Kadane's Algorithm Maximum Subarray in DSA Java - Cheat Sheet & Quick Revision

Recall & Review

beginner

What problem does Kadane's Algorithm solve?

Kadane's Algorithm finds the maximum sum of a contiguous subarray within a one-dimensional array of numbers.

beginner

What is the main idea behind Kadane's Algorithm?

It keeps track of the current subarray sum and resets it to zero if it becomes negative, while updating the maximum sum found so far.

intermediate

In Kadane's Algorithm, what does the variable 'currentSum' represent?

'currentSum' represents the sum of the current subarray being considered. It resets to zero if it becomes negative to start fresh from the next element.

intermediate

Why does Kadane's Algorithm reset the current sum to zero when it becomes negative?

Because a negative sum will only reduce the sum of any future subarray, so it's better to start a new subarray from the next element.

beginner

What is the time complexity of Kadane's Algorithm?

Kadane's Algorithm runs in O(n) time, where n is the number of elements in the array, because it processes each element once.

What does Kadane's Algorithm return?

A Maximum sum of a contiguous subarray

B Minimum sum of a contiguous subarray

C Sum of all elements in the array

D Length of the longest subarray

Kadane's Algorithm finds the maximum sum of any contiguous subarray.

What happens when the current sum becomes negative in Kadane's Algorithm?

A It is added to the maximum sum

B It is doubled

C It is reset to zero

D It is ignored and kept as is

Resetting to zero allows starting a new subarray that could have a higher sum.

What is the initial value of the maximum sum in Kadane's Algorithm?

A Negative infinity or the first element

B Zero

C Sum of all elements

D One

The maximum sum starts as the smallest possible value or the first element to handle negative numbers.

Which of these is NOT true about Kadane's Algorithm?

A It resets current sum when negative

B It works in linear time

C It updates maximum sum while iterating

D It uses extra space proportional to input size

Kadane's Algorithm uses constant extra space, not proportional to input size.

If all numbers in the array are negative, what does Kadane's Algorithm return?

A Zero

B The largest (least negative) number

C Sum of all numbers

D Minimum number

Kadane's Algorithm returns the maximum single element if all are negative.

Explain how Kadane's Algorithm finds the maximum sum of a contiguous subarray.

Think about how the algorithm decides to keep or discard parts of the subarray.

Describe the role of resetting the current sum to zero in Kadane's Algorithm and why it improves the solution.

Consider what happens if you keep adding negative sums.

Practice

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

easy

A. The minimum element in the array

B. The number of positive elements

C. The total sum of all 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 any contiguous subarray within a given array.

2. Step 2: Differentiate from other array operations

   It does not find minimum elements, total sum, or count positives, but specifically the maximum sum of a continuous segment.

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's finds max sum of continuous elements fast [OK]

Common Mistakes:

• Confusing maximum subarray sum with total array sum
• Thinking it finds minimum or count of elements
• Assuming it works on non-contiguous elements

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

easy

A. int maxSoFar = 0, maxEndingHere = 0;

B. int maxSoFar = 0, maxEndingHere = Integer.MAX_VALUE;

C. int maxSoFar = Integer.MIN_VALUE, maxEndingHere = 0;

D. int maxSoFar = Integer.MAX_VALUE, maxEndingHere = Integer.MIN_VALUE;

Solution

1. Step 1: Identify correct initial values for Kadane's variables

   maxSoFar should start at the smallest integer to handle all negative arrays; maxEndingHere starts at 0 to accumulate sums.

2. Step 2: Check each option's initialization

   int maxSoFar = Integer.MIN_VALUE, maxEndingHere = 0; correctly sets maxSoFar to Integer.MIN_VALUE and maxEndingHere to 0, which is standard practice.

3. Final Answer:

   int maxSoFar = Integer.MIN_VALUE, maxEndingHere = 0; -> Option C

4. Quick Check:

   Initialize maxSoFar low, maxEndingHere zero [OK]

Hint: Start maxSoFar at smallest int, maxEndingHere at zero [OK]

Common Mistakes:

• Initializing maxSoFar to 0 causes errors with all negative arrays
• Swapping maxSoFar and maxEndingHere initial values
• Using Integer.MAX_VALUE incorrectly

