I need to find the SUMMATION of Si * Ci (for i in range of 1 to n).

Following are the constraints:

1 ≤ n ≤ 10^6
1 ≤ si ≤ 10^6
1 ≤ ci ≤ 10^6

Here is my code.

<?php 
    for($i = 0; $i < $n; $i++){
        $total += ($s[$i] * $c[$i]);
    } 
?>

However, the code runs out of time for high input values(upto 10 ^ 6).

How to add such huge numbers with proper optimisation techniques without exceeding the execution time.

You can increase the max execution time using set_time_limit

Read more here: http://php.net/manual/en/function.set-time-limit.php

set_time_limit — Limits the maximum execution time

bool set_time_limit ( int $seconds )

If you set it to zero it will run forever until the script finishes execution or it is interrupted manually. Place this at the top of your script to do that:

set_time_limit(0);

Update:

Use a language other than PHP.

There are few aspects you should take care:

Execution Time

These operations could take lots of time, so you have to change request time limit:

set_time_limit(0);

Usage of built in function for map/reduce

PHP offers some map/reduce functions for such operations. One is array_map() and another is array_reduce()

function multiply($s, $c) {
  return $s * $c;
}
function sum($carry, $item)
{
    $carry += $item;
    return $carry;
}


$result = array_reduce(array_map('multiply', $s, $c), 'sum');

Of couse you can use anonymous functions as callbacks.

Precision / Overflow

Depending on you data you should take care of capacity/precision problems. In this case take a look at bcmath extension and bcmul() bcadd() functions.