Recursion and Dynamic Programming

Fibonacci with Memoization

Compute fib(n) recursively. Cache each fib(k) in a memo array 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 parallel `memo_value[]` / `memo_seen[]` pair indexed by `n` stores each completed subproblem. Before recursing, check `memo_seen[n]`: a hit returns immediately, a miss descends.
explicit memo state The memo lives in file-scope static arrays so the recursion stays about caching, not parameter passing — C does not need a heap map container for small `n`.

Basic Implementation

basic.c
Replay: real traced execution (multi-file project)
#include <stdio.h>
#include <stdbool.h>

#define MEMO_SIZE 16

static int memo_value[MEMO_SIZE];
static bool memo_seen[MEMO_SIZE];

int fib(int n) {
    if (memo_seen[n]) {
        return memo_value[n];
    }
    if (n < 2) {
        memo_value[n] = n;
        memo_seen[n] = true;
        return n;
    }
    int value = fib(n - 1) + fib(n - 2);
    memo_value[n] = value;
    memo_seen[n] = true;
    return value;
}

int main(void) {
    int result = fib(6);
    printf("%d\n", result);
    printf("{");
    bool first = true;
    for (int i = 0; i < MEMO_SIZE; ++i) {
        if (memo_seen[i]) {
            if (!first) printf(", ");
            printf("%d: %d", i, memo_value[i]);
            first = false;
        }
    }
    printf("}\n");
    return 0;
}

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

    17}18int value = fib(n - 1) + fib(n - 2);19memo_value[n] = value;
    values this step{}memomiss -> descend fib(5)action6n

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

    17}18int value = fib(n - 1) + fib(n - 2);19memo_value[n] = value;
    values this step{}memomiss -> descend fib(4)action5n

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

    17}18int value = fib(n - 1) + fib(n - 2);19memo_value[n] = value;
    values this step{}memomiss -> descend fib(3)action4n

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

    17}18int value = fib(n - 1) + fib(n - 2);19memo_value[n] = value;
    values this step{}memomiss -> descend fib(2)action3n

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

    17}18int value = fib(n - 1) + fib(n - 2);19memo_value[n] = value;
    values this step{}memomiss -> descend fib(1)action2n

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

    17}18int value = fib(n - 1) + fib(n - 2);19memo_value[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

    17}18int value = fib(n - 1) + fib(n - 2);19memo_value[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

    17}18int value = fib(n - 1) + fib(n - 2);19memo_value[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

    17}18int value = fib(n - 1) + fib(n - 2);19memo_value[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

    17}18int value = fib(n - 1) + fib(n - 2);19memo_value[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

    17}18int value = fib(n - 1) + fib(n - 2);19memo_value[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

    25int result = fib(6);26printf("%d\n", result);27printf("{");
    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: file-scope int memo_value[MEMO_SIZE] and bool memo_seen[MEMO_SIZE] from <stdbool.h> give the memo a deterministic iteration order keyed by n, which keeps the final printout honest without leaning on a hash-map container the standard library does not provide.
  • 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.