When does dynamic programming apply versus greedy?

Question

Accepted Answer

Both DP and greedy need **optimal substructure**. The difference: **greedy** also requires the **greedy-choice property** (a local optimum is globally optimal), while **DP** is needed when you must consider **multiple choices and overlapping subproblems**.

## The distinction

```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
```

## Classic contrast: coin change

```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
```

Greedy commits to one choice; DP explores all and keeps the best.

## When each applies

- **Greedy:** interval scheduling, Huffman, Dijkstra, MST — proven greedy-choice property.
- **DP:** knapsack, edit distance, general coin change — choices interact.

## Complexity

Greedy is usually O(n log n); DP trades more time/space (often O(n·m)) for correctness.

## Pitfalls

Using greedy without proving its property gives **wrong answers** that pass small tests. When in doubt, reach for DP or verify greedy against brute force.

## Why it matters

Picking greedy when it is valid saves enormous time; picking it when it is not produces silent bugs.

Knowing which paradigm fits — and being able to prove it — is a senior-level distinction.

It prevents both over-engineering with DP and under-delivering with an unsound greedy choice.