Find Maximum Subarray Divide and Conquer in DSA C - Practice Problems & Challenges

Challenge - 5 Problems

❓ Predict Output

intermediate

Output of Divide and Conquer Maximum Subarray Sum

What is the output of the following C code that finds the maximum subarray sum using divide and conquer?

DSA C

int max(int a, int b) { return (a > b) ? a : b; }

int maxCrossingSum(int arr[], int l, int m, int h) {
    int sum = 0;
    int left_sum = -2147483648;
    for (int i = m; i >= l; i--) {
        sum += arr[i];
        if (sum > left_sum) left_sum = sum;
    }
    sum = 0;
    int right_sum = -2147483648;
    for (int i = m + 1; i <= h; i++) {
        sum += arr[i];
        if (sum > right_sum) right_sum = sum;
    }
    return left_sum + right_sum;
}

int maxSubArraySum(int arr[], int l, int h) {
    if (l == h) return arr[l];
    int m = (l + h) / 2;
    int left_max = maxSubArraySum(arr, l, m);
    int right_max = maxSubArraySum(arr, m + 1, h);
    int cross_max = maxCrossingSum(arr, l, m, h);
    return max(max(left_max, right_max), cross_max);
}

#include <stdio.h>
int main() {
    int arr[] = {-2, 1, -3, 4, -1, 2, 1, -5, 4};
    int n = sizeof(arr) / sizeof(arr[0]);
    int max_sum = maxSubArraySum(arr, 0, n - 1);
    printf("%d", max_sum);
    return 0;
}

A 6

B 7

C 5

D 4

💡 Hint

Think about the subarray with the largest sum in the array.

The maximum subarray is [4, -1, 2, 1] which sums to 6.

🧠 Conceptual

intermediate

Key Idea Behind Divide and Conquer for Maximum Subarray

Which statement best describes the key idea behind the divide and conquer approach to find the maximum subarray?

A Sort the array and pick the largest element as the maximum subarray sum.

B Use a sliding window to find the maximum sum subarray in one pass.

C Divide the array into two halves, find maximum subarray in left, right, and crossing middle, then take the maximum.

D Calculate all possible subarrays and pick the one with the maximum sum.

💡 Hint

Think about how the problem is split into smaller parts.

Divide and conquer splits the array into two halves and considers three cases: max subarray in left half, right half, or crossing the middle.

🔧 Debug

advanced

Identify the Bug in maxCrossingSum Function

What is the bug in this maxCrossingSum function snippet?

DSA C

int maxCrossingSum(int arr[], int l, int m, int h) {
    int sum = 0;
    int left_sum = 0;
    for (int i = m; i >= l; i--) {
        sum += arr[i];
        if (sum > left_sum) left_sum = sum;
    }
    sum = 0;
    int right_sum = 0;
    for (int i = m + 1; i <= h; i++) {
        sum += arr[i];
        if (sum > right_sum) right_sum = sum;
    }
    return left_sum + right_sum;
}

A left_sum and right_sum initialized to 0 cause incorrect results when all numbers are negative.

B The loops should run from l to m and m+1 to h instead of backwards.

C sum variable should be declared inside the loops to reset automatically.

D The function should return max(left_sum, right_sum) instead of their sum.

💡 Hint

Consider what happens if all elements are negative.

Initializing left_sum and right_sum to 0 causes the function to ignore negative sums, leading to wrong max crossing sums when all elements are negative.

❓ Predict Output

advanced

Output of maxSubArraySum on All Negative Array

What is the output of the following code snippet?

DSA C

int arr[] = {-8, -3, -6, -2, -5, -4};
int n = sizeof(arr) / sizeof(arr[0]);
int max_sum = maxSubArraySum(arr, 0, n - 1);
printf("%d", max_sum);

A -8

B -3

C 0

D -2

💡 Hint

The maximum subarray in all negative numbers is the least negative single element.

The maximum subarray is [-2] with sum -2, which is the largest among all negative elements.

🚀 Application

expert

Time Complexity of Divide and Conquer Maximum Subarray

What is the time complexity of the divide and conquer algorithm for finding the maximum subarray sum for an array of size n?

A O(n^2)

B O(n log n)

C O(n)

D O(log n)

💡 Hint

Consider the recurrence relation T(n) = 2T(n/2) + O(n).

The algorithm divides the problem into two halves and combines results in linear time, leading to O(n log n) complexity.

