Largest Sum Contiguous Subarray

Difficulty Level : Medium
Last Updated : 01 Aug, 2020

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

Recommended: Please solve it on “PRACTICE ” first, before moving on to the solution.

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++

// 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; 
} 

Java

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; 
    } 
} 

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)) 
   
#This code is contributed by _Devesh Agrawal_ 

C#

// 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_ 

PHP

<?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. 
?> 

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++

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; 
} 

Java

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 

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 

C#

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 

PHP

<?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 
?> 

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++

#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; 
} 

Java

// 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 

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)) 
  
#This code is contributed by _Devesh Agrawal_ 

C#

// 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_ 

PHP

<?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) 
?> 

Output:

Maximum contiguous sum is 7

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

C++

// 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; 
} 

Java

// 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 

Python3

# 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)) 

C#

// 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. 

PHP

<?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 
?> 

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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