Fibonacci with Memoization

Compute fib(n) recursively. Cache each fib(k) in a memo map so each subproblem is solved at most once.

Algorithm

Canonical input $n = 6 produces fib(6) = 8. Replay highlights every memo write and every cache hit.

Basic Implementation

basic.pl

use strict; use warnings;

sub fib {
	my ($n, $memo_ref) = @_;
	if (exists $memo_ref->{$n}) {
		return $memo_ref->{$n};
	}
	if ($n < 2) {
		$memo_ref->{$n} = $n;
		return $n;
	}
	my $value = fib($n - 1, $memo_ref) + fib($n - 2, $memo_ref);
	$memo_ref->{$n} = $value;
	return $value;
}

my %memo = ();
my $result = fib(6, \%memo);
print "$result\n";

Complexity

Time: O(n) with memoization (vs. O(2^n) without)
Space: O(n) memo + O(n) call stack

Implementation notes

Perl: the recursion takes the memo as a hash reference argument rather than an our %memo package global, which keeps state explicit without hiding the lesson behind a shared global. The exists + arrow-deref pair stays parallel to the lesson spec instead of leaning on Perl's autovivification (which would create the slot during a probe and collapse the hit / miss branch).
The replay shows the call stack on one side and the memo map on the other so memo writes and cache hits are visually distinct.

memoization A hash reference keyed by `$n` stores each completed subproblem. Before recursing, check `exists $memo_ref->{$n}`: a hit returns immediately, a miss descends.

explicit memo state The memo is threaded through the recursion as a hash reference argument rather than a package-level `our` global, which keeps the lesson about caching, not shared state.