What is the Memory Requirements of this function in terms of Big O?

Question

Asymptotic Run-time is O(n). Not sure how to get "memory requirements" in Big O notation.

<?php

$numbers = '7316717653133062491922511967442657474235534919493496983520312774506326239578318016984801869478851843858615607891129494954595017379583319528532088055111254069874715852386305071569329096329522744304355766896648950445244523161731856403098711121722383113622298934233803081353362766142828064444866452387493035890729629049156044077239071381051585930796086670172427121883998797908792274921901699720888093776657273330010533678812202354218097512545405947522435258490771167055601360483958644670632441572215539753697817977846174064955149290862569321978468622482839722413756570560574902614079729686524145351004748216637048440319989000889524345065854122758866688116427171479924442928230863465674813919123162824586178664583591245665294765456828489128831426076900422421902267105562632111110937054421750694165896040807198403850962455444362981230987879927244284909188845801561660979191338754992005240636899125607176060588611646710940507754100225698315520005593572972571636269561882670428252483600823257530420752963450';

$length_given = 1000;//no need to use strlen()
$highest = 0;
$running_total = 1;
$first_digit = $numbers[0];//better than checking on iteration if $i==0
$seq=0;//start of sequence

for($i = 0; $i< $length_given; $i++)
{
    $current_digit = (int) $numbers[$i]; //converting to int as precaution
    if($current_digit == 0)//zero found
    {
        if($i==$length_given-1)
        {
            break;//don't bother doing anymore if last digit is zero
        }
        //reset sequence/digit counter & total and skip this digit
        $seq = 0;
        $running_total=1;
    }
    else//non-zero
    {
    /*update running total:
        * divide running total by the previous product (to remove it from total)
        * and multiple in the new product but if sequence is initializing
        * (at start or after zero found) then just multiple in product (use 1 as divisor)
        */
        $running_total = ($running_total/ ($seq<=4? 1 : $numbers[$i-5]) ) * $current_digit;

        if($running_total>$highest)
        {
            /* Always true when sequence is started/restarted after zero found because
            * the running total can't go down because the products will be greater than zero*/
            $highest = $running_total;
        }
        $seq++;
    }
}

echo $highest;
?>

Answer 1

Memory requirements captures the amount of memory needed in addition to the input data. In your case you only hold a few variables whose size does not change when the input grows larger so memory/space is fixed and O(1).