Dynamic programming reuses solutions to overlapping subproblems. Define what a state means before writing transitions or choosing a table.
Before you start
You should know arrays, loops and basic complexity notation. Start with a small input you can trace by hand. State what each variable means and which invariant is maintained, then use that invariant to explain correctness before discussing performance or writing a more compact solution.
The practical goal is to reason through this situation: For a prefix problem, dp[i] can represent the best answer using the first i elements. Read the walkthrough first, then try the interview exercise before opening its answer. The important part is explaining the decision and its consequences, rather than remembering a definition alone.
Step-by-step walkthrough
Step 1: Define the state precisely
Let dp[i] describe the best valid solution for the first i elements.
Step 2: Enumerate final choices
Relate each allowed final decision to a smaller defined state.
Step 3: Derive base cases
Empty and smallest inputs follow the same state meaning.
Worked scenario
For a prefix problem, dp[i] can represent the best answer using the first i elements.
For nonadjacent selection, the final element is either skipped, using dp[i−1], or taken with a solution ending before its neighbor, using dp[i−2]. Add its value in the taken case. Whether selecting nothing is permitted changes base values for negative input and must be specified first.
Common mistake
Memorizing a recurrence without its meaning makes base cases and bounds fragile.
Verify the behavior
Compare with exhaustive subsets on short arrays, especially empty and all-negative cases.
Interview exercise
Derive a recurrence.
Answer and reasoning
List the possible final decisions, relate each to smaller valid states and establish base values from the state’s definition.
Continue learning
Compare the scenario with the Data Structures and Algorithms interview questions and test your understanding with the Data Structures and Algorithms MCQs. For terminology and implementation details, consult the reference material.