Merge Sort Algorithm in DSA Go - 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.

How does splitting and merging affect the total work done?

Scenario Under Consideration

Analyze the time complexity of the following code snippet.

func mergeSort(arr []int) []int {
    if len(arr) <= 1 {
        return arr
    }
    mid := len(arr) / 2
    left := mergeSort(arr[:mid])
    right := mergeSort(arr[mid:])
    return merge(left, right)
}

func merge(left, right []int) []int {
    result := []int{}
    i, j := 0, 0
    for i < len(left) && j < len(right) {
        if left[i] <= right[j] {
            result = append(result, left[i])
            i++
        } else {
            result = append(result, right[j])
            j++
        }
    }
    result = append(result, left[i:]...)
    result = append(result, right[j:]...)
    return result
}

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

Identify Repeating Operations

Identify the loops, recursion, array traversals that repeat.

• Primary operation: Recursive splitting and merging of arrays.
• How many times: The array is split about log₂(n) times, and merging happens at each level over all elements.

How Execution Grows With Input

Each split divides the array in half, creating about log₂(n) levels. At each level, merging all parts touches all n elements.

Input Size (n)	Approx. Operations
10	About 10 * 4 = 40
100	About 100 * 7 = 700
1000	About 1000 * 10 = 10,000

Pattern observation: Operations grow roughly as n times log n, which grows faster than n but much slower than n squared.

Final Time Complexity

Time Complexity: O(n log n)

This means the time grows a bit faster than the size of the input but much better than simple sorting methods that check every pair.

Common Mistake

[X] Wrong: "Merge sort is just a simple loop over the array, so it must be O(n)."

[OK] Correct: Merge sort uses recursion and splits the array multiple times, so it does more work than a single loop.

Interview Connect

Understanding merge sort's time complexity helps you explain efficient sorting and recursion, a key skill in many 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. Compare each element with every other element and swap if needed

B. Pick the smallest element and place it at the beginning repeatedly

C. Divide the list into halves, sort each half, then merge them

D. Use a pivot to partition the list and sort partitions recursively

Solution

1. Step 1: Understand the divide and conquer approach

   Merge Sort splits the list into two halves recursively until single elements remain.

2. Step 2: Merge sorted halves

   It then merges these halves back together in sorted order.

3. Final Answer:

   Divide the list into halves, sort each half, then merge them -> Option C

4. Quick Check:

   Merge Sort = Divide and Conquer [OK]

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

Common Mistakes:

• Confusing Merge Sort with Bubble Sort
• Thinking Merge Sort uses a pivot like Quick Sort
• Believing Merge Sort sorts in place without extra space

2. Which of the following is the correct way to declare a function for merge sort in Go?

easy

A. func mergeSort(arr []int) []int {}

B. func mergeSort(arr int[]) int[] {}

C. func mergeSort(arr array of int) array of int {}

D. func mergeSort(arr []int) int {}

Solution

1. Step 1: Check Go function syntax for slices

   In Go, slices are declared as []int for integer slices.

2. Step 2: Verify return type matches input type

   Merge sort returns a sorted slice, so return type is also []int.

3. Final Answer:

   func mergeSort(arr []int) []int {} -> Option A

4. Quick Check:

   Go slice syntax = []int [OK]

Hint: Go slices use []type syntax, not array of type [OK]

Common Mistakes:

• Using array syntax like int[] which is invalid in Go
• Returning int instead of []int for sorted slice
• Using incorrect parameter types like 'array of int'

3. What will be the output of the following Go code snippet using merge sort?

package main
import "fmt"

func mergeSort(arr []int) []int {
    if len(arr) <= 1 {
        return arr
    }
    mid := len(arr) / 2
    left := mergeSort(arr[:mid])
    right := mergeSort(arr[mid:])
    return merge(left, right)
}

func merge(left, right []int) []int {
    result := []int{}
    i, j := 0, 0
    for i < len(left) && j < len(right) {
        if left[i] < right[j] {
            result = append(result, left[i])
            i++
        } else {
            result = append(result, right[j])
            j++
        }
    }
    result = append(result, left[i:]...)
    result = append(result, right[j:]...)
    return result
}

func main() {
    arr := []int{5, 3, 8, 4, 2}
    sorted := mergeSort(arr)
    fmt.Println(sorted)
}

medium

A. [5 3 8 4 2]

B. [2 3 4 5 8]

C. [8 5 4 3 2]

D. [2 3 5 4 8]

Solution

1. Step 1: Trace the recursive splitting

   The array splits into halves until single elements: [5,3,8,4,2] -> [5,3] and [8,4,2], then further splits.

2. Step 2: Merge sorted halves step-by-step

   Merge [5] and [3] -> [3,5]; merge [8] and [4,2] -> [2,4,8]; finally merge [3,5] and [2,4,8] -> [2,3,4,5,8].

3. Final Answer:

   [2 3 4 5 8] -> Option B

4. Quick Check:

   Sorted array = [2 3 4 5 8] [OK]

Hint: Merge halves sorted individually to get full sorted array [OK]

Common Mistakes:

• Assuming original array prints without sorting
• Mixing up merge order causing wrong output
• Forgetting to append remaining elements after loop

4. Identify the bug in this merge function snippet and select the fix:

func merge(left, right []int) []int {
    result := []int{}
    i, j := 0, 0
    for i < len(left) && j < len(right) {
        if left[i] <= right[j] {
            result = append(result, left[i])
            i++
        } else {
            result = append(result, right[j])
            j++
        }
    }
    result = append(result, left[i:]...)
    result = append(result, right[i:]...)
    return result
}

medium

A. Remove the last two append statements

B. Change comparison from <= to <

C. Initialize result with length of left + right

D. Change last append from right[i:] to right[j:]

Solution

1. Step 1: Analyze the append of remaining elements

   After merging common parts, remaining elements from left[i:] and right[j:] must be appended.

2. Step 2: Spot the incorrect slice used

   The code incorrectly appends right[i:] instead of right[j:], which can cause wrong elements or panic.

3. Final Answer:

   Change last append from right[i:] to right[j:] -> Option D

4. Quick Check:

   Append remaining right elements using j index [OK]

Hint: Use correct index variable when appending leftover elements [OK]

Common Mistakes:

• Using wrong index variable for right slice leftover
• Removing leftover appends causing incomplete merge
• Changing comparison operator unnecessarily

5. You want to sort a large list of 1 million integers using merge sort in Go. Which approach is best to optimize memory usage and performance?

hard

A. Use a bottom-up iterative merge sort to avoid deep recursion

B. Use recursive merge sort with slices copied at each step

C. Use merge sort but create new slices for every merge step

D. Use bubble sort instead for simplicity

Solution

1. Step 1: Understand recursion overhead in large data

   Recursive merge sort can cause deep call stacks and many slice copies, increasing memory use.

2. Step 2: Bottom-up iterative merge sort reduces recursion

   Bottom-up merge sort merges pairs iteratively, reducing recursion and memory overhead.

3. Final Answer:

   Use a bottom-up iterative merge sort to avoid deep recursion -> Option A

4. Quick Check:

   Iterative merge sort = better memory for large data [OK]

Hint: Iterative merge sort saves stack and memory for big lists [OK]

Common Mistakes:

• Ignoring recursion limits on large inputs
• Copying slices unnecessarily increasing memory
• Choosing inefficient sorts like bubble sort for big data