Find the N-th Value After K Seconds
Time O(n + k) · Space O(n + k) · Official statement on LeetCode
Solutions
// Time: O(n + k)
// Space: O(n + k)
// combinatorics
class Solution {
public:
int valueAfterKSeconds(int n, int k) {
return nCr((n - 1) + k, n - 1);
}
private:
int nCr(int n, int k) {
if (k < 0 || k > n) {
return 0;
}
while (size(inv_) <= n) { // lazy initialization
fact_.emplace_back(mulmod(fact_.back(), size(inv_)));
inv_.emplace_back(mulmod(inv_[MOD % size(inv_)], MOD - MOD / size(inv_))); // https://cp-algorithms.com/algebra/module-inverse.html
inv_fact_.emplace_back(mulmod(inv_fact_.back(), inv_.back()));
}
return mulmod(mulmod(fact_[n], inv_fact_[n - k]), inv_fact_[k]);
}
uint32_t addmod(uint32_t a, uint32_t b) { // avoid overflow
a %= MOD, b %= MOD;
if (MOD - a <= b) {
b -= MOD; // relied on unsigned integer overflow in order to give the expected results
}
return a + b;
}
uint32_t submod(uint32_t a, uint32_t b) {
a %= MOD, b %= MOD;
return addmod(a, MOD - b);
}
// reference: https://stackoverflow.com/questions/12168348/ways-to-do-modulo-multiplication-with-primitive-types
uint32_t mulmod(uint32_t a, uint32_t b) { // avoid overflow
a %= MOD, b %= MOD;
uint32_t result = 0;
if (a < b) {
swap(a, b);
}
while (b > 0) {
if (b % 2 == 1) {
result = addmod(result, a);
}
a = addmod(a, a);
b /= 2;
}
return result;
}
static const uint32_t MOD = 1e9 + 7;
vector<int> fact_ = {1, 1};
vector<int> inv_ = {1, 1};
vector<int> inv_fact_ = {1, 1};
};
// Time: O(n * k)
// Space: O(n)
// prefix sum
class Solution2 {
public:
int valueAfterKSeconds(int n, int k) {
static const uint32_t MOD = 1e9 + 7;
vector<int> prefix(n, 1);
for (int _ = 0; _ < k; ++_) {
for (int i = 1; i < n; ++i) {
prefix[i] = (prefix[i] + prefix[i - 1]) % MOD;
}
}
return prefix.back();
}
};
Beginner Explanation
What is Find the N-th Value After K Seconds?
Find the N-th Value After K Seconds (LeetCode #3179) 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 prefix sum.
- 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. Official solution notes mention: Prefix Sum, Combinatorics.
AlgoForge explanations are original teaching notes. Always open the official problem statement on LeetCode for constraints and examples.
Interview Walkthrough
Interview approach for Find the N-th Value After K Seconds
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 prefix sum.
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(n + k)) and space (O(n + k)) 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(n + k) time and O(n + k) space.
Pattern focus: prefix sum
Use the pattern as a checklist:
- prefix sum — confirm the invariant holds after each step
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(n + k) |
| Space | O(n + k) |
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 Find the N-th Value After K Seconds
- Skipping edge cases — empty collections, single-element inputs, max constraints.
- Wrong invariant for prefix sum — 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 prefix sum:
- 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 Find the N-th Value After K Seconds 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 prefix sum 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
Find the N-th Value After K Seconds (#3179) — Medium. Pattern: prefix sum. Complexity: O(n + k) time / O(n + k) space. Re-derive the invariant before coding.
FAQs
What is the time complexity of Find the N-th Value After K Seconds?+
The reference solutions aim for O(n + k) time and O(n + k) space. Always re-derive complexity from the code you write in the interview.
What pattern does Find the N-th Value After K Seconds use?+
It primarily maps to prefix sum, within the broader topic of math.
Is Find the N-th Value After K Seconds 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/find-the-n-th-value-after-k-seconds/