Big Oに関してこの関数のメモリ要件は何ですか？

Question

漸近実行時間はO（n）です。 Big O表記で「メモリ要件」を取得する方法がわかりません。

<?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

メモリ要件は、入力データに加えでを必要なメモリの量をキャプチャします。あなたのケースでは、入力が大きくなってメモリ/スペースが固定され、O（1）が大きくなるとサイズが変わらないいくつかの変数しか保持しません。