This document discusses finding the maximum subarray sum in a given array. It begins by defining the problem of finding the contiguous subarray with the largest sum. It then explains solving it using a dynamic programming approach. The solution loops through the array, tracking the maximum sum of subarrays from the start of the array at each index. This allows finding the maximum sum subarray in O(n) time and space complexity. In the end, the document emphasizes understanding the problem fully before coding and how dynamic programming enables efficient solutions.

1. Python Array Challenges Python Coding Challenges
Maximum Subarray Sum in Python
written by Kal Bartal February 4, 2023
Click to watch video tutorial on Maximum Subarray Sum in Python Video Tutorial on YouTube
Problem Statement:
Given an array of integers, find the contiguous subarray (containing at least one number) which
has the largest sum and return its sum.
Example:
Input: [-2, 1, -3, 4, -1, 2, 1, -5, 4]
Output: 6
Explanation: The contiguous subarray [4, -1, 2, 1] has the largest sum = 6.
Understanding the problem
This problem is all about finding the maximum sum of a continuous sequence of numbers within
an array. All we need to do is look at each contiguous subarray and find the one with the largest
sum. A contiguous subarray is just a subset of an array and must include at least one number.
We just need to make sure that we return the maximum sum.
For example, given the array [-2, 1, -3, 4, -1, 2, 1, -5, 4], the maximum sum is 6
because the subarray [4, -1, 2, 1] has the largest sum.

2. Solving this problem
Well, the first thing you should know is the concept of a subarray. A subarray is just a subset of
an array and it must include at least one number. Then, you should also understand how to find
the sum of a subarray. This can just be done by adding up all the numbers in the subarray. Lastly,
you should know how to compare different subarrays and find the one with the largest sum.
Once you understand the basics, solving this problem should be easier. You just have to make
sure that you go through each subarray and compare the sums. Then, you can return the
maximum sum.
Subarrays in Arrays
– Sum subarrays by adding their numbers.
– Maximum Subarray Sum finds max sum of subarrays.
– Subarrays must have consecutive elements.
A subarray is just a subset of an array. It must include at least one number and the elements
must be consecutive within the same array. The subarrays can be of any size, as long as the
elements are all next to each other. We will be using subarrays in the Maximum Subarray Sum
problem to find the subarray with the largest sum.
Example: If we had the array [2, 4, 6, 8], then the subarrays could be [2] [4], [6], [8],
[2, 4], [4, 6], [6, 8], [2,4,6], [4, 6, 8], and so on.
Calculating Subarray Sums
– Add up the numbers in a subarray to calculate its sum.
– Maximum Subarray Sum problem finds max sum of all subarrays.
– Sums need to be compared individually.
Finding the sum of a subarray is pretty straightforward. All you need to do is add up the numbers
in the subarray. We will use this principle to find the sum of each subarray and compare it to the
other subarrays to find the maximum sum.
Example: If we had the subarray [3, 4, 7], the sum would be 3 + 4 + 7 = 14.
Calculating the Subarray with the Largest Sum
– Calculate the sum of each subarray
– Compare the sums to all the other subarrays
– Return the subarray with the largest sum

3. Comparing different subarrays to find the one with the largest sum is pretty simple. All you need
to do is calculate the sum of each subarray and compare it to all the other subarrays. The
subarray with the largest sum will be the one you want to return.
Example: Given the array [-2, 1, -3, 4, -1, 2, 1, -5, 4], the maximum sum is 6
because the subarray [4, -1, 2, 1] has the largest sum.
Finding the Maximum Sum of a Subarray
We can use a dynamic programming approach to solve challenge. We can start by creating a list
to store the maximum sum of all subarrays from the beginning of the array to the current
position. We also need a variable to keep track of the maximum sum of all subarrays seen so
far.
We then loop through the array, and for each element, we calculate the maximum sum of the
subarray starting at the beginning, and add the current element to it. We then compare this
maximum sum to the maximum sum of all subarrays seen so far and update the maximum sum
if necessary. Once we have gone through the entire array, we will have the maximum sum of all
subarrays, which we can then return.
Here is a solution in Python that uses dynamic programming to solve the maximum subarray
sum problem:
def maxSubarraySum(arr):
# create a list to store the maximum sum of all subarrays from the beg
inning of the array to the current position
subarrays_sum = [0]
# variable to keep track of the maximum sum of all subarrays seen so f
ar
max_sum = 0
# loop through the array and calculate the maximum sum of the subarray
starting at the beginning, and add the
# current element to it
for num in arr:
subarrays_sum.append(max(subarrays_sum[-1] + num, num))
max_sum = max(max_sum, subarrays_sum[-1])
# return the maximum sum of all subarrays
return max_sum
# For example, given the array [-2, 1, -3, 4, -1, 2, 1, -5, 4], the maximu
m sum is 6 because the subarray [4, -1, 2,
# 1] has the largest sum.
print(maxSubarraySum([-2, 1, -3, 4, -1, 2, 1, -5, 4])) # 6
This code uses dynamic programming to solve the Maximum Subarray Sum problem. We start
by creating a list to store the maximum sum of all subarrays from the beginning of the array to
the current index. Then, we have a variable to keep track of the maximum sum of all subarrays
seen so far.
We then loop through the array and for each element, we calculate the maximum sum of the
subarray starting at the beginning and add the current element to it. We then compare this

4. maximum sum to the maximum sum of all subarrays seen so far and update the maximum sum
if necessary.
Once we have gone through the entire array, we will have the maximum sum of all subarrays,
which we can then return.
Time Complexity of Algorithm
– O(n), where n is the length of the array
– Loop array once, calculate max sum with constant time
– Efficient and able to handle large inputs
The time complexity of this code is O(n), where n is the length of the array. This is because we
only need to loop through the array once and for each element, the maximum sum of the
subarray is calculated with a constant time operation. This makes the time complexity of the
algorithm totally linear, making it really efficient and able to handle large input arrays without any
problems.
Space Complexity of Code
– Code runs on O(n), same as linear time
– List stores max sum of subarrays from the beginning of the array
– List size same as array length, consistent space needs
In terms of space complexity! It runs on O(n), the same as linear time. We’re using a list to store
the maximum sum of all subarrays from the beginning of the array, and the size of the list is the
same as the array’s length. This means the amount of space needed remains consistent even for
large input arrays.
Final Thoughts
– Efficient solution with time and space complexity of O(n)
– Think through a coding challenge
– Talk through the problem to ensure understanding
This is a great example of how powerful dynamic programming can be! It allows us to find the
most efficient solution for a problem, as well as giving us the chance to think about it in-depth.
Plus, a solid understanding of the problem is key, so it’s always important to ensure you
understand it before starting to code. Awesome work!