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.cs

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);
	}
}

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.

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.