Kadane's Algorithm Maximum Subarray in DSA Python - Time & Space Complexity

Time Complexity: Kadane's Algorithm Maximum Subarray

O(n)

Understanding Time Complexity

We want to understand how the time needed to find the maximum sum of a continuous subarray changes as the input size grows.

How does the number of steps grow when the array gets bigger?

Scenario Under Consideration

Analyze the time complexity of the following code snippet.

def max_subarray(arr):
    max_ending_here = max_so_far = arr[0]
    for x in arr[1:]:
        max_ending_here = max(x, max_ending_here + x)
        max_so_far = max(max_so_far, max_ending_here)
    return max_so_far

This code finds the largest sum of any continuous part of the array using Kadane's Algorithm.

Identify Repeating Operations

Identify the loops, recursion, array traversals that repeat.

• Primary operation: A single loop that goes through each element of the array once.
• How many times: Exactly once for each element, so n times if the array has n elements.

How Execution Grows With Input

As the array gets bigger, the number of steps grows directly with the number of elements.

Input Size (n)	Approx. Operations
10	About 10 steps
100	About 100 steps
1000	About 1000 steps

Pattern observation: The steps increase in a straight line as the input size grows.

Final Time Complexity

Time Complexity: O(n)

This means the time to find the maximum subarray grows directly with the size of the input array.

Common Mistake

[X] Wrong: "Since we check sums of many subarrays, the time must be quadratic O(n²)."

[OK] Correct: Kadane's Algorithm cleverly keeps track of sums as it goes, so it only needs one pass through the array, not checking all subarrays separately.

Interview Connect

Understanding this linear time solution shows you can find efficient answers by smartly reusing information, a key skill in problem solving.

Self-Check

"What if we wanted to find the maximum subarray sum but also needed to return the start and end indexes? How would the time complexity change?"

Practice

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

easy

A. The longest increasing subsequence

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: Compare with other options

   Options B, C, and D describe different problems unrelated to Kadane's Algorithm.

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 continuous parts of array [OK]

Common Mistakes:

• Confusing maximum subarray sum with total array sum
• Thinking it finds minimum or longest subsequence
• Assuming it sorts the array

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

easy

A. max_so_far = 0; max_ending_here = float('inf')

B. max_so_far = 0; max_ending_here = 0

C. max_so_far = float('-inf'); max_ending_here = 0

D. max_so_far = null; max_ending_here = null

Solution

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

   max_so_far tracks the maximum sum found so far, so it should start very low (negative infinity) to handle negative numbers. max_ending_here tracks current subarray sum and starts at 0.

2. Step 2: Check each option

   max_so_far = 0; max_ending_here = 0 initializes max_so_far to 0, which fails if all numbers are negative. max_so_far = 0; max_ending_here = float('inf') wrongly sets max_ending_here to positive infinity. max_so_far = null; max_ending_here = null uses null, which causes errors in arithmetic.

3. Final Answer:

   max_so_far = float('-inf'); max_ending_here = 0 -> Option C

4. Quick Check:

   Initialize max_so_far very low, max_ending_here zero [OK]

Hint: Start max_so_far very low to handle negatives [OK]

Common Mistakes:

• Initializing max_so_far to 0 causes wrong results with negatives
• Using null causes type errors
• Setting max_ending_here to infinity breaks logic

3. What is the output of the following code?

def max_subarray_sum(arr):
    max_so_far = float('-inf')
    max_ending_here = 0
    for x in arr:
        max_ending_here += x
        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

print(max_subarray_sum([-2, 3, 2, -1, 4, -5]))

medium

A. 8

B. 7

C. 6

D. 5

Solution

1. Step 1: Trace the loop and update sums

   Start max_so_far = float('-inf'), max_ending_here = 0
   Iteration 1: x=-2, max_ending_here= -2; max_so_far updated to -2; -2<0 reset to 0 (max_so_far=-2)
   Iteration 2: x=3, max_ending_here=3 > -2, max_so_far=3
   Iteration 3: x=2, max_ending_here=5 >3, max_so_far=5
   Iteration 4: x=-1, max_ending_here=4 <5 no update, 4>=0 no reset
   Iteration 5: x=4, max_ending_here=8 >5, max_so_far=8
   Iteration 6: x=-5, max_ending_here=3 <8 no update, 3>=0 no reset

2. Step 2: Final return value

   max_so_far is 8, so the function returns 8.

3. Final Answer:

   8 -> Option A

4. Quick Check:

   Max subarray sum = 8 [OK]

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

Common Mistakes:

• Not resetting max_ending_here when negative
• Updating max_so_far incorrectly
• Miscounting sums during iteration

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

def max_subarray(arr):
    max_so_far = 0
    max_ending_here = 0
    for i in range(len(arr)):
        max_ending_here = max_ending_here + arr[i]
        if max_so_far < max_ending_here:
            max_so_far = max_ending_here
        elif max_ending_here < 0:
            max_ending_here = 0
    return max_so_far

print(max_subarray([-3, -2, -1]))

medium

A. max_so_far initialized to 0 causes wrong result for all negatives

B. Using elif instead of if causes skipping reset of max_ending_here

C. Loop should start from 1, not 0

D. max_ending_here should be initialized to float('-inf')

Solution

1. Step 1: Trace execution on [-3, -2, -1]

   max_so_far=0, max_ending_here=0.
   i=0, x=-3: meh=-3; 0 < -3? no; elif -3<0: yes, meh=0.
   i=1, x=-2: meh=-2; no update; reset meh=0.
   i=2, x=-1: meh=-1; no update; reset.
   Returns 0, but correct max subarray sum is -1.

2. Step 2: Why A and not others

   Init max_so_far=0 prevents update when all negative (meh never >0). B: elif ok here (if never true, always resets). C: range correct. D: meh=0 correct.

3. Final Answer:

   max_so_far initialized to 0 causes wrong result for all negatives -> Option A

4. Quick Check:

   Init max_so_far to float('-inf') for negatives [OK]

Hint: Initialize max_so_far to float('-inf') to handle all-negative arrays [OK]

Common Mistakes:

• Initializing max_so_far to 0 fails with all negatives
• Using elif causes reset to be skipped
• Wrong loop range causes index errors

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 best approach.

hard

A. Use a nested loop to check all subarrays and record indices

B. Track current start index when resetting max_ending_here; update global start and end when max_so_far changes

C. Store all subarrays and pick the max sum after the loop ends

D. Return max_so_far only; indices cannot be found with Kadane's Algorithm

Solution

1. Step 1: Understand Kadane's Algorithm extension

   To find indices, track the start index of the current subarray when max_ending_here resets to 0. When max_so_far updates, record current start and end indices.

2. Step 2: Compare with other options

   Store all subarrays and pick the max sum after the loop ends is inefficient and not Kadane's approach. Use a nested loop to check all subarrays and record indices is brute force, not Kadane's. Return max_so_far only; indices cannot be found with Kadane's Algorithm is incorrect; indices can be found.

3. Final Answer:

   Track current start index when resetting max_ending_here; update global start and end when max_so_far changes -> Option B

4. Quick Check:

   Track start on reset, update indices on max update [OK]

Hint: Track start index on reset, update when max_so_far changes [OK]

Common Mistakes:

• Ignoring index tracking in Kadane's Algorithm
• Using brute force instead of Kadane's logic
• Assuming indices can't be found with Kadane's