Merge Two Sorted Arrays Without Extra Space in DSA Python - Cheat Sheet & Quick Revision

Recall & Review

beginner

What does it mean to merge two sorted arrays without extra space?

It means combining two sorted arrays into one sorted sequence without using any additional array or storage space beyond the given arrays.

beginner

Why can't we simply concatenate two sorted arrays and sort them to merge without extra space?

Concatenating and sorting requires extra space or time. The goal is to merge efficiently without extra space and ideally in linear or near-linear time.

intermediate

What is the 'gap method' used in merging two sorted arrays without extra space?

The gap method compares elements at a certain gap distance and swaps them if out of order, reducing the gap gradually until it becomes 1, resulting in a merged sorted sequence.

intermediate

In the gap method, how is the gap value updated in each iteration?

The gap is updated by dividing it by 2 and rounding up (gap = ceil(gap / 2)) until it becomes 1.

intermediate

What is the time complexity of merging two sorted arrays without extra space using the gap method?

The time complexity is approximately O((n + m) log(n + m)), where n and m are the sizes of the two arrays.

What is the main challenge when merging two sorted arrays without extra space?

A Reversing the arrays

B Sorting each array separately

C Using recursion

D Avoiding use of extra storage space

The main challenge is to merge without using any extra storage space beyond the given arrays.

Which method is commonly used to merge two sorted arrays without extra space?

A Gap method

B Bubble sort

C Quick sort

D Insertion sort

The gap method efficiently merges two sorted arrays without extra space.

How is the gap value changed in the gap method during merging?

A Decreased by half and rounded up

B Increased by one

C Kept constant

D Doubled each time

The gap is decreased by half and rounded up each iteration until it reaches 1.

What happens when two elements compared at the gap distance are out of order?

A The gap is reset

B They are swapped

C They are ignored

D The arrays are reversed

If elements are out of order, they are swapped to move towards a sorted sequence.

What is the approximate time complexity of the gap method for merging two arrays of sizes n and m?

A O(log(n + m))

B O(n * m)

C O((n + m) log(n + m))

D O(n + m)

The gap method runs in about O((n + m) log(n + m)) time.

Explain the gap method for merging two sorted arrays without extra space.

Think about how the gap shrinks and elements are swapped to sort.

Describe why merging two sorted arrays without extra space is useful and what challenges it solves.

Consider situations with limited memory or large data.

Practice

1. What is the main goal when merging two sorted arrays without extra space?

easy

A. Combine both arrays into one sorted array without using extra memory

B. Sort each array separately using extra space

C. Copy elements from one array to another without sorting

D. Create a new array to hold merged elements

Solution

1. Step 1: Understand the problem requirement

   The problem asks to merge two sorted arrays into a single sorted sequence without using extra space, meaning no new arrays can be created.

2. Step 2: Identify the correct approach

   We must rearrange elements within the given arrays to maintain sorted order without additional memory.

3. Final Answer:

   Combine both arrays into one sorted array without using extra memory -> Option A

4. Quick Check:

   Merge without extra space = Combine in-place [OK]

Hint: Merge by swapping elements in place without extra arrays [OK]

Common Mistakes:

• Trying to create a new array for merged result
• Sorting arrays separately without merging
• Ignoring the no extra space constraint

2. Which of the following Python code snippets correctly swaps elements between two arrays arr1 and arr2 without extra space?

easy

A. arr1[i] = arr1[i] + arr2[j]; arr1[i] = arr1[i] - arr2[j]; arr2[j] = arr1[i]

B. arr1[i] = arr2[j]; arr2[j] = arr1[i]

C. temp = arr1[i]; arr1[i] = arr2[j]; arr2[j] = temp

D. arr1[i], arr2[j] = arr2[j], arr1[i]

Solution

1. Step 1: Check swapping syntax in Python

   Python allows swapping two variables in one line using tuple unpacking: a, b = b, a.

2. Step 2: Identify correct swap without extra variable

   arr1[i], arr2[j] = arr2[j], arr1[i] uses this syntax correctly to swap arr1[i] and arr2[j] without extra space.

