# Types of Dynamic Programming Questions

Dynamic programming is probably the trickiest and most-feared interview question type. The hardest parts are 1) to know it’s a dynamic programming question to begin with 2) to find the subproblem.

We looked at a ton of dynamic programming questions and summarized common patterns and subproblems.

We also highlighted the keywords that indicate it’s likely a dynamic programming problem.

# Sequence

This is the most common type of DP problem and a good place to get a feel of dynamic programming. In the recurrence relation,**dp[i]**** normally means max/min/best value for the sequence ending at index i.**

- House robber — find
**maximum**amount of loot - Coin change — find
**minimum**amount of coins needed to make up an amount

# Grid

This is the 2D version of the sequence DP. **dp[i][j]**** means max/min/best value for matrix cell ending at index i, j.**

- Robot unique paths —
**number of ways**for robot to move from top left to bottom right - Min path sum — find path in a grid with
**minimum**cost - Maximal square — find
**maximal**square of 1s in a grid of 0s and 1s

# Dynamic number of subproblems

This is similar to “Sequence DP” except `dp[i]`

depends on a dynamic number of subproblems, e.g. `dp[i] = max(d[j]..) for j from 0 to i`

.

- Longest Increasing Subsequence — find the
**longest**increasing subsequence of an array of numbers - Buy/sell stock with at most K transactions —
**maximize**profit by buying and selling stocks using at most K transaction

# Partition

This is a continuation of DFS + memoization problems. These problems are easier to reason and solve with a top-down approach. The key to solve these problems is to draw the state-space tree and then traverse it.

- Decode ways —
**how many ways**to decode a string - Word break — partition a word into words in a dictionary
- Triangle — find the
**smallest**sum path to traverse a triangle of numbers from top to bottom - Partition to Equal Sum Subsets —
**partition**a set of numbers into two equal-sum subsets

# Two Sequences

This type of problem has two sequences in their problem statement. **dp[i][j]**** represents the max/min/best value for the first sequence ending in index i and second sequence ending in index j.**

- Edit distance — find the
**minimum**distance to edit one string to another - Longest common subsequence — find the
**longest**common subsequence that is common in two sequences

# Game theory

This type of problem asks for whether a player can win a decision game. The key to solving game theory problems is to identify winning state, and formulating a winning state as a state that returns a losing state to the opponent

- Coins in a line
- Divisor game
- Stone game

Learn more about each type with detailed analysis and illustrations at AlgoMonster: