Kadane's Algorithm Maximum Subarray in DSA C++

Mental Model

Find the largest sum of a continuous part of an array by keeping track of the best sum ending at each position.

Analogy: Imagine walking along a path with ups and downs in height. You want to find the highest hill segment you can stand on continuously without jumping off.

Array: [ -2, 1, -3, 4, -1, 2, 1, -5, 4 ]
Indices:  0   1   2   3   4   5   6   7   8

Dry Run Walkthrough

Input: array: [-2, 1, -3, 4, -1, 2, 1, -5, 4]

Goal: Find the maximum sum of any continuous subarray

Step 1: start with first element as current and max sum

curr_sum = -2, max_sum = -2

Why: Initialize sums with first element to start comparisons

Step 2: move to element 1: decide to add or start new subarray

curr_sum = max(1, -2 + 1) = 1, max_sum = max(-2, 1) = 1

Why: Choose best sum ending here: either start fresh or continue

Step 3: move to element 2: update current and max sums

curr_sum = max(-3, 1 + -3) = -2, max_sum = max(1, -2) = 1

Why: Current sum dropped, but max sum stays from before

Step 4: move to element 3: update sums

curr_sum = max(4, -2 + 4) = 4, max_sum = max(1, 4) = 4

Why: New subarray starting at 4 is better than continuing

Step 5: move to element 4: update sums

curr_sum = max(-1, 4 + -1) = 3, max_sum = max(4, 3) = 4

Why: Continuing subarray still good but max sum unchanged

Step 6: move to element 5: update sums

curr_sum = max(2, 3 + 2) = 5, max_sum = max(4, 5) = 5

Why: Adding 2 improves current sum and max sum

Step 7: move to element 6: update sums

curr_sum = max(1, 5 + 1) = 6, max_sum = max(5, 6) = 6

Why: Adding 1 further improves sums

Step 8: move to element 7: update sums

curr_sum = max(-5, 6 + -5) = 1, max_sum = max(6, 1) = 6

Why: Current sum drops but max sum stays

Step 9: move to element 8: update sums

curr_sum = max(4, 1 + 4) = 5, max_sum = max(6, 5) = 6

Why: Current sum improves but max sum remains max

Result:

Maximum subarray sum = 6

Annotated Code

DSA C++

#include <iostream>
#include <vector>
#include <algorithm>

int maxSubArray(const std::vector<int>& nums) {
    int curr_sum = nums[0];
    int max_sum = nums[0];
    for (size_t i = 1; i < nums.size(); ++i) {
        curr_sum = std::max(nums[i], curr_sum + nums[i]);
        max_sum = std::max(max_sum, curr_sum);
    }
    return max_sum;
}

int main() {
    std::vector<int> nums = {-2, 1, -3, 4, -1, 2, 1, -5, 4};
    std::cout << "Maximum subarray sum = " << maxSubArray(nums) << std::endl;
    return 0;
}

curr_sum = std::max(nums[i], curr_sum + nums[i]);

choose best sum ending at current index: start new or continue

max_sum = std::max(max_sum, curr_sum);

update max sum found so far

Output

Maximum subarray sum = 6

Complexity Analysis

Time: O(n) because we scan the array once updating sums

Space: O(1) because only a few variables are used regardless of input size

vs Alternative: Compared to checking all subarrays (O(n^2)), Kadane's is much faster and uses less memory

Edge Cases

array with all negative numbers

algorithm returns the largest (least negative) single element

DSA C++

int curr_sum = nums[0];

array with one element

returns that element as max sum

DSA C++

int curr_sum = nums[0];

array with all positive numbers

returns sum of entire array

DSA C++

max_sum = std::max(max_sum, curr_sum);

When to Use This Pattern

When asked to find the maximum sum of a continuous subarray, reach for Kadane's algorithm because it efficiently tracks best sums in one pass.

Common Mistakes

Mistake: Not resetting current sum when it becomes negative
Fix: Use max between current element and current sum plus element to decide whether to start fresh

Mistake: Initializing max sum to zero instead of first element
Fix: Initialize max sum and current sum to first element to handle negative arrays correctly

Summary

Finds the maximum sum of any continuous subarray in an array.

Use when you need the largest sum of consecutive elements efficiently.

Keep track of the best sum ending at each position to avoid checking all subarrays.

Practice

1. What does Kadane's Algorithm primarily find in an array of integers?

easy

A. The total sum of all elements

B. The minimum element in the array

C. The maximum product of any two elements

D. The maximum sum of a contiguous subarray

Solution

1. Step 1: Understand the problem Kadane's Algorithm solves

   Kadane's Algorithm is designed to find the maximum sum of a contiguous subarray within a one-dimensional array of numbers.

2. Step 2: Compare options with the algorithm's purpose

   Options A, B, and D do not describe the main goal of Kadane's Algorithm. Only The maximum sum of a contiguous subarray correctly states it finds the maximum sum of a contiguous subarray.

