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

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_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:

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; 
    } 
      
    // Drive 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; 
    } 
  
    // Drive 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.

My Personal Notes

Oct	NOV	Dec
	16
2018	2019	2020

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