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A dynamic-programming state is a precisely defined subproblem: its parameters identify the case, and its meaning says what value the state stores. For example, dp[i] could mean the best score obtainable from position i onward—but the index alone does not tell you that. The recurrence is a separate rule for computing that value from other states. In an interview, explain the state in words before writing the recurrence, then make the base cases, evaluation order, final answer, and complexity consistent with that definition.
What a DP state means
MIT OpenCourseWare’s 6.006 Lecture 15 recommends describing a subproblem in words and in terms of its parameters. A state therefore has two parts: the information that identifies a subproblem, and the semantic contract that says what its value represents. An index, grid coordinate, or capacity is not a definition by itself.
For example, dp[i] might mean the length of the longest increasing subsequence ending at index i, or the maximum score obtainable from index i through the end. Both use one index, but they describe different subproblems and imply different transitions. Always state the meaning explicitly.
The state must contain enough information to determine the subproblem’s answer and make its transitions valid. If future choices depend on remaining capacity, for instance, the index alone may be insufficient; a state such as dp[i][capacity] can represent a prefix together with the available capacity. Too few parameters can combine cases that have different valid futures. Unnecessary parameters inflate the number of states and the memory required.
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State and recurrence are different
The state defines what a subproblem asks. The recurrence explains how to compute its answer from smaller or otherwise related subproblems. A recurrence only makes sense relative to a clear state definition: if you cannot say what each value means, you cannot reliably judge whether a transition is valid.
For a suffix-based bowling problem, MIT defines B(i) as the best score starting at pin i and continuing to the end. The recurrence can compare choices such as skipping the pin, taking one pin, or taking two adjacent pins, with each choice leading to a smaller suffix. The state meaning makes clear why those options cover the decisions at position i and why their successor states move forward.
A complete formulation: Fibonacci
Define F[i] as the ith Fibonacci number. With the first two values as base cases, the recurrence is F[i] = F[i-1] + F[i-2]. Computing in increasing order of i respects the dependencies, and the requested result is F[n].
Naive recursion repeats work for the same indices. Memoization stores a computed value and reuses it; bottom-up tabulation fills values in dependency order. The University of Texas at Austin’s Lecture 11 describes the memoized approach as storing a linear number of values. The usual linear-time claim counts arithmetic operations under a unit-cost integer model; for very large Fibonacci numbers, the cost of operating on increasingly long integers also matters.
State structure determines evaluation order
DAG shortest paths
For shortest paths from source s in a directed acyclic graph, let the state for vertex v be the shortest distance from s to v. Its recurrence takes the minimum, over incoming predecessors, of the predecessor’s distance plus the connecting edge’s weight; the source distance is zero. A topological ordering ensures predecessors are processed before the vertices that depend on them. MIT OpenCourseWare gives the running time as O(|V| + |E|) for this recurrence.
This example also shows that a state need not be an array position: it can be a graph vertex. The dependency structure—not a generic rule to count upward—determines a valid order.
Prefixes, suffixes, and other subproblems
Sequence problems often use prefixes, suffixes, or contiguous substrings. The choice should match the question and make dependencies manageable. In MIT’s bowling example, the suffix definition yields Θ(n) subproblems and Θ(1) work per subproblem, for Θ(n) total time under memoization, as described in the 2020 lecture. A different decomposition may be correct but less direct or more expensive.
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How to explain a DP solution in an interview
- Establish why dynamic programming may fit. Look for overlapping subproblems and a way for their answers to contribute to a larger answer. Optimization or counting language can be a clue, not proof; optimal substructure alone does not mean every recursive algorithm is DP. The University of Texas at Austin lecture describes the reuse principle this way: “Dynamic programming works by eliminating this redundancy.”
- Define the subproblem in words. State exactly what a value represents and list every parameter needed to identify it. Say whether an index refers to a prefix, suffix, endpoint, or another interpretation.
- Derive the recurrence from choices or predecessors. Explain why the alternatives cover the possibilities and how each transition maps to a state. Test the definition and recurrence on a tiny input before coding.
- Give base cases and a computation strategy. For tabulation, explain an order in which dependencies are ready. For top-down recursion, explain the cache and the stopping cases.
- Identify how to obtain the requested result. It may be one cell, such as
dp[n], or an aggregate over multiple states. If the problem asks for an actual path or set of choices rather than only its value, store enough parent or decision information to reconstruct a witness. - Analyze time and space. Count reachable states and the work performed per state; the basic time bound is their product. Account for stored states and any recursion depth. If you compress space, first show which previous values the recurrence actually needs.
- Present the model before the code. Saying the state and recurrence aloud lets the interviewer check the assumptions and semantics before implementation details obscure them.
Choosing between candidate state designs
When more than one formulation appears plausible, compare them on practical and proof-related grounds rather than assuming one style is always best:
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- Semantic clarity: Can you state the meaning of every value in one unambiguous sentence?
- Completeness: Does the state include every detail that affects future choices?
- Size and transition cost: How many reachable states are there, and how much work does each transition require?
- Proof and order: Is it clear why the recurrence is correct and how its dependencies can be evaluated?
- Implementation constraints: Would recursion depth be a concern, or can the required previous values be kept in a rolling array?
- Reconstruction needs: Does the task ask only for an optimal value, or must you recover the choices that produced it?
Top-down memoization follows the recursive dependencies and can avoid states that are never needed. Bottom-up tabulation avoids recursion depth and can make a compact dependency window easier to use. These are trade-offs, not a universal ranking; the state semantics should remain the same whichever evaluation style you choose.
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