Recursion and Dynamic Programming

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.

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.

Basic Implementation

basic.pl
Replay: real traced execution (multi-file project)
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";

  1. memo ← {}, action ← miss -> descend fib(5)

    11}12my $value = fib($n - 1, $memo_ref) + fib($n - 2, $memo_ref);13$memo_ref->{$n} = $value;
    values this step{}memomiss -> descend fib(5)action6n

  2. memo ← {}, action ← miss -> descend fib(4)

    11}12my $value = fib($n - 1, $memo_ref) + fib($n - 2, $memo_ref);13$memo_ref->{$n} = $value;
    values this step{}memomiss -> descend fib(4)action5n

  3. memo ← {}, action ← miss -> descend fib(3)

    11}12my $value = fib($n - 1, $memo_ref) + fib($n - 2, $memo_ref);13$memo_ref->{$n} = $value;
    values this step{}memomiss -> descend fib(3)action4n

  4. memo ← {}, action ← miss -> descend fib(2)

    11}12my $value = fib($n - 1, $memo_ref) + fib($n - 2, $memo_ref);13$memo_ref->{$n} = $value;
    values this step{}memomiss -> descend fib(2)action3n

  5. memo ← {}, action ← miss -> descend fib(1)

    11}12my $value = fib($n - 1, $memo_ref) + fib($n - 2, $memo_ref);13$memo_ref->{$n} = $value;
    values this step{}memomiss -> descend fib(1)action2n

  6. memo ← {1: 1}, action ← base 1; memo{1} = 1; return

    11}12my $value = fib($n - 1, $memo_ref) + fib($n - 2, $memo_ref);13$memo_ref->{$n} = $value;
    values this step{1: 1}memobase 1; memo{1} = 1; returnaction1n

  7. memo ← {0: 0, 1: 1}, action ← base 0; memo{0} = 0; fib(2)=1; memo{2} = 1

    11}12my $value = fib($n - 1, $memo_ref) + fib($n - 2, $memo_ref);13$memo_ref->{$n} = $value;
    values this step{0: 0, 1: 1}memobase 0; memo{0} = 0; fib(2)=1; memo{2} = 1action0n

  8. memo ← {0: 0, 1: 1, 2: 1, 3: 2}, action ← hit 1; fib(3)=2; memo{3} = 2

    11}12my $value = fib($n - 1, $memo_ref) + fib($n - 2, $memo_ref);13$memo_ref->{$n} = $value;
    values this step{0: 0, 1: 1, 2: 1, 3: 2}memohit 1; fib(3)=2; memo{3} = 2action1n

  9. memo ← {0: 0, 1: 1, 2: 1, 3: 2, 4: 3}, action ← hit 1; fib(4)=3; memo{4} = 3

    11}12my $value = fib($n - 1, $memo_ref) + fib($n - 2, $memo_ref);13$memo_ref->{$n} = $value;
    values this step{0: 0, 1: 1, 2: 1, 3: 2, 4: 3}memohit 1; fib(4)=3; memo{4} = 3action2n

  10. memo ← {0: 0, 1: 1, 2: 1, 3: 2, 4: 3, 5: 5}, action ← hit 2; fib(5)=5; memo{5} = 5

    11}12my $value = fib($n - 1, $memo_ref) + fib($n - 2, $memo_ref);13$memo_ref->{$n} = $value;
    values this step{0: 0, 1: 1, 2: 1, 3: 2, 4: 3, 5: 5}memohit 2; fib(5)=5; memo{5} = 5action3n

  11. memo ← {0: 0, 1: 1, 2: 1, 3: 2, 4: 3, 5: 5, 6: 8}, action ← hit 3; fib(6)=8; memo{6} = 8

    11}12my $value = fib($n - 1, $memo_ref) + fib($n - 2, $memo_ref);13$memo_ref->{$n} = $value;
    values this step{0: 0, 1: 1, 2: 1, 3: 2, 4: 3, 5: 5, 6: 8}memohit 3; fib(6)=8; memo{6} = 8action4n

  12. stdout ← 8

    18my $result = fib(6, \%memo);19print "$result\n";
    values this step8stdout8result

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.