Merge Sort Algorithm in DSA C++ - Time & Space Complexity

Time Complexity: Merge Sort Algorithm

O(n log n)

Understanding Time Complexity

We want to understand how the time taken by merge sort changes as the list size grows.

Specifically, how many steps does merge sort take to sort a list of numbers?

Scenario Under Consideration

Analyze the time complexity of the following code snippet.

void merge(int arr[], int left, int mid, int right) {
    int n1 = mid - left + 1;
    int n2 = right - mid;
    int L[n1], R[n2];
    for (int i = 0; i < n1; i++) L[i] = arr[left + i];
    for (int j = 0; j < n2; j++) R[j] = arr[mid + 1 + j];
    int i = 0, j = 0, k = left;
    while (i < n1 && j < n2) {
        if (L[i] <= R[j]) arr[k++] = L[i++];
        else arr[k++] = R[j++];
    }
    while (i < n1) arr[k++] = L[i++];
    while (j < n2) arr[k++] = R[j++];
}

void mergeSort(int arr[], int left, int right) {
    if (left < right) {
        int mid = left + (right - left) / 2;
        mergeSort(arr, left, mid);
        mergeSort(arr, mid + 1, right);
        merge(arr, left, mid, right);
    }
}
    

This code splits the array into halves recursively and merges them back in sorted order.

Identify Repeating Operations

Identify the loops, recursion, array traversals that repeat.

• Primary operation: The merge function that combines two sorted halves by comparing elements.
• How many times: Merge is called once for each level of recursion, and recursion depth is about log n.
• Each merge processes all elements in the current segment, so total work per level is proportional to n.

How Execution Grows With Input

As the list size doubles, the number of steps grows a bit more than double but less than square.

Input Size (n)	Approx. Operations
10	About 40 steps
100	About 700 steps
1000	About 10,000 steps

Pattern observation: The steps grow roughly like n times log n, which grows faster than n but slower than n squared.

Final Time Complexity

Time Complexity: O(n log n)

This means if the list size doubles, the time taken grows a bit more than double but much less than square.

Common Mistake

[X] Wrong: "Merge sort is always slower than simple sorting because it uses recursion and extra arrays."

[OK] Correct: Merge sort's smart splitting and merging keep the total steps low, making it faster on large lists than simple methods like bubble sort.

Interview Connect

Understanding merge sort's time complexity shows you can analyze recursive algorithms and compare sorting methods, a key skill in coding interviews.

Self-Check

"What if we changed merge sort to split the array into three parts instead of two? How would the time complexity change?"

Practice

1. What is the main idea behind the Merge Sort Algorithm?

easy

A. Select the smallest element and place it at the beginning

B. Divide the array into halves, sort each half, then merge them

C. Sort the array by repeatedly swapping adjacent elements

D. Build a heap and extract elements in sorted order

Solution

1. Step 1: Understand the divide and conquer approach

   Merge sort splits the array into two halves recursively until single elements remain.

2. Step 2: Merge sorted halves

   It then merges these sorted halves to produce a fully sorted array.

3. Final Answer:

   Divide the array into halves, sort each half, then merge them -> Option B

4. Quick Check:

   Merge Sort = Divide, Sort, Merge [OK]

Hint: Remember: split, sort, then merge [OK]

Common Mistakes:

• Confusing merge sort with bubble sort
• Thinking merge sort swaps adjacent elements
• Mixing merge sort with selection sort

2. Which of the following is the correct way to declare the merge function header in C++ for merge sort?

easy

A. int mergeSort(int arr[], int left, int right);

B. int merge(int arr[], int left, int mid, int right);

C. void mergeSort(int arr[], int left, int right);

D. void merge(int arr[], int left, int mid, int right);

Solution

1. Step 1: Identify the merge function role

   The merge function combines two sorted parts; it does not return a value, so it should be void.

2. Step 2: Check parameter list

   It takes the array and three indices: left, mid, and right to merge subarrays.

3. Final Answer:

   void merge(int arr[], int left, int mid, int right); -> Option D

4. Quick Check:

   Merge function = void with arr and indices [OK]

Hint: Merge combines parts, returns nothing [OK]

Common Mistakes:

• Using int return type for merge
• Confusing merge with mergeSort function
• Wrong parameter order or missing parameters

3. What will be the output of the following code snippet after applying merge sort on the array?

int arr[] = {5, 2, 9, 1};
mergeSort(arr, 0, 3);
for(int i = 0; i < 4; i++) std::cout << arr[i] << " ";

medium

A. 5 2 9 1

B. 9 5 2 1

C. 1 2 5 9

D. 2 1 5 9

Solution

1. Step 1: Apply merge sort on the array

   The array {5, 2, 9, 1} is split and sorted recursively to {1, 2, 5, 9}.

2. Step 2: Print the sorted array

   The for loop prints elements in sorted order: 1 2 5 9.

3. Final Answer:

   1 2 5 9 -> Option C

4. Quick Check:

   Sorted array = 1 2 5 9 [OK]

Hint: Merge sort outputs sorted array ascending [OK]

Common Mistakes:

• Printing original array without sorting
• Confusing merge sort with reverse sort
• Off-by-one errors in indices

4. Identify the error in the following merge sort code snippet:

void merge(int arr[], int left, int mid, int right) {
  int n1 = mid - left;
  int n2 = right - mid;
  int L[n1], R[n2];
  for(int i = 0; i < n1; i++) L[i] = arr[left + i];
  for(int j = 0; j < n2; j++) R[j] = arr[mid + 1 + j];
  // merging logic...
}

medium

A. Incorrect calculation of n1 and n2 sizes

B. Wrong indexing in copying elements to L array

C. Using variable-length arrays which is invalid in C++

D. Missing return statement in void function

Solution

1. Step 1: Check size calculations

   n1 = mid - left is missing +1 (should be mid - left + 1) to include the mid element; n2 = right - mid is correct.

2. Step 2: Check copying to L array

   Copying starts at left + i for i < n1, which goes up to mid-1, missing arr[mid].

3. Step 3: Identify the main error

   The error causes wrong indexing/range when copying elements to L array.

4. Final Answer:

   Incorrect calculation of n1 and n2 sizes -> Option A

5. Quick Check:

   Subarray size calculation off by one [OK]

Hint: Check array copy indices carefully [OK]

Common Mistakes:

• Off-by-one errors in subarray sizes
• Using variable-length arrays in C++
• Ignoring void function return type

5. Given an array with many duplicate elements, how can merge sort be adapted to maintain the relative order of equal elements (stable sort)?

hard

A. Always pick the element from the left subarray first when merging equals

B. Always pick the element from the right subarray first when merging equals

C. Swap equal elements randomly during merge

D. Use quicksort instead of merge sort

Solution

1. Step 1: Understand stability in sorting

   A stable sort keeps equal elements in their original relative order.

2. Step 2: Apply stability in merge step

   When merging, if elements are equal, always choose the element from the left subarray first to keep order.

3. Final Answer:

   Always pick the element from the left subarray first when merging equals -> Option A

4. Quick Check:

   Stable merge = left element first on equals [OK]

Hint: Pick left element first on ties to keep order [OK]

Common Mistakes:

• Picking right element first breaks stability
• Random swaps destroy order
• Confusing merge sort with quicksort stability