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.java

import java.util.HashMap;
import java.util.Map;

public class Basic {
    public static void main(String[] args) {
        Map<Integer, Integer> memo = new HashMap<>();
        int result = fib(6, memo);
        System.out.println(result);
        System.out.println(memo);
    }

    private static int fib(int n, Map<Integer, Integer> memo) {
        if (memo.containsKey(n)) {
            return memo.get(n);
        }
        if (n < 2) {
            memo.put(n, n);
            return n;
        }
        int value = fib(n - 1, memo) + fib(n - 2, memo);
        memo.put(n, value);
        return value;
    }
}

Complexity

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

Implementation notes

Java: Map<Integer, Integer> memo = new HashMap<>(); passed explicitly to fib(int n, Map<Integer, Integer> memo).
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 `HashMap<Integer, Integer>` cache stores each completed subproblem. Before recursing, check the memo: a hit returns immediately, a miss descends.

explicit memo parameter Pass the memo as an explicit parameter so the lesson stays about caching, not language-level scoping.