Practice

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

easy

A. Use a sliding window to find the maximum sum subarray in one pass

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

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

D. Calculate sum of all subarrays and pick the maximum

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 left, right, and crossing subarrays

   We find max subarray sums in left half, right half, and subarrays crossing the middle, then take the maximum of these.

3. Final Answer:

   Divide the array into two halves, find max subarray in left, right, and crossing middle, then combine results -> Option C

4. Quick Check:

   Divide and conquer = split, solve, combine [OK]

Hint: Split array, solve halves, check crossing sum, combine [OK]

Common Mistakes:

• Thinking sorting helps find max subarray sum
• Ignoring subarrays crossing the middle
• Trying to sum all subarrays directly

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

easy

A. if (low == high) return 0;

B. if (low > high) return 0;

C. if (low < high) return arr[low] + arr[high];

D. if (low == high) return arr[low];

Solution

1. Step 1: Identify base case meaning

   When the subarray has only one element (low equals high), the max subarray sum is that element itself.

2. Step 2: Match base case condition

   The condition if (low == high) return arr[low]; correctly returns the single element's value.

3. Final Answer:

   if (low == high) return arr[low]; -> Option D

4. Quick Check:

   Single element subarray returns element value [OK]

Hint: Base case: one element means return that element [OK]

Common Mistakes:

• Returning 0 when low == high
• Using incorrect comparison operators
• Returning sum of two elements instead of one

3. What is the output of the following C code snippet using divide and conquer to find max subarray sum?

int arr[] = {-2, 1, -3, 4, -1, 2, 1, -5, 4};
int result = maxSubArray(arr, 0, 8);
printf("%d", result);

Assume maxSubArray is correctly implemented divide and conquer function.

medium

A. 6

B. 7

C. 5

D. 4

Solution

1. Step 1: Identify max subarray in given array

   The maximum subarray is [4, -1, 2, 1] with sum 4 + (-1) + 2 + 1 = 6, but check if including more elements helps.

2. Step 2: Check if extending subarray increases sum

   Including the element before 4 is -3 (reduces sum), after 1 is -5 (reduces sum), so max sum is 6.

3. Final Answer:

   6 -> Option A

4. Quick Check:

   Max subarray sum = 6 [OK]

Hint: Find max sum of continuous elements [OK]

Common Mistakes:

• Counting non-continuous elements
• Miscomputing sums with negative numbers
• Assuming max element alone is answer

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

int maxCrossingSum(int arr[], int low, int mid, int high) {
    int left_sum = 0, sum = 0;
    for (int i = mid; i >= low; i--) {
        sum += arr[i];
        if (sum > left_sum)
            left_sum = sum;
    }
    int right_sum = 0;
    sum = 0;
    for (int i = mid + 1; i <= high; i++) {
        sum += arr[i];
        if (sum > right_sum)
            right_sum = sum;
    }
    return left_sum + right_sum;
}

medium

A. left_sum and right_sum should be initialized to INT_MIN, not 0

B. The loops should start from low and high, not mid

C. sum variable should not be reset between loops

D. Return statement should subtract left_sum from right_sum

Solution

1. Step 1: Understand initialization of sums

   Initializing left_sum and right_sum to 0 causes errors if all numbers are negative, as max sum could be less than 0.

2. Step 2: Correct initialization

   They should be initialized to the smallest integer (INT_MIN) to correctly track max sums even if negative.

3. Final Answer:

   left_sum and right_sum should be initialized to INT_MIN, not 0 -> Option A

4. Quick Check:

   Initialize max sums to INT_MIN to handle negatives [OK]

Hint: Use INT_MIN for max sums to handle negative arrays [OK]

Common Mistakes:

• Initializing max sums to zero causes wrong results
• Not resetting sum variable properly
• Incorrect loop ranges

5. Given an array arr = [-1, -2, -3, -4], what will the divide and conquer max subarray function return and why?

hard

A. 0 because empty subarray sum is considered

B. -1 because the max subarray is the single largest element

C. -10 because sum of all elements is max

D. -4 because last element is max

Solution

1. Step 1: Analyze array with all negative numbers

   All elements are negative, so the max subarray is the one with the least negative value (largest number).

2. Step 2: Identify max element

   The largest element is -1, so max subarray sum is -1 (single element subarray).

3. Final Answer:

   -1 because the max subarray is the single largest element -> Option B

4. Quick Check:

   Max subarray in negatives = max single element [OK]

Hint: Max subarray in all negatives is max single element [OK]

Common Mistakes:

• Assuming sum of all negatives is max
• Considering empty subarray sum as zero incorrectly
• Picking last element without comparison