# Python Program - Maximum Subarray Problem

Kadane's algorithm is used to find the maximum sum of a contiguous subarray. Kadane's algorithm is based on the idea of looking for all positive contiguous subarray and find the maximum sum of a contiguous subarray.

In this algorithm, a variable called max_sum is created to store maximum sum of the positive contiguous subarray till current iterated element and a variable called current_sum is created to store sum of the positive subarray which ends at current iterated element. In each iteration, current_sum is compared with max_sum, to update max_sum if it is greater than max_sum.

### Example:

To understand the kadane's algorithm, lets consider an array $Array = [-3, 1, -8, 12, 0, -3, 5, -9, 4]$ and discuss each step taken to find the maximum sum of all positive contiguous subarray.

  max_sum = current_sum = 0

Step 1: i = 0, Array[0] =  -3
current_sum = current_sum + (-3) = -3
Set current_sum = 0 because current_sum < 0

Step 2: i = 1, Array[0] =  1
current_sum = current_sum + 1 = 1
update max_sum = 1 because current_sum > max_sum

Step 3: i = 2, Array[0] =  -8
current_sum = current_sum + (-8) = -7
Set current_sum = 0 because current_sum < 0

Step 4: i = 3, Array[0] =  12
current_sum = current_sum + 12 = 12
update max_sum = 12 because current_sum > max_sum

Step 5: i = 4, Array[0] =  0
current_sum = current_sum + 0 = 12

Step 6: i = 5, Array[0] =  -3
current_sum = current_sum + (-3) = 9

Step 7: i = 6, Array[0] =  5
current_sum = current_sum + 5 = 14
update max_sum = 14 because current_sum > max_sum

Step 8: i = 7, Array[0] =  -9
current_sum = current_sum + (-9) = 5

Step 9: i = 8, Array[0] =  4
current_sum = current_sum + 4 = 9


Hence, after all iterations, the value of max_sum is 14. The stating index point and end index point of this subarray are 3 and 6 respectively.

# function for kadane's algorithm
max_sum = 0
current_sum = 0
for i in MyList:
current_sum = current_sum + i
if current_sum < 0:
current_sum = 0
if max_sum < current_sum:
max_sum = current_sum
return max_sum

MyList = [-3, 1, -8, 12, 0, -3, 5, -9, 4]


Output

Maximum SubArray is: 14


To get the location of maximum subarray, variables max_start and max_end are maintained with the help of variables current_start and current_end.

# function for kadane's algorithm
max_sum = 0
current_sum = 0

max_start = 0
max_end = 0
current_start = 0
current_end = 0

for i in range(len(MyList)):
current_sum = current_sum + MyList[i]
current_end = i
if current_sum < 0:
current_sum = 0
# Start a new sequence from next element
current_start = current_end + 1

if max_sum < current_sum:
max_sum = current_sum
max_start = current_start
max_end = current_end

print("Maximum SubArray is:", max_sum)
print("Start index of max_Sum:", max_start)
print("End index of max_Sum:", max_end)

MyList = [-3, 1, -8, 12, 0, -3, 5, -9, 4]


Output

Maximum SubArray is: 14
Start index of max_Sum: 3
End index of max_Sum: 6


### Time Complexity:

The time complexity of Kadane's algorithm is $\mathcal{O}(N)$.