Airplane Seat Assignment Probability
Time O(1) · Space O(1) · Official statement on LeetCode
Solutions
// Time: O(1)
// Space: O(1)
class Solution {
public:
double nthPersonGetsNthSeat(int n) {
// p(k) = 1 * (prob that 1th passenger takes his own seat) +
// 0 * (prob that 1th passenger takes kth one's seat) +
// 1 * (prob that 1th passenger takes the others' seat) *
// (prob that the first k-1 passengers get a seat
// which is not kth one's seat)
// = 1/k + p(k-1)*(k-2)/k
//
// p(1) = 1
// p(2) = 1/2 + p(1) * (2-2)/2 = 1/2
// p(3) = 1/3 + p(2) * (3-2)/3 = 1/3 + 1/2 * (3-2)/3 = 1/2
// ...
// p(n) = 1/n + 1/2 * (n-2)/n = (2+n-2)/(2n) = 1/2
return n != 1 ? 0.5 : 1.0;
}
};
// Time: O(n)
// Space: O(1)
class Solution2 {
public:
double nthPersonGetsNthSeat(int n) {
vector<double> dp(2);
dp[0] = 1.0; // zero-indexed
for (int i = 2; i <= n; ++i) {
dp[(i - 1) % 2] = 1.0 / i + dp[(i - 2) % 2] * (i - 2) / i;
}
return dp[(n - 1) % 2];
}
};
Beginner Explanation
What is Airplane Seat Assignment Probability?
Airplane Seat Assignment Probability (LeetCode #1227) is a Medium problem that primarily trains math.
How to think about it
- Restate the goal in your own words before coding.
- Work a tiny example by hand so the invariant becomes obvious.
- Identify the pattern — this problem aligns with general problem-solving.
- Only then translate the idea into code.
Why this problem matters
It sits in the sweet spot of interview difficulty: multiple valid approaches, clear trade-offs.
AlgoForge explanations are original teaching notes. Always open the official problem statement on LeetCode for constraints and examples.
Interview Walkthrough
Interview approach for Airplane Seat Assignment Probability
Opening (30–60 seconds)
- Clarify inputs/outputs and edge cases (empty input, single element, duplicates, overflow).
- State a brute force so the interviewer knows you can solve it naively.
- Propose the optimal direction tied to general problem-solving.
Core solution narrative
- Define the state you track (pointers, DP cell, set membership, stack top, etc.).
- Explain the transition when you process the next element.
- Call out time (O(1)) and space (O(1)) before coding.
- Code cleanly; narrate variable names.
What interviewers listen for
- Correctness on edge cases
- Complexity honesty
- Ability to discuss trade-offs (e.g., hash map space vs. sort + two pointers)
Follow-up questions they may ask
- Can you solve it with less memory?
- What if the input stream is infinite / doesn't fit in RAM?
- How would tests look for adversarial inputs?
Optimized Approach
Optimized solution notes
The reference solutions on AlgoForge target O(1) time and O(1) space.
Pattern focus: general problem-solving
Use the pattern as a checklist:
- Identify the dominant pattern and stick to one clear invariant
Multiple methods appear in the source solutions — compare them and explain when each is preferable.
Implementation tips
- Prefer readable names over micro-optimizations in interviews.
- Extract helpers only when they clarify (e.g., expand-around-center, DFS visit).
- After AC-level logic, re-scan for off-by-one and null checks.
Complexity Analysis
Complexity
| Measure | Bound |
|---|---|
| Time | O(1) |
| Space | O(1) |
How to justify this in an interview
- Time: count loops, map/set operations, and recursive branching; state average vs worst case if relevant.
- Space: include hash maps, recursion stack, and output allocation when the problem asks for it.
If your implementation differs from the reference, re-derive big-O from your code — never memorize a complexity you cannot defend.
Common Mistakes
Common mistakes on Airplane Seat Assignment Probability
- Skipping edge cases — empty collections, single-element inputs, max constraints.
- Wrong invariant for general problem-solving — updating state too early or too late.
- Mutating input unexpectedly when the problem forbids it.
- Off-by-one in windows, ranges, or binary search bounds.
- Ignoring overflow / precision for integer arithmetic problems.
- Overengineering — jumping to an advanced structure when a simpler approach works.
Alternative Approaches
Alternatives
The source file includes more than one method. Compare:
- Primary optimized path — best complexity for typical interviews.
- Secondary approach — often brute force, sorting-based, or space-optimized variant.
Practice articulating when you would pick each (constraints, readability, follow-ups).
Edge Cases
Edge cases checklist
- Minimum input size
- Maximum input size / time limits
- Duplicates and already-sorted input
- Negative numbers / zeros (if applicable)
- Disconnected structures (graphs/trees)
- Single path vs branching recursion depth
Pattern Recognition
Spotting this pattern
Signal phrases that point to general problem-solving:
- Sorted input or ability to sort without changing the answer class
- Need for contiguous subarray / substring → consider sliding window
- Need for O(1) membership → hash set/map
- Optimal substructure + overlapping subproblems → DP
- Connectivity / components → graph DFS/BFS or Union-Find
Primary topics: math.
Follow-up Interview Questions
Follow-ups
- How does the solution change if the input is a stream?
- Can you solve it in-place?
- What if duplicates must be handled differently?
- How would you parallelize the approach?
- Design tests that would break a buggy implementation.
Practice Recommendations
What to practice next
- Re-solve Airplane Seat Assignment Probability in a second language (cpp, python).
- Drill 3–5 more problems tagged math.
- Teach the solution out loud in under 5 minutes.
- Add this problem to your revision calendar in 3 days and 14 days.
Visualization
Study checklist
- Read the official problem statement on LeetCode
- Solve on paper / whiteboard first
- Implement the general problem-solving approach
- Verify edge cases from the checklist
- State time and space complexity aloud
- Compare with the AlgoForge reference solution
- Schedule a revision session
Revision notes
Airplane Seat Assignment Probability (#1227) — Medium. Pattern: general problem-solving. Complexity: O(1) time / O(1) space. Re-derive the invariant before coding.
FAQs
What is the time complexity of Airplane Seat Assignment Probability?+
The reference solutions aim for O(1) time and O(1) space. Always re-derive complexity from the code you write in the interview.
What pattern does Airplane Seat Assignment Probability use?+
It primarily maps to general problem-solving, within the broader topic of math.
Is Airplane Seat Assignment Probability good for interviews?+
Yes — as a Medium problem it is a solid practice target. Pair it with related problems in the same pattern family for spaced repetition.
Where can I read the official statement?+
Open the official LeetCode page for constraints and examples: https://leetcode.com/problems/airplane-seat-assignment-probability/