DP Base Cases and Overflow: Silent Bugs in Your First Rows

Originally published on LeetCopilot Blog

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Wrong base cases, missing sentinels, or overflow when numbers grow fast—here's how to set base states safely.

Many DP bugs hide in the first few rows of your table: wrong base cases, missing sentinel values, or overflow when numbers grow fast. This guide shows how to set base states safely, prevent overflow, and articulate your choices in interviews.

TL;DR

• Topic: configuring DP base cases and overflow protections before writing transitions.
• Why it matters: a single bad base row can cascade into every cell; overflow can turn correct logic into wrong answers.
• Core steps: define the problem’s zero/one-size answers, pick safe sentinel values, choose numeric types that won’t overflow, and assert boundaries in transitions.
• Common confusion: copying template bases without matching the recurrence or using INT_MAX without guarding addition.
• You’ll learn: setup patterns, a micro trace, a safe initialization snippet, and a checklist to explain in interviews.

Beginner-Friendly Explanation: What Base Cases Represent

• Zero-size answer: what is the answer when no items or length zero? This seeds your table.
• One-size answer: the simplest non-empty case that validates your recurrence direction.
• Sentinels: large or small placeholders that mean “unreachable”; they must avoid overflow when combined with valid costs.
• This builds on the setup patterns in Dynamic Programming Base Cases for Grid Problems.

Step-by-Step Learning Guidance

1) Define reachable states explicitly

List which indices are valid; mark others as INF or -INF so you never combine them accidentally.

2) Choose overflow-safe sentinels

When using addition, avoid INF that will overflow: use a bounded large value (e.g., 10**15 in Python) or checks before adding.

3) Seed the table before loops

Initialize base rows/cols prior to transitions. Validate they match problem guarantees, as reinforced in Dynamic Programming Tabulation Checklist for Beginners.

4) Apply guarded transitions

Before combining prev + cost, verify prev is reachable. This mirrors the table-audit habit from Dynamic Programming State Transition Table Practice.

5) Trace a tiny example

Run a 2x2 or size-3 case and confirm the base seeds propagate as expected.

Visualizable Example: Coin Change (Min Coins)

Amount: 0 1 2 3
Base:   0 INF INF INF
Coin 1: 0 1 2 3
Coin 2: 0 1 1 2

Notice how INF is never added unless the previous state was reachable.

Code Example: Overflow-Safe Initialization

INF = 10**15

def min_coins(amount, coins):
    dp = [INF] * (amount + 1)
    dp[0] = 0
    for coin in coins:
        for x in range(coin, amount + 1):
            if dp[x - coin] != INF:
                dp[x] = min(dp[x], dp[x - coin] + 1)
    return dp[amount] if dp[amount] != INF else -1

Guarding against INF before addition prevents overflow and incorrect minima.

Practical Preparation Strategies

• Type sizing: In typed languages, pick 64-bit integers for large counts or long paths.
• Clamp additions: When near limits, clamp sums or short-circuit unreachable states.
• Row-by-row audits: After each outer loop, print the row to see whether sentinels are leaking into valid cells.
• Targeted practice: Compare memoization seeds with tabulation seeds to see how initialization differs.
• Assistive checks: LeetCopilot can highlight unreachable states and flag transitions that add to INF so you catch overflow early.

Common Mistakes to Avoid

Adding to infinity without checks

INF + cost can wrap or remain INF incorrectly; always guard before arithmetic.

Misaligned base directions

Filling from top-left while recurrence expects bottom-up leaves cells uninitialized.

Ignoring problem-specific floors

Some problems need -INF for maximization; using 0 accidentally allows unearned paths.

Forgetting modulo or caps

When outputs require modulo, apply it after safe addition to avoid intermediate overflow.

FAQ

• How do I pick sentinel values? Choose a value larger than any possible answer but small enough to avoid overflow when added once.
• What should I practice before this? Review the recurrence identification steps in How to Know When Dynamic Programming Is Needed to ensure the base cases align with the recurrence.
• Is overflow still a concern in Python? Python integers expand, but overflow-like bugs still happen if you use extreme sentinels that distort minima.
• How do I explain this in interviews? State the base cases, show the guard condition before transitions, and mention how your sentinel prevents bogus paths.

Conclusion

Sound DP solutions start with sound base cases. By defining reachable states, guarding against overflow, and tracing a tiny example, you keep the table honest and earn trust during interviews.

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