PWC153 - Left Factorials


Here we are with TASK #1 from The Weekly Challenge #153. Enjoy!

The challenge

Write a script to compute Left Factorials of 1 to 10. Please refer OEIS A003422 for more information.

Expected Output:

1, 2, 4, 10, 34, 154, 874, 5914, 46234, 409114

The questions

Our fine host is a fox disguised as an innocent lamb!

After having endured countless petty nitpicks (by many, yours truly included) about missing stuff in formulating the challenges, we are gently redirected to a place that leaves no doubt about what we’re asked to do:

\[!n = \sum_{k = 0}^{n-1} k!\]

Well played indeed!

The solution

Why provide a solution when we can provide… three?

Raku goes first, with two of them that will be printed one along the other, for comparison:

#!/usr/bin/env raku
use v6;
sub MAIN (Int:D $min = 1, Int:D $max = 10) {
   ($min .. $max).map({left-factorial($_)}).join(', ').put;
   ($min .. $max).map({left-factorial-cached($_)}).join(', ').put;

The first takes advantage of the multi mechanism, separating lower values from higher ones.

multi sub left-factorial (Int:D $n where 0 <= * <= 2) { $n }
multi sub left-factorial (Int:D $n where * >  2) {
   my $f = 1;
   1 + (1 ..^ $n).map({$f *= $^x}).sum;

These higher ones are always calculated over and over, so in this case we’re not reusing any previous calculation, i.e. calculating left-factorial for 5 does not leverage our previous calculation for 4.

This is a totally unacceptable loss of performance, of course. 🙄

In this case, it makes total sense to keep those previous values around, because we’re requested to print a sequence of consecutive values. This leads us to the second solution, which keeps track of previous values in a few state variables:

sub left-factorial-cached (Int:D $n where * >= 0) {
   state $factorial = 1;
   state $k = 1;
   state @left-factorials = 0, 1, 2;
   while $n > @left-factorials.end {
      $factorial *= ++$k;
      @left-factorials.push: @left-factorials[*-1] + $factorial;
   return @left-factorials[$n];

This last solution might be readily translated into Perl, but we take a lazy detour here and let Perl manage caching for us. This goes at the expense of over-caching, because values for the factorial functions are all cached too, while this is not strictly necessary:

#!/usr/bin/env perl
use v5.24;
use warnings;
use experimental 'signatures';
no warnings 'experimental::signatures';
use Memoize;

my $min = shift // 1;
my $max = shift // 10;
say join ', ', map { left_factorial($_) } $min .. $max;

sub left_factorial ($n) {
   return $n if $n <= 2;
   return factorial($n - 1) + left_factorial($n - 1);

sub factorial ($n) {
   return 1 if $n < 2;
   return $n * factorial($n - 1);

Readability is probably better here, though. This solution makes it clear the recursive nature of the approach, while at the same time acknowledging that some dynamic programming can help speed things up as inputs go high.

So… what do you prefer?

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