Largest Sum Contiguous Subarray

Write an efficient 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_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:
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:



C++

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// C++ program to print largest contiguous array sum
#include<iostream>
#include<climits>
using namespace std;
  
int maxSubArraySum(int a[], int size)
{
    int max_so_far = INT_MIN, max_ending_here = 0;
  
    for (int i = 0; i < size; i++)
    {
        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 program to test maxSubArraySum*/
int main()
{
    int a[] = {-2, -3, 4, -1, -2, 1, 5, -3};
    int n = sizeof(a)/sizeof(a[0]);
    int max_sum = maxSubArraySum(a, n);
    cout << "Maximum contiguous sum is " << max_sum;
    return 0;
}

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Java

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import java.io.*;
// Java program to print largest contiguous array sum
import java.util.*;
  
class Kadane
{
    public static void main (String[] args)
    {
        int [] a = {-2, -3, 4, -1, -2, 1, 5, -3};
        System.out.println("Maximum contiguous sum is " +
                                       maxSubArraySum(a));
    }
  
    static int maxSubArraySum(int a[])
    {
        int size = a.length;
        int max_so_far = Integer.MIN_VALUE, max_ending_here = 0;
  
        for (int i = 0; i < size; i++)
        {
            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;
    }
}

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Python

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# 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))
   
#This code is contributed by _Devesh Agrawal_

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C#

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// C# program to print largest 
// contiguous array sum
using System;
  
class GFG
{
    static int maxSubArraySum(int []a)
    {
        int size = a.Length;
        int max_so_far = int.MinValue, 
            max_ending_here = 0;
  
        for (int i = 0; i < size; i++)
        {
            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 code 
    public static void Main ()
    {
        int [] a = {-2, -3, 4, -1, -2, 1, 5, -3};
        Console.Write("Maximum contiguous sum is " +
                                maxSubArraySum(a));
    }
  
}
  
// This code is contributed by Sam007_

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PHP

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<?php
// PHP program to print largest
// contiguous array sum
  
function maxSubArraySum($a, $size)
{
    $max_so_far = PHP_INT_MIN; 
    $max_ending_here = 0;
  
    for ($i = 0; $i < $size; $i++)
    {
        $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 code
$a = array(-2, -3, 4, -1,
           -2, 1, 5, -3);
$n = count($a);
$max_sum = maxSubArraySum($a, $n);
echo "Maximum contiguous sum is "
                          $max_sum;
  
// This code is contributed by anuj_67.
?>

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Output:

Maximum contiguous sum is 7


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.

C++

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int maxSubArraySum(int a[], int size)
{
   int max_so_far = 0, max_ending_here = 0;
   for (int i = 0; i < size; i++)
   {
       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 */
       else if (max_so_far < max_ending_here)
           max_so_far = max_ending_here;
   }
   return max_so_far;
}

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Java

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static int maxSubArraySum(int a[],int size) 
      
    int max_so_far = 0, max_ending_here = 0
  
    for (int i = 0; i < size; i++) 
    
        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 */
        else if (max_so_far < max_ending_here) 
            max_so_far = max_ending_here; 
          
    
    return max_so_far; 
  
// This code is contributed by ANKITRAI1

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Python

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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

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C#

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static int maxSubArraySum(int[] a,
                          int size) 
int max_so_far = 0, 
    max_ending_here = 0; 
  
for (int i = 0; i < size; i++) 
    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 */
    else if (max_so_far < max_ending_here) 
        max_so_far = max_ending_here; 
return max_so_far; 
  
// This code is contributed
// by ChitraNayal

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PHP

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<?php 
function maxSubArraySum(&$a, $size)
{
$max_so_far = 0;
$max_ending_here = 0;
for ($i = 0; $i < $size; $i++)
{
    $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 */
    else if ($max_so_far < $max_ending_here)
        $max_so_far = $max_ending_here;
}
return $max_so_far;
  
// This code is contributed
// by ChitraNayal
?>

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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 array are negative.

C++

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#include<iostream>
using namespace std;
  
int maxSubArraySum(int a[], int size)
{
   int max_so_far = a[0];
   int curr_max = a[0];
  
   for (int i = 1; i < size; i++)
   {
        curr_max = max(a[i], curr_max+a[i]);
        max_so_far = max(max_so_far, curr_max);
   }
   return max_so_far;
}
  
/* Driver program to test maxSubArraySum */
int main()
{
   int a[] =  {-2, -3, 4, -1, -2, 1, 5, -3};
   int n = sizeof(a)/sizeof(a[0]);
   int max_sum = maxSubArraySum(a, n);
   cout << "Maximum contiguous sum is " << max_sum;
   return 0;
}

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Java

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// Java program to print largest contiguous
// array sum
import java.io.*;
  
class GFG {
  
    static int maxSubArraySum(int a[], int size)
    {
    int max_so_far = a[0];
    int curr_max = a[0];
  
    for (int i = 1; i < size; i++)
    {
           curr_max = Math.max(a[i], curr_max+a[i]);
        max_so_far = Math.max(max_so_far, curr_max);
    }
    return max_so_far;
    }
  
    /* Driver program to test maxSubArraySum */
    public static void main(String[] args)
    {
    int a[] = {-2, -3, 4, -1, -2, 1, 5, -3};
    int n = a.length;   
    int max_sum = maxSubArraySum(a, n);
    System.out.println("Maximum contiguous sum is " 
                       + max_sum);
    }
}
  