3. Final Answer:

   The maximum sum of a contiguous subarray -> Option D

4. Quick Check:

   Maximum subarray sum = maximum sum of contiguous elements [OK]

Hint: Kadane finds max sum in a continuous slice [OK]

Common Mistakes:

• Confusing maximum sum with maximum product
• Thinking it finds minimum element
• Assuming it sums all elements

2. Which of the following is the correct initialization for Kadane's Algorithm variables in C++?

easy

A. int max_so_far = INT_MIN, max_ending_here = 0;

B. int max_so_far = INT_MAX, max_ending_here = INT_MAX;

C. int max_so_far = 0, max_ending_here = INT_MIN;

D. int max_so_far = 0, max_ending_here = 0;

Solution

1. Step 1: Understand variable roles in Kadane's Algorithm

   max_so_far stores the maximum sum found so far, so it should start very low (INT_MIN) to allow any sum to replace it. max_ending_here tracks the current subarray sum and starts at 0.

2. Step 2: Check each option's initialization

   int max_so_far = INT_MIN, max_ending_here = 0; correctly initializes max_so_far to INT_MIN and max_ending_here to 0. Other options either start max_so_far at 0 or INT_MAX, which can cause incorrect results.

3. Final Answer:

   int max_so_far = INT_MIN, max_ending_here = 0; -> Option A

4. Quick Check:

   max_so_far = INT_MIN, max_ending_here = 0 [OK]

Hint: Start max_so_far at INT_MIN to handle negatives [OK]

Common Mistakes:

• Initializing max_so_far to 0 causes wrong results with negatives
• Setting max_ending_here to INT_MIN breaks logic
• Using INT_MAX causes overflow issues

3. What is the output of the following C++ code using Kadane's Algorithm?

int arr[] = {-2, 1, -3, 4, -1, 2, 1, -5, 4};
int n = 9;
int max_so_far = INT_MIN, max_ending_here = 0;
for (int i = 0; i < n; i++) {
    max_ending_here += arr[i];
    if (max_so_far < max_ending_here)
        max_so_far = max_ending_here;
    if (max_ending_here < 0)
        max_ending_here = 0;
}
std::cout << max_so_far << std::endl;

medium

A. 6

B. 7

C. 5

D. 4

Solution

1. Step 1: Trace Kadane's Algorithm on the array

   Iterate through arr: max_ending_here and max_so_far update as follows:
   - i=0: max_ending_here=-2, max_so_far=-2, reset max_ending_here=0
   - i=1: max_ending_here=1, max_so_far=1
   - i=2: max_ending_here=-2, max_so_far=1, reset max_ending_here=0
   - i=3: max_ending_here=4, max_so_far=4
   - i=4: max_ending_here=3, max_so_far=4
   - i=5: max_ending_here=5, max_so_far=5
   - i=6: max_ending_here=6, max_so_far=6
   - i=7: max_ending_here=1, max_so_far=6
   - i=8: max_ending_here=5, max_so_far=6

2. Step 2: Identify maximum subarray sum

   The maximum sum found is 6, which corresponds to subarray [4, -1, 2, 1].

3. Final Answer:

   6 -> Option A

4. Quick Check:

   Max subarray sum = 6 [OK]

Hint: Add elements, reset sum if negative, track max [OK]

Common Mistakes:

• Not resetting max_ending_here when negative
• Confusing max_so_far with current sum
• Miscounting subarray sums

4. Identify the bug in this Kadane's Algorithm implementation and select the fix:

int maxSubArraySum(int arr[], int n) {
    int max_so_far = 0, max_ending_here = 0;
    for (int i = 0; i < n; i++) {
        max_ending_here += arr[i];
        if (max_so_far < max_ending_here)
            max_so_far = max_ending_here;
        if (max_ending_here < 0)
            max_ending_here = 0;
    }
    return max_so_far;
}

medium

A. Remove resetting max_ending_here to 0

B. Initialize max_ending_here to INT_MIN instead of 0

C. Initialize max_so_far to INT_MIN instead of 0

D. Change loop to start from i = 1

Solution

1. Step 1: Analyze initialization impact on negative arrays

   Starting max_so_far at 0 causes incorrect results if all array elements are negative, as max_so_far will never update to a negative sum.

2. Step 2: Fix initialization to handle all negative values

   Setting max_so_far to INT_MIN allows it to update correctly even if all elements are negative, ensuring the maximum subarray sum is found.

3. Final Answer:

   Initialize max_so_far to INT_MIN instead of 0 -> Option C

4. Quick Check:

   max_so_far = INT_MIN fixes negative-only arrays [OK]

Hint: Use INT_MIN for max_so_far to handle negatives [OK]

Common Mistakes:

• Leaving max_so_far at 0 causes wrong max for negatives
• Removing reset breaks algorithm logic
• Changing loop start misses first element

5. Given an array with all negative numbers: int arr[] = {-8, -3, -6, -2, -5, -4}; What is the maximum subarray sum using Kadane's Algorithm?

hard

A. 0

B. -2

C. -3

D. -8

Solution

1. Step 1: Understand Kadane's behavior with all negative numbers

   When all numbers are negative, resetting max_ending_here to 0 causes the algorithm to miss the largest (least negative) number. Proper initialization with max_so_far = INT_MIN helps track the max single element.

2. Step 2: Identify the maximum single element

   The largest number in the array is -2, which is the maximum subarray sum since any longer subarray sum will be smaller.

3. Final Answer:

   -2 -> Option B

4. Quick Check:

   Max single negative element = -2 [OK]

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

Common Mistakes:

• Assuming max sum is 0 by resetting sums
• Ignoring single element max in negative arrays
• Confusing minimum with maximum