1D Dynamic Programming Problems

Recognizing a Dynamic Programming Problem

A problem is likely to be solvable using DP if:

• You're asked to count the number of ways to reach a goal.

• You're asked to find the minimum or maximum value to achieve a goal.

• You must try all possible ways/choices and select the best.

• The solution can be defined in terms of recurrence (overlapping subproblems).

• The brute-force or recursive solution has repeating subproblems.

How to Approach a DP Problem

1. Define the State:

• Choose a parameter (usually index, value, etc.) that represents a subproblem.

• Example: dp[i] might represent the minimum cost to reach step i.

2. Identify the Base Case:

• What is the result of the smallest input?

• Example: dp[0] = 0, dp[1] = 1, etc.

3. Define the Recurrence Relation:

• What decisions can you make at each step?

• Example: dp[i] = min(dp[i-1], dp[i-2]) + cost[i]

4. Choose a Method:

• Top-Down (Memoization): Write a recursive function and cache results.

• Bottom-Up (Tabulation): Fill a DP table iteratively.

• Space Optimization: Use variables instead of a full array if only a few past states are needed.

5. Implement & Test:

• Start with simple inputs and edge cases.

• Compare your DP solution with brute-force for validation.

Common 1D DP Problem Formats

Tips for Mastering DP

• Recursion First: Always try solving recursively before optimizing.

• Visualize the DP Array/Table: Helps debug and understand transitions.

• Practice Common Patterns: Count ways, max/min path, subset sum, etc.

• Talk It Out or Write Down Transitions: Articulating the recurrence helps. dp-185