It should be noted that many algorithm's O time can be reduced. For instance, many O(n^2) algorithms can be reduced to O(n log(n)) by finding a way to cut the inner loop short. A famous example is bubblesort: 

# O(n^2)
for my $i (0..$#a-1) {
    for (0..$#a-1-$i) {
        ($a[$_],$a[$_+1]) = ($a[$_+1],$a[$_])
            if ($a[$_+1] < $a[$_]);
  }
}

# O(n log(n))
for my $i (0..$#a-1) {

    # looping over already-sorted items is not needed
    for (0..$#a-1-$i) {
        ($a[$_],$a[$_+1]) = ($a[$_+1],$a[$_])
            if ($a[$_+1] < $a[$_]);
  }
}

[download]

O(n) algorithms that involve a search via a loop can sometimes be reduced from O(n) to O(log(n)) by breaking out of the loop once a "result" is found: 

# O(n)
my $truestatus;
foreach my $item (@list) {
    $truestatus = 1 if (some_condition);
}


# O(log(n))
my $truestatus;
foreach my $item (@list) {
    
    # if some_condition is going to be valid for at least one $item in
    # @list, then the limit of the average runtime will approach
    # O(log n).
    if (some_condition) {
        $truestatus = 1;
        last
    }
}

[download]

Of course, the secret is to know where to apply these optimizations, which is often harder than it seems :)

In reply to Re: An informal introduction to O(N) notation by jryan
in thread An informal introduction to O(N) notation by dws

Title:
Use:  <p> text here (a paragraph) </p>
and:  <code> code here </code>
to format your post, it's "PerlMonks-approved HTML":


  • Posts are HTML formatted. Put <p> </p> tags around your paragraphs. Put <code> </code> tags around your code and data!
  • Titles consisting of a single word are discouraged, and in most cases are disallowed outright.
  • Read Where should I post X? if you're not absolutely sure you're posting in the right place.
  • Please read these before you post! —
  • Posts may use any of the Perl Monks Approved HTML tags:
    a, abbr, b, big, blockquote, br, caption, center, col, colgroup, dd, del, details, div, dl, dt, em, font, h1, h2, h3, h4, h5, h6, hr, i, ins, li, ol, p, pre, readmore, small, span, spoiler, strike, strong, sub, summary, sup, table, tbody, td, tfoot, th, thead, tr, tt, u, ul, wbr
  • You may need to use entities for some characters, as follows. (Exception: Within code tags, you can put the characters literally.)
            For:     Use:
    & &amp;
    < &lt;
    > &gt;
    [ &#91;
    ] &#93;
  • Link using PerlMonks shortcuts! What shortcuts can I use for linking?
  • See Writeup Formatting Tips and other pages linked from there for more info.