Kadane's Algorithm Maximum Subarray in DSA Typescript - 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.

beginner

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

'currentSum' represents the sum of the current subarray being considered. If it drops below zero, it is reset to zero to start a new subarray.

intermediate

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

Because a negative sum will only decrease the sum of any future subarray, so it's better to start fresh 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 Minimum sum of any contiguous subarray

B Sum of all elements in the array

C Maximum sum of any contiguous subarray

D Length of the longest subarray

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

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

A It is doubled

B It is reset to zero

C It is ignored and kept as is

D Algorithm stops

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 Sum of all elements

B Zero

C First element of the array

D Negative infinity or very small number

Starting with negative infinity ensures any sum found will be larger initially.

Kadane's Algorithm is an example of which programming technique?

A Dynamic Programming

B Divide and Conquer

C Greedy Algorithm

D Backtracking

Kadane's Algorithm uses dynamic programming by building solutions from previous results.

What is the time complexity of Kadane's Algorithm?

A O(n)

B O(n^2)

C O(log n)

D O(n!)

Kadane's Algorithm processes each element once, so it runs in linear time.

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 why Kadane's Algorithm is efficient compared to checking all subarrays.

Consider the difference between brute force and Kadane's approach.

Practice

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

easy

A. The number of positive elements

B. The minimum element in the array

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 largest sum of a continuous segment.

3. Final Answer:

   The maximum sum of a contiguous subarray -> Option D

4. Quick Check:

   Maximum subarray sum = The maximum sum of a contiguous subarray [OK]

Hint: Kadane finds max sum in continuous parts of array [OK]

Common Mistakes:

• Confusing max sum with total sum of array
• Thinking it finds minimum values
• Assuming it counts positive numbers

2. Which of the following is the correct initialization step in Kadane's Algorithm in TypeScript?

easy

A. let maxSoFar = Number.MIN_SAFE_INTEGER, maxEndingHere = 0;

B. let maxSoFar = [], maxEndingHere = [];

C. let maxSoFar = 0, maxEndingHere = 0;

D. let maxSoFar = null, maxEndingHere = null;

Solution

1. Step 1: Understand initialization for Kadane's Algorithm

   We initialize maxSoFar to the smallest safe integer to handle arrays with negative numbers correctly, and maxEndingHere to 0.

2. Step 2: Check each option's correctness

   let maxSoFar = Number.MIN_SAFE_INTEGER, maxEndingHere = 0; correctly uses Number.MIN_SAFE_INTEGER for maxSoFar and 0 for maxEndingHere. Other options use incorrect types or values.

3. Final Answer:

   let maxSoFar = Number.MIN_SAFE_INTEGER, maxEndingHere = 0; -> Option A

4. Quick Check:

   Initialization with min safe integer = let maxSoFar = Number.MIN_SAFE_INTEGER, maxEndingHere = 0; [OK]

Hint: Initialize maxSoFar with smallest integer to handle negatives [OK]

Common Mistakes:

• Initializing maxSoFar to 0 causes wrong results with all negatives
• Using arrays instead of numbers for maxSoFar
• Setting variables to null instead of numbers

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

function maxSubArray(nums: number[]): number {
  let maxSoFar = Number.MIN_SAFE_INTEGER;
  let maxEndingHere = 0;
  for (const num of nums) {
    maxEndingHere = Math.max(num, maxEndingHere + num);
    maxSoFar = Math.max(maxSoFar, maxEndingHere);
  }
  return maxSoFar;
}

console.log(maxSubArray([-2,1,-3,4,-1,2,1,-5,4]));

medium

A. 6

B. 7

C. 5

D. 4

Solution

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

   Iterate through [-2,1,-3,4,-1,2,1,-5,4]: max subarray sums update as -2,1, -2,4,3,5,6,1,5.

2. Step 2: Identify the maximum sum found

   The maximum sum found during iteration 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: Track maxEndingHere and maxSoFar updates carefully [OK]

Common Mistakes:

• Adding all positive numbers only
• Ignoring negative numbers impact
• Confusing maxSoFar with maxEndingHere

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

function maxSubArray(nums: number[]): number {
  let maxSoFar = 0;
  let maxEndingHere = 0;
  for (const num of nums) {
    maxEndingHere = Math.max(num, maxEndingHere + num);
    maxSoFar = Math.max(maxSoFar, maxEndingHere);
  }
  return maxSoFar;
}

medium

A. Remove resetting maxEndingHere to 0 when negative

B. Initialize maxSoFar with Number.MIN_SAFE_INTEGER instead of 0

C. Change maxEndingHere += num to maxEndingHere = num

D. Return maxEndingHere instead of maxSoFar

Solution

1. Step 1: Analyze initialization problem

   Initializing maxSoFar to 0 causes wrong results if all numbers are negative, as maxSoFar never updates below 0.

2. Step 2: Fix initialization to handle negative arrays

   Setting maxSoFar to Number.MIN_SAFE_INTEGER allows correct tracking of max sums even if all are negative.

3. Final Answer:

   Initialize maxSoFar with Number.MIN_SAFE_INTEGER instead of 0 -> Option B

4. Quick Check:

   Proper initialization fixes negative-only arrays [OK]

Hint: Use smallest integer for maxSoFar to handle negatives [OK]

Common Mistakes:

• Leaving maxSoFar as 0 causes wrong max for negatives
• Removing reset of maxEndingHere breaks algorithm logic
• Returning wrong variable from function

5. Given an array of integers, how can Kadane's Algorithm be adapted to also return the start and end indices of the maximum sum subarray?
Choose the correct approach.

hard

A. Use two nested loops to find max sum and indices

B. Store all subarrays and pick the max sum one at the end

C. Track start index when maxEndingHere resets; update end index when maxSoFar updates

D. Return the first positive element's index as start and end

Solution

1. Step 1: Understand Kadane's Algorithm extension for indices

   When maxEndingHere resets to current element, update start index; when maxSoFar updates, update end index to current position.

2. Step 2: Compare with other approaches

   Storing all subarrays or using nested loops is inefficient; returning first positive index is incorrect.

3. Final Answer:

   Track start index when maxEndingHere resets; update end index when maxSoFar updates -> Option C

4. Quick Check:

   Index tracking with resets and updates = Track start index when maxEndingHere resets; update end index when maxSoFar updates [OK]

Hint: Update start on reset, end on max update for indices [OK]

Common Mistakes:

• Using nested loops loses Kadane's efficiency
• Ignoring index updates when maxEndingHere resets
• Assuming first positive element is max subarray start