Find Maximum Subarray Divide and Conquer in DSA C - Build from Scratch

Find Maximum Subarray using Divide and Conquer

📖 Scenario: You are analyzing daily temperature changes to find the longest period of increasing temperature. This can be represented as finding the maximum sum of a contiguous subarray in an array of integers.

🎯 Goal: Build a program in C that uses the divide and conquer method to find the maximum sum of a contiguous subarray in a given integer array.

📋 What You'll Learn

Create an integer array called arr with the exact values: {2, -4, 3, -1, 5, -6, 1, 4, -2, 3}

Create two integer variables called low and high to represent the start and end indices of the array

Write a function called maxCrossingSum that finds the maximum subarray sum crossing the midpoint

Write a recursive function called maxSubArraySum that uses divide and conquer to find the maximum subarray sum

Print the maximum subarray sum found by calling maxSubArraySum(arr, low, high)

💡 Why This Matters

🌍 Real World

Finding maximum subarrays is useful in financial analysis to find the best time to buy and sell stocks for maximum profit.

💼 Career

Understanding divide and conquer algorithms and recursion is essential for software engineering roles that require problem-solving and optimization skills.

1

Create the integer array

Create an integer array called arr with these exact values: {2, -4, 3, -1, 5, -6, 1, 4, -2, 3}

DSA C

# Create the integer array 'arr' with the given values
// Your code here

Hint

Use int arr[] = {2, -4, 3, -1, 5, -6, 1, 4, -2, 3}; to create the array.

2

Set the low and high indices

Create two integer variables called low and high. Set low to 0 and high to 9 (the last index of arr).

DSA C

#include 

int arr[] = {2, -4, 3, -1, 5, -6, 1, 4, -2, 3};

// Create low and high indices
// Your code here

Hint

Set low to 0 and high to 9 because the array has 10 elements indexed 0 to 9.

3

Write the divide and conquer functions

Write the function maxCrossingSum that takes arr, low, mid, and high and returns the maximum sum of a subarray crossing mid. Then write the recursive function maxSubArraySum that takes arr, low, and high and returns the maximum subarray sum using divide and conquer.

DSA C

#include 

int arr[] = {2, -4, 3, -1, 5, -6, 1, 4, -2, 3};

int low = 0;
int high = 9;

// Write maxCrossingSum function
// Write maxSubArraySum function
// Your code here

Hint

Use two loops in maxCrossingSum to find max sums on left and right of mid. Use recursion in maxSubArraySum to divide the array.

4

Print the maximum subarray sum

Write a main function that calls maxSubArraySum(arr, low, high) and prints the result with printf.

DSA C

#include 

int arr[] = {2, -4, 3, -1, 5, -6, 1, 4, -2, 3};

int low = 0;
int high = 9;

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

int maxCrossingSum(int arr[], int low, int mid, int high) {
    int sum = 0;
    int left_sum = -2147483648; // very small number
    for (int i = mid; i >= low; i--) {
        sum += arr[i];
        if (sum > left_sum)
            left_sum = sum;
    }

    sum = 0;
    int right_sum = -2147483648;
    for (int i = mid + 1; i  right_sum)
            right_sum = sum;
    }

    return left_sum + right_sum;
}

int maxSubArraySum(int arr[], int low, int high) {
    if (low == high)
        return arr[low];

    int mid = (low + high) / 2;

    int left_max = maxSubArraySum(arr, low, mid);
    int right_max = maxSubArraySum(arr, mid + 1, high);
    int cross_max = maxCrossingSum(arr, low, mid, high);

    return max(max(left_max, right_max), cross_max);
}

// Write main function to print the maximum subarray sum
// Your code here

Hint

Call maxSubArraySum(arr, low, high) and print the result using printf.

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