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 " ,
$max_sum;
// This code is contributed by anuj_67.
?>
<?php// PHP program to print largest// contiguous array sumfunction maxSubArraySum($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 $max_so_far;
// This code is contributed
// by ChitraNayal
?>
<?php function maxSubArraySum(&$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 " .
$max_sum;
// This code is contributed
// by Akanksha Rai(Abby_akku)
?>
<?phpfunction maxSubArraySum($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, $n);
// This code is contributed
// by ChitraNayal
?>
<?php // PHP program to print largest // contiguous array sumfunction maxSubArraySum($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.
