Coding Dynamic Programming

Largest Sum Contiguous Subarray

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Python Articles and errors
Python Programming - Largest Sum Contiguous Subarray - Dynamic Programming Write program to find the sum of contiguous subarray within one-dimensional array

Largest Sum Contiguous Subarray

Write an efficient Python program to find the sum of contiguous subarray within a one-dimensional array of numbers which has the largest sum.

kadane Algorithm

 

 

 

 

 

 

 

 

 

 

 

Kadane’s Algorithm:

Initialize:
    max_so_far = 0
    max_ending_here = 0

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

Explanation:

Simple idea of the 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 = 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

    for i=1,  a[1] =  -3
    max_ending_here = max_ending_here + (-3)
    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 0 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 for Largest Sum Contiguous Subarray

Python

# 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 -3

Above program can be optimized further, if we compare max_so_far with max_ending_here only if max_ending_here is greater than 0.

READ  C program print integer

Program

Python

def maxSubArraySum(a,size): 
      
    max_so_far = 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

The implementation handles the case when all numbers in array are negative.

Program

Python

# 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 
  
# Driver function to check the above function  
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.

Program

Python
# Python program to print largest contiguous array sum 
  
from sys import maxsize 
  
# Function to find the maximum contiguous subarray 
# and print its starting and end index 
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)) 
  
# Driver program to test maxSubArraySum 
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

About the author

Venkatesan Prabu

Venkatesan Prabu

Wikitechy Founder, Author, International Speaker, and Job Consultant. My role as the CEO of Wikitechy, I help businesses build their next generation digital platforms and help with their product innovation and growth strategy. I'm a frequent speaker at tech conferences and events.

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