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 `Dictionary<int, int>` keyed by `n` stores each completed subproblem. Before recursing, check `memo.ContainsKey(n)`: a hit returns immediately, a miss descends.
explicit memo state The memo is threaded through the recursion as `Dictionary<int, int> memo` so the lesson stays about caching, not global state.

Basic Implementation

basic.cs
Replay: real traced execution (multi-file project)
using System;
using System.Collections.Generic;

class Program {
	static int Fib(int n, Dictionary<int, int> memo) {
		if (memo.ContainsKey(n)) {
			return memo[n];
		}
		if (n < 2) {
			memo[n] = n;
			return n;
		}
		int value = Fib(n - 1, memo) + Fib(n - 2, memo);
		memo[n] = value;
		return value;
	}

	static void Main() {
		Dictionary<int, int> memo = new Dictionary<int, int>();
		int result = Fib(6, memo);
		Console.WriteLine(result);
	}
}

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

    12}13int value = Fib(n - 1, memo) + Fib(n - 2, memo);14memo[n] = value;
    values this step{}memomiss -> descend Fib(5)action6n

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

    12}13int value = Fib(n - 1, memo) + Fib(n - 2, memo);14memo[n] = value;
    values this step{}memomiss -> descend Fib(4)action5n

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

    12}13int value = Fib(n - 1, memo) + Fib(n - 2, memo);14memo[n] = value;
    values this step{}memomiss -> descend Fib(3)action4n

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

    12}13int value = Fib(n - 1, memo) + Fib(n - 2, memo);14memo[n] = value;
    values this step{}memomiss -> descend Fib(2)action3n

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

    12}13int value = Fib(n - 1, memo) + Fib(n - 2, memo);14memo[n] = value;
    values this step{}memomiss -> descend Fib(1)action2n

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

    12}13int value = Fib(n - 1, memo) + Fib(n - 2, memo);14memo[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

    12}13int value = Fib(n - 1, memo) + Fib(n - 2, memo);14memo[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

    12}13int value = Fib(n - 1, memo) + Fib(n - 2, memo);14memo[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

    12}13int value = Fib(n - 1, memo) + Fib(n - 2, memo);14memo[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

    12}13int value = Fib(n - 1, memo) + Fib(n - 2, memo);14memo[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

    12}13int value = Fib(n - 1, memo) + Fib(n - 2, memo);14memo[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

    20	int result = Fib(6, memo);21	Console.WriteLine(result);22}
    values this step8stdout8result

Complexity

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

Implementation notes

  • The memo is a Dictionary<int, int> allocated in Main and passed by reference through every recursive call, so all stack frames share one CLR-managed hash table that is reclaimed by GC.
  • The checked-in code intentionally uses memo.ContainsKey(n) followed by memo[n] instead of TryGetValue, making memo hits replay-visible as a lookup and immediate return. Misses recurse until n < 2, then write memo[n] = n.
  • Fibonacci values are int values copied into the dictionary. After Fib(n - 1, memo) + Fib(n - 2, memo) returns, memo[n] = value records the completed subproblem; the trace separates those writes from later hits.