// This code is contributd by Prerna Saini

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Python

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# 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))
  
#This code is contributed by _Devesh Agrawal_

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C#

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// C# program to print largest 
// contiguous array sum
using System;
  
class GFG
{
    static int maxSubArraySum(int []a, int size)
    {
    int max_so_far = a[0];
    int curr_max = a[0];
  
    for (int i = 1; i < size; i++)
    {
        curr_max = Math.Max(a[i], curr_max+a[i]);
        max_so_far = Math.Max(max_so_far, curr_max);
    }
  
    return max_so_far;
    }
  
    // Driver code 
    public static void Main ()
    {
        int []a = {-2, -3, 4, -1, -2, 1, 5, -3};
        int n = a.Length; 
        Console.Write("Maximum contiguous sum is "
                           + maxSubArraySum(a, n));
    }
  
}
  
// This code is contributed by Sam007_

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PHP

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<?php
function maxSubArraySum($a, $size)
{
    $max_so_far = $a[0];
    $curr_max = $a[0];
      
    for ($i = 1; $i < $size; $i++)
    {
        $curr_max = max($a[$i], 
                        $curr_max + $a[$i]);
        $max_so_far = max($max_so_far
                          $curr_max);
    }
    return $max_so_far;
}
  
// Driver Code
$a = array(-2, -3, 4, -1, 
           -2, 1, 5, -3);
$n = sizeof($a);
$max_sum = maxSubArraySum($a, $n);
echo "Maximum contiguous sum is " .
                          $max_sum;
  
// This code is contributed 
// by Akanksha Rai(Abby_akku)
?>

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Output:

Maximum contiguous sum is 7

To print the subarray with the maximum sum, we maintain indices whenever we get the maximum sum.

C++

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// C++ program to print largest contiguous array sum
#include<iostream>
#include<climits>
using namespace std;
  
int maxSubArraySum(int a[], int size)
{
    int max_so_far = INT_MIN, max_ending_here = 0,
       start =0, end = 0, s=0;
  
    for (int i=0; i< size; i++ )
    {
        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;
        }
    }
    cout << "Maximum contiguous sum is "
        << max_so_far << endl;
    cout << "Starting index "<< start
        << endl << "Ending index "<< end << endl;
}
  
/*Driver program to test maxSubArraySum*/
int main()
{
    int a[] = {-2, -3, 4, -1, -2, 1, 5, -3};
    int n = sizeof(a)/sizeof(a[0]);
    int max_sum = maxSubArraySum(a, n);
    return 0;
}

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Java

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// Java program to print largest 
// contiguous array sum
class GFG {
  
    static void maxSubArraySum(int a[], int size)
    {
        int max_so_far = Integer.MIN_VALUE,
        max_ending_here = 0,start = 0,
        end = 0, s = 0;
  
        for (int i = 0; i < size; i++) 
        {
            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;
            }
        }
        System.out.println("Maximum contiguous sum is " 
                           + max_so_far);
        System.out.println("Starting index " + start);
        System.out.println("Ending index " + end);
    }
  
    // Driver code
    public static void main(String[] args)
    {
        int a[] = { -2, -3, 4, -1, -2, 1, 5, -3 };
        int n = a.length;
        maxSubArraySum(a, n);
    }
}
  
// This code is contributed by  prerna saini

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Python3

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# 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))

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C#

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// C# program to print largest 
// contiguous array sum
using System;
  
class GFG 
{
    static void maxSubArraySum(int []a, 
                               int size)
    {
        int max_so_far = int.MinValue,
        max_ending_here = 0, start = 0,
        end = 0, s = 0;
  
        for (int i = 0; i < size; i++) 
        {
            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;
            }
        }
        Console.WriteLine("Maximum contiguous "
                         "sum is " + max_so_far);
        Console.WriteLine("Starting index "
                                      start);
        Console.WriteLine("Ending index "
                                      end);
    }
  
    // Driver code
    public static void Main()
    {
        int []a = {-2, -3, 4, -1, 
                   -2, 1, 5, -3};
        int n = a.Length;
        maxSubArraySum(a, n);
    }
}
  
// This code is contributed
// by anuj_67.

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PHP

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<?php 
// PHP program to print largest 
// contiguous array sum
  
function maxSubArraySum($a, $size)
{
    $max_so_far = PHP_INT_MIN;
    $max_ending_here = 0;
    $start = 0;
    $end = 0;
    $s = 0;
  
    for ($i = 0; $i < $size; $i++)
    {
        $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;
        }
    }
    echo "Maximum contiguous sum is ".
                     $max_so_far."\n";
    echo "Starting index ". $start . "\n".
            "Ending index " . $end . "\n";
}
  
// Driver Code
$a = array(-2, -3, 4, -1, -2, 1, 5, -3);
$n = sizeof($a);
$max_sum = maxSubArraySum($a, $n);
  
// This code is contributed
// by ChitraNayal
?>

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Output:

Maximum contiguous sum is 7
Starting index 2
Ending index 6

Now try below question
Given an array of integers (possibly some of the elements negative), write a C program to find out the *maximum product* possible by multiplying ‘n’ consecutive integers in the array where n <= ARRAY_SIZE. Also print the starting point of maximum product subarray.


References:
http://en.wikipedia.org/wiki/Kadane%27s_Algorithm

Please write comments if you find anything incorrect, or you want to share more information about the topic discussed above.

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