3. Final Answer:

   arr1[i], arr2[j] = arr2[j], arr1[i] -> Option D

4. Quick Check:

   Python swap syntax = arr1[i], arr2[j] = arr2[j], arr1[i] [OK]

Hint: Use Python tuple unpacking for swap: a, b = b, a [OK]

Common Mistakes:

• Assigning arr1[i] before arr2[j] loses original value
• Using temp variable unnecessarily
• Using arithmetic swap which can cause errors

3. What will be the content of arr1 and arr2 after running the following code?

arr1 = [1, 5, 9, 10, 15, 20]
arr2 = [2, 3, 8, 13]
i = 5
j = 0
while i >= 0 and j < len(arr2):
    if arr1[i] > arr2[j]:
        arr1[i], arr2[j] = arr2[j], arr1[i]
        i -= 1
        j += 1
    else:
        i -= 1
arr1.sort()
arr2.sort()
print(arr1)
print(arr2)

medium

A. [1, 5, 8, 10, 13, 15] and [2, 3, 9, 20]

B. [1, 2, 3, 5, 8, 9] and [10, 13, 15, 20]

C. [1, 5, 9, 8, 3, 2] and [20, 15, 10, 13]

D. [1, 3, 5, 8, 9, 10] and [2, 13, 15, 20]

Solution

1. Step 1: Trace the while loop swaps

   Start with i=5 (arr1[5]=20), j=0 (arr2[0]=2). Since 20 > 2, swap: arr1[5]=2, arr2[0]=20; i=4, j=1.
   Next i=4 (15), j=1 (3): 15 > 3, swap: arr1[4]=3, arr2[1]=15; i=3, j=2.
   i=3 (10), j=2 (8): 10 > 8, swap: arr1[3]=8, arr2[2]=10; i=2, j=3.
   i=2 (9), j=3 (13): 9 > 13? No, so i=1.
   i=1 (5), j=3 (13): 5 > 13? No, so i=0.
   i=0 (1), j=3 (13): 1 > 13? No, so i=-1 loop ends.
   After swaps: arr1 = [1, 5, 9, 8, 3, 2], arr2 = [20, 15, 10, 13].

2. Step 2: Sort both arrays after swapping

   The code sorts both arrays:
   arr1.sort() -> [1, 2, 3, 5, 8, 9]
   arr2.sort() -> [10, 13, 15, 20]

3. Final Answer:

   [1, 2, 3, 5, 8, 9] and [10, 13, 15, 20] -> Option B

4. Quick Check:

   Swapped and sorted arrays = [1,2,3,5,8,9], [10,13,15,20] [OK]

Hint: Swap bigger in arr1 with smaller in arr2, then sort both [OK]

Common Mistakes:

• Not decrementing i or incrementing j correctly
• Forgetting to sort arrays after swapping
• Confusing indices during swaps

4. Identify the error in the following code snippet intended to merge two sorted arrays arr1 and arr2 without extra space:

def merge(arr1, arr2):
    i = 0
    j = 0
    while i < len(arr1) and j < len(arr2):
        if arr1[i] > arr2[j]:
            arr1[i], arr2[j] = arr2[j], arr1[i]
        i += 1
        j += 1
    arr1.sort()
    arr2.sort()

medium

A. Sorting arrays after swapping is unnecessary

B. Swapping elements without checking if arr1[i] is smaller

C. Incrementing both i and j in every iteration causes skipping elements

D. Using while loop instead of for loop causes infinite loop

Solution

1. Step 1: Analyze loop increments

   Both i and j are incremented every iteration regardless of swap condition, which can skip elements that need comparison.

2. Step 2: Explain why skipping elements is wrong

   To merge correctly, we must compare all relevant pairs. Incrementing both pointers blindly misses some comparisons, causing incorrect merge.

3. Final Answer:

   Incrementing both i and j in every iteration causes skipping elements -> Option C

4. Quick Check:

   Increment both pointers blindly = skip elements [OK]

Hint: Increment pointers only when appropriate to avoid skipping [OK]

Common Mistakes:

• Incrementing both indices unconditionally
• Not sorting arrays after swaps
• Assuming while loop causes infinite loop here

5. Given two sorted arrays arr1 = [1, 4, 7, 8, 10] and arr2 = [2, 3, 9], how can you merge them without extra space so that after merging, arr1 contains the first 5 smallest elements and arr2 contains the remaining elements, both sorted?

hard

A. Use the gap method: compare elements at gap distance and swap if needed, reduce gap until 0

B. Concatenate both arrays and sort the combined array

C. Swap elements only if arr1[i] is greater than arr2[j], then sort both arrays

D. Copy arr2 elements into arr1 and sort arr1

Solution

1. Step 1: Understand the gap method

   The gap method starts with a gap equal to the sum of lengths divided by 2, compares elements at gap distance across both arrays, and swaps if out of order. The gap reduces each iteration until it becomes zero.

2. Step 2: Apply gap method to merge without extra space

   This method efficiently merges two sorted arrays in place without extra memory, ensuring both arrays are sorted and merged correctly.

3. Final Answer:

   Use the gap method: compare elements at gap distance and swap if needed, reduce gap until 0 -> Option A

4. Quick Check:

   Gap method merges in place without extra space [OK]

Hint: Use gap method to merge by reducing gap and swapping [OK]

Common Mistakes:

• Trying to concatenate arrays which uses extra space
• Sorting arrays separately without merging
• Swapping without systematic gap reduction