Apa iku dynamic programming (memoization vs tabulation)?

Question

Accepted Answer

**Dynamic programming (DP)** ngresleki masalah kanthi **overlapping subproblems** lan **optimal substructure** kanthi ngehitung saben subproblem sapisan lan nganggo maneh hasilne. Rong gaya iku **memoization** (top-down) lan **tabulation** (bottom-up).

## Gagasan pokok

Rekursi naif ngreklakake subproblem sing sama kanthi eksponensial. DP ngesinau menawa, murni pekerjaan eksponensial dadi polinomial.

## Memoization (top-down)

```python
from functools import lru_cache

@lru_cache(maxsize=None)               # cache results automatically
def fib(n):
    if n < 2:
        return n
    return fib(n - 1) + fib(n - 2)     # each n computed once
```

## Tabulation (bottom-up)

```python
def fib_table(n):
    if n < 2:
        return n
    dp = [0, 1]
    for i in range(2, n + 1):
        dp.append(dp[i - 1] + dp[i - 2])  # build up from base cases
    return dp[n]
```

## Memoization vs tabulation

| | Memoization | Tabulation |
|---|------------|------------|
| Arah | rekursi top-down | iterasi bottom-up |
| Nghitung | mung state sing perlu | kabeh state |
| Risiko | stack overflow | ora ana |
| Ruang | bisa dipurno | asring bisa dikecilake menyang O(1) baris |

## Kerumitan

Fibonacci: saka O(2ⁿ) naif dadi **O(n)** wektu, O(n) (utawa O(1)) ruang.

## Kapan nganggo / jebakan

Gunakake DP nalika subproblem **anggone nindih**. Dhefinasiaken state lan rekurens kanthi tliti — iku bagean sing angel. Yen subproblem ora nindih, divide-and-conquer awon cukup.

## Mengapa iku penting

DP ngganti masalah eksponensial sing angel — path paling cekak, jarak sunting, ransel — dadi sing efisien.

Karep kanggo ngenali state lan rekurens minangka salah siji sing paling dihargakake ing interview algoritma.

Iku ndasari sistem nyata saka tool diff nganti alignment urutan bioinformatika.

	Memoization	Tabulation
Arah	rekursi top-down	iterasi bottom-up
Nghitung	mung state sing perlu	kabeh state
Risiko	stack overflow	ora ana
Ruang	bisa dipurno	asring bisa dikecilake menyang O(1) baris