Dynamic Programming Interviews and the Meaning of a State

Dynamic Programming Interviews and the Meaning of a State Define the state and the stopping point

Let dp[a] mean the fewest coins needed to make exactly amount a using the permitted denominations. The word exactly matters. A solution that exceeds the target does not satisfy this problem.

The base case is dp[0] equal to zero because making zero requires no coins. Other states begin as unreachable. You can represent that with a suitable sentinel, but explain how your arithmetic avoids treating the sentinel as an ordinary count.

Assume the denominations are positive integers and the target is a nonnegative integer. A zero or negative denomination would undermine the simple dependency order used by this solution. Clarifying that assumption is part of defining the problem.

Derive the recurrence from the last coin

Imagine an optimal solution for amount a and consider its last coin c. The remaining coins must make a minus c. If that remainder could be made with fewer coins than the solution used, replacing it would improve the original solution as well.

This motivates trying every allowed last coin and taking the smallest reachable value of dp[a minus c] plus one. Only consider coins that do not exceed a. The recurrence is a consequence of the choice you are enumerating, rather than a formula that appears without explanation.

For this example, the answers for amounts zero through six are 0, 1, 2, 1, 1, 2, and 2. At amount six, using a final coin of three reaches dp[3] plus one, giving two coins.

Evaluate the states in a safe order

Fill amounts from zero upward. Every positive coin leads to a smaller remainder, so the states needed for amount a have already been computed. If the interviewer changes the problem to limited coin quantities, this state definition may no longer contain enough information.

Distinguish minimum coin count from the number of ways to form an amount. Both questions mention coins, but they aggregate possibilities differently. Reusing a remembered loop without checking the objective can solve a different problem convincingly.

Handle missing answers and reconstruct a solution

Try denominations 3 and 4 with target 2. No combination works, so the state stays unreachable. Your final output should report that outcome explicitly instead of returning the sentinel as a coin count.

If the interviewer wants the actual coins, record which choice produced each reachable minimum. Starting at the target, follow those choices back to zero. Tie-breaking can be arbitrary unless the output contract requires a specific combination.

You can use PhantomCodeAI[https://www.phantomcodeai.com/download] as one preparation resource, but ask for a critique of your state definition before requesting completed code. A useful follow-up is to change the supply from unlimited to limited and explain why the original state needs reconsideration.

With target T and K denominations, this bottom-up method examines up to K choices for each amount, using O(TK) time and O(T) storage. State those parameters clearly. The explanation should connect the state, choices, base case, and evaluation order into one argument.