Kadane's Algorithm

Kadane’s Algorithm:

Initialize:
    max_so_far = INT_MIN
    max_ending_here = 0
Loop for each element of the array
(a) max_ending_here = max_ending_here + a[i]
(b) if(max_so_far < max_ending_here)
max_so_far = max_ending_here
(c) if(max_ending_here < 0)
max_ending_here = 0
return max_so_far

Explanation:
The simple idea of Kadane’s algorithm is to look for all positive contiguous segments of the array (max_ending_here is used for this). And keep track of maximum sum contiguous segment among all positive segments (max_so_far is used for this). Each time we get a positive sum compare it with max_so_far and update max_so_far if it is greater than max_so_far

Lets take the example:
{-2, -3, 4, -1, -2, 1, 5, -3}
max_so_far = INT_MIN
max_ending_here = 0

for i=0,  a[0] =  -2
max_ending_here = max_ending_here + (-2)
Set max_ending_here = 0 because max_ending_here < 0
and set max_so_far = -2

for i=1,  a[1] =  -3
max_ending_here = max_ending_here + (-3)
Since max_ending_here = -3 and max_so_far = -2, max_so_far will remain -2
Set max_ending_here = 0 because max_ending_here < 0
  

for i=2,  a[2] =  4
max_ending_here = max_ending_here + (4)
max_ending_here = 4
max_so_far is updated to 4 because max_ending_here greater 
than max_so_far which was -2 till now

for i=3,  a[3] =  -1
max_ending_here = max_ending_here + (-1)
max_ending_here = 3

for i=4,  a[4] =  -2
max_ending_here = max_ending_here + (-2)
max_ending_here = 1

for i=5,  a[5] =  1
max_ending_here = max_ending_here + (1)
max_ending_here = 2

for i=6,  a[6] =  5
max_ending_here = max_ending_here + (5)
max_ending_here = 7
max_so_far is updated to 7 because max_ending_here is 
greater than max_so_far

for i=7,  a[7] =  -3
max_ending_here = max_ending_here + (-3)
max_ending_here = 4

Program:
# Python program to find maximum contiguous subarray
# Function to find the maximum contiguous subarray

from sys import maxint
def maxSubArraySum(a,size):     
    max_so_far = -maxint - 1
    max_ending_here = 0     
    for i in range(0, size):
        max_ending_here = max_ending_here + a[i] 
        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

# Driver function to check the above function
a = [-13, -3, -25, -20, -3, -16, -23, -12, -5, -22, -15, -4, -7]
print "Maximum contiguous sum is", maxSubArraySum(a,len(a))

Output:

Maximum contiguous sum is 7

Another approach:

def maxSubArraySum(a,size):
    max_so_far = a[0]
    max_ending_here = 0
    for i in range(0, size):
        max_ending_here = max_ending_here + a[i]
        if max_ending_here < 0:
            max_ending_here = 0    
        # Do not compare for all elements. Compare only  
        # when  max_ending_here > 0
        elif (max_so_far < max_ending_here):
            max_so_far = max_ending_here         
    return max_so_far

Time Complexity: O(n)

Algorithmic Paradigm: Dynamic Programming
Following is another simple implementation suggested by Mohit Kumar. The implementation handles the case
when all numbers in the array are negative.

Python program to find maximum contiguous subarray

def maxSubArraySum(a,size):
    max_so_far =a[0]
    curr_max = a[0]
    for i in range(1,size):
        curr_max = max(a[i], curr_max + a[i])
        max_so_far = max(max_so_far,curr_max)
    return max_so_far
a = [-2, -3, 4, -1, -2, 1, 5, -3]
print"Maximum contiguous sum is" , maxSubArraySum(a,len(a))

Output:

Maximum contiguous sum is 7
To print the subarray with the maximum sum, we maintain indices whenever we get the maximum sum.

Python program to print largest contiguous array sum

from sys import maxsize
def maxSubArraySum(a,size):
    max_so_far = -maxsize - 1
    max_ending_here = 0
    start = 0
    end = 0
    s = 0
    for i in range(0,size):
        max_ending_here += a[i]
        if max_so_far <  max_ending_here:
            max_so_far = max_ending_here
            start = s
            end = i
        if max_ending_here <  0:
            max_ending_here = 0
            s = i+1
    print ("Maximum contiguous sum is %d"%(max_so_far))
    print ("Starting Index %d"%(start))
    print ("Ending Index %d"%(end))
a = [-2, -3, 4, -1, -2, 1, 5, -3] 
maxSubArraySum(a,len(a))

Output:

Maximum contiguous sum is 7

Starting index 2
Ending index 6

Kadane’s Algorithm can be viewed both as a greedy and DP. As we can see that we are keeping a running sum
of integers and when it becomes less than 0, we reset it to 0 (Greedy Part). This is because
continuing with a negative sum is way more worse than restarting with a new range. Now it can also be
viewed as a DP, at each stage we have 2 choices: Either take the current element and continue with
previous sum OR restart a new range. These both choices are being taken care of in the implementation.

Time Complexity: O(n)
Auxiliary Space: O(1)