Ni lini gani programu ya nguvu inayolinganisha na algorithm yenye tamaa?

Question

Accepted Answer

Wote DP na algorithm yenye tamaa wanahitaji **muundo mzuri wa subproblem**. Tofauti: **algorithm yenye tamaa** pia inahitaji **sifa ya chaguo lenye tamaa** (optimum ya ndani ni optimum ya ulimwengu), ilhali **DP** inahitajika wakati unapaswa kufikiri juu ya **chaguo nyingi na subproblemu zinazosambazana**.

## Tofauti

```text
Optimal substructure?  -- both need this
  + greedy-choice property holds?  -> GREEDY (fast, one pass of choices)
  + must compare many sub-solutions / they overlap?  -> DYNAMIC PROGRAMMING
```

## Kulinganisha kwa kawaida: mabadiliko ya sarafu

```python
# Greedy FAILS for coins {1,3,4}, amount 6 -> 4+1+1 (3 coins)
# DP finds the true optimum: 3+3 (2 coins)
def min_coins(amount, coins):
    INF = float('inf')
    dp = [0] + [INF] * amount
    for a in range(1, amount + 1):
        for c in coins:
            if c <= a:
                dp[a] = min(dp[a], dp[a - c] + 1)  # try every coin
    return dp[amount]

min_coins(6, [1, 3, 4])               # -> 2
```

Algorithm yenye tamaa inajitolea kwa chaguo moja; DP inachunguza zote na kutumia nzuri.

## Wakati kila moja inatumika

- **Algorithm yenye tamaa:** ratiba ya vipindi, Huffman, Dijkstra, MST — sifa ya chaguo lenye tamaa iliyothibitishwa.
- **DP:** knapsack, umbali wa hariri, mabadiliko ya sarafu ya kawaida — machaguo yanaingiliana.

## Kiwango cha jumbo

Algorithm yenye tamaa kawaida ni O(n log n); DP inabadilisha wakati/nafasi zaidi (mara nyingi O(n·m)) kwa usahihi.

## Mitego

Kutumia algorithm yenye tamaa bila kuthibitisha sifa yake kunatoa **majibu yasiyosahihi** ambayo yanafaulu mitihani ndogo. Ikiwa una shaka, chukua DP au thibitisha algorithm yenye tamaa dhidi ya nguvu mbaya.

## Kwa nini ni muhimu

Kuchagua algorithm yenye tamaa wakati ni halali huokoa wakati mkubwa; kuchagua wakati sio hutengeneza matatizo ya kimya.

Kujua ni paradigma gani inayolingana — na kuweza kuandika — ni tofauti ya kiwango cha juu.

Inakamatia wote kukamatia mifano na DP na kutokuwa na kutokutoa kwa chaguo lenye tamaa lenye matatizo.