Find Maximum Subarray Divide and Conquer in DSA Javascript - Cheat Sheet & Quick Revision

Recall & Review

beginner

What is the main idea behind the Divide and Conquer approach for finding the maximum subarray?

Split the array into two halves, find the maximum subarray in the left half, the right half, and the maximum subarray crossing the middle, then return the largest of these three.

intermediate

In the Divide and Conquer method, how do we find the maximum subarray crossing the middle point?

We find the maximum sum subarray ending at the middle from the left side and the maximum sum subarray starting at the middle+1 from the right side, then add them together.

intermediate

What is the time complexity of the Divide and Conquer algorithm for maximum subarray?

O(n log n) because the array is divided into halves recursively and each merge step takes linear time to find the crossing subarray.

beginner

Why do we consider three cases (left, right, crossing) in the Divide and Conquer maximum subarray problem?

Because the maximum subarray can be fully in the left half, fully in the right half, or crossing the middle point between halves.

beginner

Show the base case for the Divide and Conquer maximum subarray algorithm.

When the array has only one element, the maximum subarray is that element itself.

What does the Divide and Conquer maximum subarray algorithm do first?

A Find the maximum element

B Sort the array

C Reverse the array

D Split the array into two halves

The algorithm splits the array into two halves to solve the problem recursively.

How do we find the maximum subarray crossing the middle?

A By finding max sum ending at middle and max sum starting at middle+1

B By checking all subarrays in the array

C By sorting the array

D By taking the maximum element only

We find the max sum subarray ending at middle and max sum subarray starting at middle+1, then add them.

What is the time complexity of the Divide and Conquer maximum subarray algorithm?

A O(n log n)

B O(n^2)

C O(n)

D O(log n)

The algorithm divides the array recursively and merges results in linear time, resulting in O(n log n).

Which case is NOT considered in the Divide and Conquer maximum subarray approach?

A Maximum subarray in the right half

B Maximum subarray after sorting

C Maximum subarray crossing the middle

D Maximum subarray in the left half

Sorting is not part of the maximum subarray divide and conquer approach.

What is the base case for the recursive maximum subarray function?

A Array length is zero

B Array has negative numbers

C Array length is one

D Array is sorted

When the array has one element, that element is the maximum subarray.

Explain how the Divide and Conquer approach finds the maximum subarray in an array.

Think about breaking the problem into smaller parts and combining results.

Describe how to find the maximum subarray that crosses the middle point in the Divide and Conquer method.

Focus on sums around the middle index.

Practice

1. What is the main idea behind the Divide and Conquer approach to find the maximum subarray sum?

easy

A. Split the array into two halves, find max subarray in left, right, and crossing middle, then combine results

B. Sort the array and pick the largest element as the max subarray sum

C. Use a sliding window to find the max sum by expanding and shrinking the window

D. Calculate the sum of all elements and return it as the max subarray sum

Solution

1. Step 1: Understand divide and conquer strategy

   The array is split into two halves recursively to simplify the problem.

2. Step 2: Combine results from subproblems

   Find max subarray in left half, right half, and crossing the middle, then take the maximum of these three.

3. Final Answer:

   Split array, find max left, right, and crossing sums, then combine -> Option A

4. Quick Check:

   Divide and conquer = split and combine max sums [OK]

Hint: Remember: divide array, conquer halves, combine max sums [OK]

Common Mistakes:

• Thinking sorting helps find max subarray sum
• Ignoring the crossing middle subarray
• Assuming max subarray is always the largest single element

2. Which of the following is the correct base case condition in a divide and conquer function to find the maximum subarray?

easy

A. If the array length is 1, return the single element

B. If the array length is 0, return 0

C. If the array length is 2, return the sum of both elements

D. If the array length is 1, return 0

Solution

1. Step 1: Identify base case for recursion

   When the array has only one element, that element itself is the max subarray sum.

2. Step 2: Return the single element value

   Return the element value directly as the max sum for that subarray.

