My first thought would be the log function (log(1) = 0, and log(1e9) = 20.7232658369464). Otherwise, you have 1000 ranges of size n .. n + ( 999_999 ), so you could just divide by 1_000_000.

Hope that helps.

Update: 2015-02-09
Thanks to choroba for catching my error in linking to the log function documentation.

baxy77bax:

You might consider simply using a hash. You don't specify any requirements on how the mapping is done, so it's possible you don't actually care about the actual number you're mapping to. If so, then there's little difference between:

my ($idx, @array, $a_big_num, $some_data);

# Store the data
$idx = map_func($a_big_num);
$array[$idx] = $some_data;

# Fetch the data
$idx = map_func($a_big_num);
$some_data = $array[$idx];
[download]

and

my (%hash, $a_big_num, $some_data);

# Store
$hash{$a_big_num} = $some_data;

# Fetch
$some_data = $hash{$a_big_num};
[download]

The only real differences between them pop up if (a) you need to retain the order of insertion/deletion or (b) if you actually need to know the slot number ($idx) somewhere in your code. I'm guessing you don't need either, though, because you have absolutely no stated requirements on your mapping function that indicate it.

In fact, you have *so* *few) requirements specified on your mapping function, that this one would meet all stated requirements:

sub map_func { return 7 }
[download]

Before you complain about the fact that there are collisions, you'd have to deal with that in any case because when you map a billion different integers into a thousand, your mapping function will always return collisions unless it's something like:

sub map_func {
   state %indexes;
   state $idx;
   my $a_big_num = shift;
   return $indexes{$a_big_num} if exists $indexes{$a_big_num};
   die "Ran out of slots!" if keys %indexes > 1000;
   return $indexes{$a_big_num}=$idx++;
}
[download]

But in this case, we're using a hash anyway, so there's little to be gained over just using the hash directly. (Again, assuming you don't need the aforementioned conditions (a) or (b).)

...roboticus

When your only tool is a hammer, all problems look like your thumb.

If your logic can work on a hash, I also suggest using a hash as a "sparse array". This also has the advantage that your indices are not restricted to positive integers. You can use anything that stringifies. Hashes are maybe a little slower than arrays, but with hashes you do not need the mapping and reverse mapping. In most cases the hash will be not (much) slower than map+array+remap.

An example of sparse vector cosine calculation:
Bag::Similarity::Cosine.

An example of a matrix with mixed negative and positive indices:
Align::Sequence::Ukkonen. See the "matrix" $F implemented as a hashref.

Helmut Wollmersdorfer

What is the intended application? Are you willing to share?