0/1 & Unbounded Knapsack, LCS/Edit Distance, Interval DP, Bitmask DP, and space-reduction templates.
dp[w] = \max\bigl(dp[w],\; dp[w - \text{weight}_i] + \text{val}_i\bigr) \quad \text{for } w = W \text{ down to } \text{weight}_i
Iterating capacity w backwards prevents using the same item multiple times in the same step.
dp[i][j] = \begin{cases} dp[i-1][j-1] & s_1[i]=s_2[j] \\ 1 + \min(dp[i-1][j],\, dp[i][j-1],\, dp[i-1][j-1]) & \text{otherwise} \end{cases}
Models minimum insertions, deletions, and substitutions to transform s_1 into s_2.
Every DP formulation requires three components: (1) minimal state variables that satisfy the Markov property, (2) base cases, and (3) a topological evaluation order over the state DAG.
| Archetype | State Definition | Loop Direction | Classic Problems |
|---|---|---|---|
| 0/1 Knapsack | dp[w] = max value with capacity w | Outer: items; Inner: w from W down to wt[i] | Subset Sum, Partition Equal Subset |
| Unbounded Knapsack | dp[w] = min/max ways for capacity w | Outer: items; Inner: w from wt[i] up to W | Coin Change, Rod Cutting |
| Two-Sequence Alignment | dp[i][j] on prefixes A[:i], B[:j] | i from 1..m, j from 1..n (rolling 2 rows) | LCS, Edit Distance, Regex Match |
| Interval DP | dp[i][j] on subarray s[i..j] | Length len from 2..n; left i from 0..n-len | Matrix Chain Multiplication, Burst Balloons |
| Bitmask DP | dp[mask][u] over visited subset mask | mask from 0..(1<<n)-1 | Traveling Salesperson (TSP) in O(n^2 2^n) |
def knapsack_01(weights: list[int], values: list[int], capacity: int) -> int:
dp = [0] * (capacity + 1)
for wt, val in zip(weights, values):
for w in range(capacity, wt - 1, -1): # Reverse prevents reuse
dp[w] = max(dp[w], dp[w - wt] + val)
return dp[capacity]
def min_coins_unbounded(coins: list[int], amount: int) -> int:
INF = amount + 1
dp = [0] + [INF] * amount
for c in coins:
for a in range(c, amount + 1): # Forward allows unlimited reuse
dp[a] = min(dp[a], dp[a - c] + 1)
return dp[amount] if dp[amount] < INF else -1When the state space is sparse and only a small fraction of all possible states are reachable from the initial query, top-down memoization skips unreachable states entirely.