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 `Map<number, number>` 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.

Basic Implementation

basic.ts
Replay: real traced execution (multi-file project)
function fib(n: number, memo: Map<number, number>): number {
    if (memo.has(n)) {
        return memo.get(n) as number;
    }
    if (n < 2) {
        memo.set(n, n);
        return n;
    }
    const value: number = fib(n - 1, memo) + fib(n - 2, memo);
    memo.set(n, value);
    return value;
}

const memo: Map<number, number> = new Map();
const result: number = fib(6, memo);
console.log(result);
console.log(JSON.stringify(Object.fromEntries(memo)));

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

    8}9const value: number = fib(n - 1, memo) + fib(n - 2, memo);10memo.set(n, value);
    values this step{}memomiss -> descend fib(5)action6n

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

    8}9const value: number = fib(n - 1, memo) + fib(n - 2, memo);10memo.set(n, value);
    values this step{}memomiss -> descend fib(4)action5n

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

    8}9const value: number = fib(n - 1, memo) + fib(n - 2, memo);10memo.set(n, value);
    values this step{}memomiss -> descend fib(3)action4n

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

    8}9const value: number = fib(n - 1, memo) + fib(n - 2, memo);10memo.set(n, value);
    values this step{}memomiss -> descend fib(2)action3n

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

    8}9const value: number = fib(n - 1, memo) + fib(n - 2, memo);10memo.set(n, value);
    values this step{}memomiss -> descend fib(1)action2n

  6. memo ← {1: 1}, action ← base 1; memo.set(1, 1); return

    8}9const value: number = fib(n - 1, memo) + fib(n - 2, memo);10memo.set(n, value);
    values this step{1: 1}memobase 1; memo.set(1, 1); returnaction1n

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

    8}9const value: number = fib(n - 1, memo) + fib(n - 2, memo);10memo.set(n, value);
    values this step{0: 0, 1: 1}memobase 0; memo.set(0, 0); fib(2)=1; memo.set(2, 1)action0n

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

    8}9const value: number = fib(n - 1, memo) + fib(n - 2, memo);10memo.set(n, value);
    values this step{0: 0, 1: 1, 2: 1, 3: 2}memohit 1; fib(3)=2; memo.set(3, 2)action1n

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

    8}9const value: number = fib(n - 1, memo) + fib(n - 2, memo);10memo.set(n, value);
    values this step{0: 0, 1: 1, 2: 1, 3: 2, 4: 3}memohit 1; fib(4)=3; memo.set(4, 3)action2n

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

    8}9const value: number = fib(n - 1, memo) + fib(n - 2, memo);10memo.set(n, value);
    values this step{0: 0, 1: 1, 2: 1, 3: 2, 4: 3, 5: 5}memohit 2; fib(5)=5; memo.set(5, 5)action3n

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

    8}9const value: number = fib(n - 1, memo) + fib(n - 2, memo);10memo.set(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.set(6, 8)action4n

  12. stdout ← 8

    15const result: number = fib(6, memo);16console.log(result);17console.log(JSON.stringify(Object.fromEntries(memo)));
    values this step8stdout8result

Complexity

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

Implementation notes

  • TypeScript: const memo: Map<number, number> = new Map(); passed explicitly to fib(n, memo). The Map<number, number> annotation documents the key/value contract.
  • 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.