PHP Program for Largest Sum Contiguous Subarray

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

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 = 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: 

<?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

Another approach:

<?php 
function maxSubArraySum(&$a, $size)
{
$max_so_far = $a[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 the array are negative. 

<?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.  

<?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

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)

Now try the below question 
Given an array of integers (possibly some 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 the maximum product subarray.

kartik

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