last element of first sublist,
     end : last element of second sublist)
       
       // Temporary vector of size (end - start + 1)
       vector<int> temp(end - start + 1);
       
       i ← start
       j ← mid + 1
       k ← 0
       
       while i <= mid and j <= end
           if array[i] < array[j]
               temp[k] ← array[i]
               i ← i + 1
           else
               temp[k] ← array[j]
               j ← j + 1
           end if
           k ← k + 1
       end while
       
       while i <= mid
           temp[k] ← array[i]
           i ← i + 1
           k ← k + 1
       end while
       
       while j <= end
           temp[k] ← array[j]
           j ← j + 1
           k ← k + 1
       end while
       
       Copy the elements of the vector temp back into array
       
   end procedure

   Example

   In this example, the merge sort merge() function is called with the arguments array (the array to partition), 0 (the subscript of the first element of the array), and 11 (the subscript of the last element of the array).

   Begin merge

   i ← start
   j ← mid + 1
   k ← 0
   

   
   

   	[0]	[1]	[2]	[3]	[4]	[5]	[6]	[7]	[8]	[9]	[10]	[11]
   array	13	22	31	48	67	81	14	45	58	72	79	93
   	start i					mid	j					end

   [0]

   [1]

   [2]

   [3]

   [4]

   [5]

   [6]

   [7]

   [8]

   [9]

   [10]

   [11]

   temp

   k

   
   

   i <= mid and j <= end; loop entered. array[i] < array[j]

   temp[k] ← array[i]
   i ← i + 1
   k ← k + 1
   

   
   

   	[0]	[1]	[2]	[3]	[4]	[5]	[6]	[7]	[8]	[9]	[10]	[11]
   array	13	22	31	48	67	81	14	45	58	72	79	93
   	start	i				mid	j					end

   [0]

   [1]

   [2]

   [3]

   [4]

   [5]

   [6]

   [7]

   [8]

   [9]

   [10]

   [11]

   temp

   13

   k

   
   

   i <= mid and j <= end; loop continues. array[i] not < array[j]

   temp[k] ← array[j]
   j ← j + 1
   k ← k + 1
   

   
   

   	[0]	[1]	[2]	[3]	[4]	[5]	[6]	[7]	[8]	[9]	[10]	[11]
   array	13	22	31	48	67	81	14	45	58	72	79	93
   	start	i				mid		j				end

   [0]

   [1]

   [2]

   [3]

   [4]

   [5]

   [6]

   [7]

   [8]

   [9]

   [10]

   [11]

   temp

   13

   14

   k

   
   

   i <= mid and j <= end; loop continues. array[i] < array[j]

   temp[k] ← array[i]
   i ← i + 1
   k ← k + 1
   

   
   

   	[0]	[1]	[2]	[3]	[4]	[5]	[6]	[7]	[8]	[9]	[10]	[11]
   array	13	22	31	48	67	81	14	45	58	72	79	93
   	start		i			mid		j				end

   [0]

   [1]

   [2]

   [3]

   [4]

   [5]

   [6]

   [7]

   [8]

   [9]

   [10]

   [11]

   temp

   13

   14

   22

   k

   
   

   i <= mid and j <= end; loop continues. array[i] < array[j]

   temp[k] ← array[i]
   i ← i + 1
   k ← k + 1
   

   
   

   	[0]	[1]	[2]	[3]	[4]	[5]	[6]	[7]	[8]	[9]	[10]	[11]
   array	13	22	31	48	67	81	14	45	58	72	79	93
   	start			i		mid		j				end

   [0]

   [1]

   [2]

   [3]

   [4]

   [5]

   [6]

   [7]

   [8]

   [9]

   [10]

   [11]

   temp

   13

   14

   22

   31

   k

   
   

   i <= mid and j <= end; loop continues. array[i] not < array[j]

   temp[k] ← array[j]
   j ← j + 1
   k ← k + 1
   

   
   

   	[0]	[1]	[2]	[3]	[4]	[5]	[6]	[7]	[8]	[9]	[10]	[11]
   array	13	22	31	48	67	81	14	45	58	72	79	93
   	start			i		mid			j			end

   [0]

   [1]

   [2]

   [3]

   [4]

   [5]

   [6]

   [7]

   [8]

   [9]

   [10]

