You are possibly right. You are just as possibly wrong. There are several things that we don't know, such as:

• Average line length. Shorter lines means more lowlevel iterations.
• Average file length. Longer files will require more memory - but that is about all.
• Average hit count. How often is the string found in the file.
• Average hit placement. How often does the string end up at the beginning or the end.
• Implementation issues. Is the string passed in already in one chunk or do we have access to a file handle.

There are just too many unknowns to use blanket statements as to which algorithm is best.

But one thing that is a major issue is that the special regex capture variables shouldn't be used. They impose too much penalty. Instead though you can use @- and @+ which have no penalty. As in the following:

my $str = "1 one
2 two
3 one
4 four
5 one
6 five";
my $last_pos = 0;
my $newlines = 1;
while ($str =~ /(one)/g) {
    $newlines += substr($str, $last_pos, $-[0] - $last_pos) =~ tr/\n//
+;
    $last_pos = $-[0];
    print "Found on line $newlines\n";
}
# prints 
# Found on line 1
# Found on line 3
# Found on line 5
[download]

Notice the optimization that only counts newlines from the previous match.