3. Final Answer:

   Return the single element when array length is 1 -> Option A

4. Quick Check:

   Base case single element = element value [OK]

Hint: Base case: single element array returns that element [OK]

Common Mistakes:

• Returning 0 instead of the element for single element array
• Returning sum of two elements as base case
• Not defining a base case causing infinite recursion

3. What is the output of the following code snippet?

function maxCrossingSum(arr, left, mid, right) {
  let sum = 0, leftSum = -Infinity;
  for (let i = mid; i >= left; i--) {
    sum += arr[i];
    if (sum > leftSum) leftSum = sum;
  }
  sum = 0;
  let rightSum = -Infinity;
  for (let i = mid + 1; i <= right; i++) {
    sum += arr[i];
    if (sum > rightSum) rightSum = sum;
  }
  return leftSum + rightSum;
}

const arr = [2, -1, 3, -4, 5];
console.log(maxCrossingSum(arr, 0, 2, 4));

medium

A. 9

B. 7

C. 3

D. 5

Solution

1. Step 1: Calculate left sum from mid to left

   Starting at index 2 (value 3), sum = 3; then index 1 (-1), sum=2; index 0 (2), sum=4. Max leftSum = 4.

2. Step 2: Calculate right sum from mid+1 to right

   Starting at index 3 (-4), sum = -4; index 4 (5), sum=1. Max rightSum = 1.

3. Step 3: Add leftSum and rightSum

   4 + 1 = 5.

4. Final Answer:

   5 -> Option D

5. Quick Check:

   Left max 4 + Right max 1 = 5 [OK]

Hint: Sum left max from mid down, right max from mid+1 up, add [OK]

Common Mistakes:

• Adding wrong subarray sums crossing mid
• Mixing indices for left and right sums
• Using max element instead of max crossing sum

4. Identify the bug in this divide and conquer max subarray function:

function maxSubArray(arr, left, right) {
  if (left === right) return arr[left];
  const mid = Math.floor((left + right) / 2);
  const leftMax = maxSubArray(arr, left, mid - 1);
  const rightMax = maxSubArray(arr, mid + 1, right);
  const crossMax = maxCrossingSum(arr, left, mid, right);
  return Math.max(leftMax, rightMax, crossMax);
}

medium

A. The base case should return 0 instead of arr[left]

B. The left recursive call should be maxSubArray(arr, left, mid)

C. The right recursive call should be maxSubArray(arr, mid, right)

D. The mid calculation is incorrect

Solution

1. Step 1: Check recursive calls range

   Left recursive call uses mid - 1, which excludes mid element incorrectly.

2. Step 2: Correct left recursive call to include mid

   Change left recursive call to maxSubArray(arr, left, mid) to include mid element.

3. Final Answer:

   Left recursive call should be maxSubArray(arr, left, mid) -> Option B

4. Quick Check:

   Include mid in left recursion [OK]

Hint: Include mid index in left recursion call [OK]

Common Mistakes:

• Excluding mid element in recursive calls
• Returning 0 instead of element in base case
• Incorrect mid calculation causing infinite recursion

5. Given the array [-2, 1, -3, 4, -1, 2, 1, -5, 4], what is the maximum subarray sum found using divide and conquer?

hard

A. 7

B. 5

C. 6

D. 4

Solution

1. Step 1: Identify the max subarray

   The subarray [4, -1, 2, 1] (indices 3-6) sums to 4 + (-1) + 2 + 1 = 6.

2. Step 2: Check extending right

   [4, -1, 2, 1, -5] = 1 < 6; [4, -1, 2, 1, -5, 4] = 5 < 6.

3. Step 3: Check extending left

   [-3, 4, -1, 2, 1] = 3 < 6; further left smaller.

4. Final Answer:

   6 -> Option C

5. Quick Check:

   Max subarray sum = 6 [OK]

Hint: Check sums of all subarrays crossing mid carefully [OK]

Common Mistakes:

• Miscomputing as 7 by skipping the -3
• Choosing sum 5 by including the end
• Picking single largest element 4