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

int[] arr = {-2, 1, -3, 4, -1, 2, 1, -5, 4};
int maxSoFar = Integer.MIN_VALUE, maxEndingHere = 0;
for (int num : arr) {
    maxEndingHere += num;
    if (maxSoFar < maxEndingHere) maxSoFar = maxEndingHere;
    if (maxEndingHere < 0) maxEndingHere = 0;
}
System.out.println(maxSoFar);

medium

A. 6

B. 7

C. 5

D. 4

Solution

1. Step 1: Trace the loop and update sums

   Iterate through arr, updating maxEndingHere and maxSoFar:
   - Start: maxSoFar = -∞, maxEndingHere = 0
   - Add -2: maxEndingHere = -2 (reset to 0)
   - Add 1: maxEndingHere = 1, maxSoFar = 1
   - Add -3: maxEndingHere = -2 (reset to 0)
   - Add 4: maxEndingHere = 4, maxSoFar = 4
   - Add -1: maxEndingHere = 3
   - Add 2: maxEndingHere = 5, maxSoFar = 5
   - Add 1: maxEndingHere = 6, maxSoFar = 6
   - Add -5: maxEndingHere = 1
   - Add 4: maxEndingHere = 5

2. Step 2: Identify maximum sum found

   The maximum sum recorded is 6, but check carefully:
   At step adding 1 after 2, maxEndingHere = 6, maxSoFar = 6
   At last step, maxEndingHere = 5, maxSoFar remains 6
   Wait, re-check:
   Actually, after adding 1 (step 7), maxEndingHere = 6
   Then adding -5 (step 8), maxEndingHere = 1
   Then adding 4 (step 9), maxEndingHere = 5
   So maxSoFar = 6

3. Correction:

   Re-examining the sums, the maximum subarray is [4, -1, 2, 1] which sums to 6, but the subarray [4, -1, 2, 1] sums to 6, so 6 is correct.
   However, the standard Kadane's algorithm for this array returns 6, but the known maximum subarray sum is 6.
   But the options include 7, which is incorrect.
   Therefore, the original answer C (6) is correct.

4. Final Answer:

   6 -> Option A

5. Quick Check:

   Max subarray sum = 6 [OK]

Hint: Add and reset sum; track max sum found [OK]

Common Mistakes:

• Forgetting to reset maxEndingHere when negative
• Miscounting sums during iteration
• Confusing maxSoFar with maxEndingHere

4. Identify the bug in this Kadane's Algorithm Java code snippet:

int maxSoFar = 0, maxEndingHere = 0;
for (int num : arr) {
    maxEndingHere = Math.max(num, maxEndingHere + num);
    maxSoFar = Math.max(maxSoFar, maxEndingHere);
}
System.out.println(maxSoFar);

medium

A. maxEndingHere should be initialized to Integer.MAX_VALUE

B. maxSoFar should start at Integer.MIN_VALUE, not 0

C. The loop should start from index 1, not 0

D. Using Math.max is incorrect for Kadane's Algorithm

Solution

1. Step 1: Analyze initialization of maxSoFar

   Starting maxSoFar at 0 causes incorrect results if all numbers are negative, as maxSoFar will never update to a negative max subarray.

2. Step 2: Correct initialization for negative arrays

   maxSoFar should be initialized to Integer.MIN_VALUE to correctly track maximum sums even if all are negative.

3. Final Answer:

   maxSoFar should start at Integer.MIN_VALUE, not 0 -> Option B

4. Quick Check:

   Initialize maxSoFar low to handle negatives [OK]

Hint: Initialize maxSoFar to smallest int to handle negatives [OK]

Common Mistakes:

• Initializing maxSoFar to 0 causes wrong max for negative arrays
• Misunderstanding Math.max usage in Kadane's
• Incorrect loop indexing assumptions

5. Given an array int[] arr = {-1, -2, -3, -4}, what will Kadane's Algorithm return as the maximum subarray sum if initialized with maxSoFar = Integer.MIN_VALUE and maxEndingHere = 0?

hard

A. -1

B. 0

C. -10

D. 4

Solution

1. Step 1: Trace Kadane's Algorithm on all negative array

   maxEndingHere resets to 0 when negative, but maxSoFar tracks max subarray sum:
   - Add -1: maxEndingHere = -1 (reset to 0), maxSoFar = -1
   - Add -2: maxEndingHere = -2 (reset to 0), maxSoFar = -1
   - Add -3: maxEndingHere = -3 (reset to 0), maxSoFar = -1
   - Add -4: maxEndingHere = -4 (reset to 0), maxSoFar = -1

2. Step 2: Identify maximum subarray sum

   The maximum subarray sum is -1, the largest single element in the array.

3. Final Answer:

   -1 -> Option A

4. Quick Check:

   Max subarray sum in negatives = max single element [OK]

Hint: Max sum in all negatives is largest single element [OK]

Common Mistakes:

• Assuming max sum is 0 due to resets
• Summing all negatives incorrectly
• Ignoring initialization of maxSoFar