   [11]

   temp

   13

   14

   22

   31

   45

   k

   
   

   i <= mid and j <= end; loop continues. array[i] < array[j]

   temp[k] ← array[i]
   i ← i + 1
   k ← k + 1
   

   
   

   	[0]	[1]	[2]	[3]	[4]	[5]	[6]	[7]	[8]	[9]	[10]	[11]
   array	13	22	31	48	67	81	14	45	58	72	79	93
   	start				i	mid			j			end

   [0]

   [1]

   [2]

   [3]

   [4]

   [5]

   [6]

   [7]

   [8]

   [9]

   [10]

   [11]

   temp

   13

   14

   22

   31

   45

   48

   k

   
   

   i <= mid and j <= end; loop continues. array[i] not < array[j]

   temp[k] ← array[j]
   j ← j + 1
   k ← k + 1
   

   
   

   	[0]	[1]	[2]	[3]	[4]	[5]	[6]	[7]	[8]	[9]	[10]	[11]
   array	13	22	31	48	67	81	14	45	58	72	79	93
   	start				i	mid				j		end

   [0]

   [1]

   [2]

   [3]

   [4]

   [5]

   [6]

   [7]

   [8]

   [9]

   [10]

   [11]

   temp

   13

   14

   22

   31

   45

   48

   58

   k

   
   

   i <= mid and j <= end; loop continues. array[i] < array[j]

   temp[k] ← array[i]
   i ← i + 1
   k ← k + 1
   

   
   

   	[0]	[1]	[2]	[3]	[4]	[5]	[6]	[7]	[8]	[9]	[10]	[11]
   array	13	22	31	48	67	81	14	45	58	72	79	93
   	start					mid i				j		end

   [0]

   [1]

   [2]

   [3]

   [4]

   [5]

   [6]

   [7]

   [8]

   [9]

   [10]

   [11]

   temp

   13

   14

   22

   31

   45

   48

   58

   67

   k

   
   

   i <= mid and j <= end; loop continues. array[i] < not array[j]

   temp[k] ← array[j]
   j ← j + 1
   k ← k + 1
   

   
   

   	[0]	[1]	[2]	[3]	[4]	[5]	[6]	[7]	[8]	[9]	[10]	[11]
   array	13	22	31	48	67	81	14	45	58	72	79	93
   	start					mid i					j	end

   [0]

   [1]

   [2]

   [3]

   [4]

   [5]

   [6]

   [7]

   [8]

   [9]

   [10]

   [11]

   temp

   13

   14

   22

   31

   45

   48

   58

   67

   72

   k

   
   

   i <= mid and j <= end; loop continues. array[i] < not array[j]

   temp[k] ← array[j]
   j ← j + 1
   k ← k + 1
   

   
   

   	[0]	[1]	[2]	[3]	[4]	[5]	[6]	[7]	[8]	[9]	[10]	[11]
   array	13	22	31	48	67	81	14	45	58	72	79	93
   	start					mid i						end j

   	[0]	[1]	[2]	[3]	[4]	[5]	[6]	[7]	[8]	[9]	[10]	[11]
   temp	13	14	22	31	45	48	58	67	72	79
   											k

   
   

   i <= mid and j <= end; loop continues. array[i] < array[j]

   temp[k] ← array[i]
   i ← i + 1
   k ← k + 1
   

   
   

   	[0]	[1]	[2]	[3]	[4]	[5]	[6]	[7]	[8]	[9]	[10]	[11]
   array	13	22	31	48	67	81	14	45	58	72	79	93
   	start					mid	i					end j

   	[0]	[1]	[2]	[3]	[4]	[5]	[6]	[7]	[8]	[9]	[10]	[11]
   temp	13	14	22	31	45	48	58	67	72	79	81
   												k

   
   

   i not <= mid; loop ends. i not <= mid; loop not entered. j <= end; loop entered.

   temp[k] ← array[j]
   j ← j + 1
   k ← k + 1
   

   
   

   	[0]	[1]	[2]	[3]	[4]	[5]	[6]	[7]	[8]	[9]	[10]	[11]
   array	13	22	31	48	67	81	14	45	58	72	79	93
   	start					mid i						end	j

   	[0]	[1]	[2]	[3]	[4]	[5]	[6]	[7]	[8]	[9]	[10]	[11]
   temp	13	14	22	31	45	48	58	67	72	79	81	93
   													k

   Complexity

   Time Complexity: O(n log n)

   Space Complexity: O(n)