# Max Sum Contiguous Subarray

This is a maximum sum contiguous problem from interviewbit.com

Problem : Find the contiguous subarray within an array (containing at least one number) which has the largest sum.

For example: Given the array [-2,1,-3,4,-1,2,1,-5,4], the contiguous subarray [4,-1,2,1] has the largest sum = 6.

This is my solution :

def max_sub_array(array):
""" Finds msub-array with maximum sum and returns the maximum sum and sub-array"""
max_start, max_stop = 0, 1 # Start, Stop of maximum sub-array
curr = 0                   # pointer to current array
max_sum = array[0]         # Sum of maximum array
current_sum = 0            # sum of current array

for i, elem in enumerate(array):
current_sum +=  elem

if max_sum < current_sum:
max_sum = current_sum
max_start = curr
max_stop = i  + 1
if current_sum < 0:
current_sum = 0
curr = i + 1

return  max_sum , array[max_start:max_stop]


checking test cases:

assert max_sub_array([-4,-2,-3,-4,-5]) == (-2,[-2]), "Wrong evaluation"
assert max_sub_array([-1]) == (-1,[-1]), "Wrong evaluation"
assert max_sub_array([-5, 1, -3, 7, -1, 2, 1, -4, 6]) == (11,[7, -1, 2, 1, -4, 6]), "Wrong evaluation"
assert max_sub_array([-5, 1, -3, 7, -1, 2, 1, -6, 5]) == (9, [7, -1, 2, 1]), "Wrong evaluation"
assert max_sub_array( [6, -3, -2, 7, -5, 2, 1, -7, 6]) == (8,[6, -3, -2, 7]), "Wrong evaluation"
assert max_sub_array([-5, -2, -1, -4, -7]) == (-1,[-1]), "Wrong evaluation"
assert max_sub_array( [4, 1, 1, 4, -4, 10, -4, 10, 3, -3, -9, -8, 2, -6, -6, -5, -1, -7, 7, 8]) == (25,[4, 1, 1, 4, -4, 10, -4, 10, 3]), "Wrong evaluation"
assert max_sub_array([4, -5, -1, 0, -2, 20, -4, -3, -2, 8, -1, 10, -1, -1 ]) ==  (28, [20, -4, -3, -2, 8, -1, 10]), "Wrong evaluation"


How can this code be optimised ?

• It is a known problem in computer science as @kaushal says. Kadane's algorithm is $O(n)$ (linear run time complexity) and is probably as good as it gets. – esote Jul 10 '18 at 2:27

Expanding upon my comment:

def max_subarray(arr):
max_ending = max_current = arr[0]

for i in arr[1:]:
max_ending = max(i, max_ending + i)
max_current = max(max_current, max_ending)

return max_current

print(max_subarray([-4, -2, -3, -4, 5])) # 5

print(max_subarray([4, -5, -1, 0, -2, 20, -4, -3, -2, 8, -1, 10, -1, -1])) # 28


Like your algorithm, it is $O(n)$. However, it does fewer operations while looping. You could then alter it to return the array which gives that sum, which shouldn't be too hard.

It goes without saying that this is a well known problem. ref

Your code is already linear in the input, takes a single pass over the array, and does the minimum possible checks (if statements) and update assignments for the involved variables.

It cannot be optimised any further.

• Welcome to Code Review! Whilst this may theoretically answer the question, it would be preferable to include the essential parts of the answer here, and provide the link for reference. – Dannnno May 10 '18 at 22:49