PWC108 - Bell Numbers


On with TASK #2 from the Perl Weekly Challenge #108. Enjoy!

The challenge

Write a script to display top 10 Bell Numbers. Please refer to wikipedia page for more information.

The questions

A fundamental question in this challenge is what do you mean by display?

I mean, displaying a number is printing it, right? Or should we actually display all possible arrangements leading to that number?.

The solution

We’ll take the easy route in this, but still keep the door open for possible future wastAHEMimprovements.

As it is so often, the wikipedia page provides a neat way to calculate the numbers iteratively. This implies a time complexity of $O(n^2)$ and a space complexity of $O(n)$, which is not bad considering that we’re asked to address only a handful of numbers and that the time complexity actually accounts for computing all $n$ numbers (so, on average, the complexity of any single of them is $O(n)$, assuming we can rely on the previous calculations).

Enough talking, let’s get to the code:

#!/usr/bin/env perl
use 5.024;
use warnings;
use experimental qw< postderef signatures >;
no warnings qw< experimental::postderef experimental::signatures >;

sub bell_number ($N) {
   state $cache = [1];
   state $line = [1];
   while ($cache->$#* < $N) {
      my @previous_line = $line->@*;
      $line->@* = $previous_line[-1];
      push $line->@*, $_ + $line->[-1] for @previous_line;
      push $cache->@*, $line->[0];
   return $cache->[$N];

printf "B%d: %d\n", $_, bell_number($_) for 0 .. 9;

We keep two cache variables: one for the sequence itself (so that we can reuse results if we need), one for the previous line in the triangle to calculate the numbers. If you don’t know what triangle I’m talking about, look at the Triangle scheme for calculations.

I guess this is all for this post 😅

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