Re: An informal introduction to O(N) notation

[jryan]

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] < $a[$_]);
  }
}

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
    }
}

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