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 `std::map<int, int>` 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 reference parameter so the lesson stays about caching, not language-level scoping.

Basic Implementation

basic.cpp
Replay: real traced execution (multi-file project)
#include <iostream>
#include <map>

int fib(int n, std::map<int, int>& memo) {
    auto it = memo.find(n);
    if (it != memo.end()) {
        return it->second;
    }
    if (n < 2) {
        memo[n] = n;
        return n;
    }
    int value = fib(n - 1, memo) + fib(n - 2, memo);
    memo[n] = value;
    return value;
}

int main() {
    std::map<int, int> memo;
    int result = fib(6, memo);
    std::cout << result << std::endl;
    std::cout << "{";
    bool first = true;
    for (const auto& kv : memo) {
        if (!first) std::cout << ", ";
        std::cout << kv.first << ": " << kv.second;
        first = false;
    }
    std::cout << "}" << std::endl;
    return 0;
}

  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

    20int result = fib(6, memo);21std::cout << result << std::endl;22std::cout << "{";
    values this step8stdout8result

Complexity

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

Implementation notes

  • C++: std::map<int, int> memo; passed by reference to fib(int n, std::map<int, int>& memo). The ordered map keeps the printout deterministic so the lesson can show the final memo state honestly.
  • 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.