# PHP Program - Maximum Subarray Problem

Kadane's algorithm is used to find the maximum sum of a contiguous subarray. Kadane's algorithm is based on the idea of looking for all positive contiguous subarray and find the maximum sum of a contiguous subarray.

In this algorithm, a variable called max_sum is created to store maximum sum of the positive contiguous subarray till current iterated element and a variable called current_sum is created to store sum of the positive subarray which ends at current iterated element. In each iteration, current_sum is compared with max_sum, to update max_sum if it is greater than max_sum.

### Example:

To understand the kadane's algorithm, lets consider an array Array = [-3, 1, -8, 12, 0, -3, 5, -9, 4] and discuss each step taken to find the maximum sum of all positive contiguous subarray. ```  max_sum = current_sum = 0

Step 1: i = 0, Array =  -3
current_sum = current_sum + (-3) = -3
Set current_sum = 0 because current_sum < 0

Step 2: i = 1, Array =  1
current_sum = current_sum + 1 = 1
update max_sum = 1 because current_sum > max_sum

Step 3: i = 2, Array =  -8
current_sum = current_sum + (-8) = -7
Set current_sum = 0 because current_sum < 0

Step 4: i = 3, Array =  12
current_sum = current_sum + 12 = 12
update max_sum = 12 because current_sum > max_sum

Step 5: i = 4, Array =  0
current_sum = current_sum + 0 = 12

Step 6: i = 5, Array =  -3
current_sum = current_sum + (-3) = 9

Step 7: i = 6, Array =  5
current_sum = current_sum + 5 = 14
update max_sum = 14 because current_sum > max_sum

Step 8: i = 7, Array =  -9
current_sum = current_sum + (-9) = 5

Step 9: i = 8, Array =  4
current_sum = current_sum + 4 = 9
```

Hence, after all iterations, the value of max_sum is 14. The stating index point and end index point of this subarray are 3 and 6 respectively.

```<?php
\$max_sum = 0;
\$current_sum = 0;
for(\$i=0; \$i<\$n; \$i++) {
\$current_sum = \$current_sum + \$Array[\$i];

if (\$current_sum < 0)
\$current_sum = 0;

if(\$max_sum < \$current_sum)
\$max_sum = \$current_sum;
}
return \$max_sum;
}

// test the code
\$MyArray = array(-3, 1, -8, 12, 0, -3, 5, -9, 4);
\$n = sizeof(\$MyArray);
echo "Maximum SubArray is: ".kadane(\$MyArray, \$n);
?>
```

The above code will give the following output:

```Maximum SubArray is: 14
```

To get the location of maximum subarray, variables max_start and max_end are maintained with the help of variables current_start and current_end.

```<?php
\$max_sum = 0;
\$current_sum = 0;

\$max_start = 0;
\$max_end = 0;
\$current_start = 0;
\$current_end = 0;

for(\$i=0; \$i<\$n; \$i++) {
\$current_sum = \$current_sum + \$Array[\$i];
\$current_end = \$i;

if (\$current_sum < 0) {
\$current_sum = 0;
//Start a new sequence from next element
\$current_start = \$current_end + 1;
}

if(\$max_sum < \$current_sum) {
\$max_sum = \$current_sum;
\$max_start = \$current_start;
\$max_end = \$current_end;
}
}

echo "Maximum SubArray is: ".\$max_sum."\n";
echo "Start index of max_Sum: ".\$max_start."\n";
echo "End index of max_Sum: ".\$max_end."\n";
}

// test the code
\$MyArray = array(-3, 1, -8, 12, 0, -3, 5, -9, 4);
\$n = sizeof(\$MyArray);
?>
```

The above code will give the following output:

```Maximum SubArray is: 14
Start index of max_Sum: 3
End index of max_Sum: 6
```

### Time Complexity:

The time complexity of Kadane's algorithm is Θ(N).

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