Count K-Reducible Numbers Less Than N
Time O(n^2) · Space O(n) · Official statement on LeetCode
Solutions
// Time: O(n^2)
// Space: O(n)
// dp
vector<int> cnt = {0, 0};
class Solution {
public:
int countKReducibleNumbers(string s, int k) {
static const int MOD = 1e9 + 7;
while (size(s) - 1 >= size(cnt)) { // cached
cnt.emplace_back(cnt[__builtin_popcount(size(cnt))] + 1);
}
vector<int> dp(size(s));
for (int i = 0, curr = 0; i < size(s); ++i) {
for (int j = i - 1; j >= 0; --j) {
dp[j + 1] = (dp[j + 1] + dp[j]) % MOD;
}
if (s[i] != '1') {
continue;
}
dp[curr] = (dp[curr] + 1) % MOD;
++curr;
}
int result = 0;
for (int i = 1; i < size(s); ++i) {
if (cnt[i] < k) {
result = (result + dp[i]) % MOD;
}
}
return result;
}
};
// Time: O(n^2)
// Space: O(n)
// dp, combinatorics
// vector<int> cnt = {0, 0};
class Solution2 {
public:
int countKReducibleNumbers(string s, int k) {
static const int MOD = 1e9 + 7;
vector<int> fact = {1, 1};
vector<int> inv = {1, 1};
vector<int> inv_fact = {1, 1};
const auto& nCr = [&](int n, int k) {
while (size(inv) <= n) { // lazy initialization
fact.emplace_back((static_cast<int64_t>(fact.back()) * size(inv)) % MOD);
inv.emplace_back((static_cast<int64_t>(inv[MOD % size(inv)]) * (MOD - MOD / size(inv))) % MOD); // https://cp-algorithms.com/algebra/module-inverse.html
inv_fact.emplace_back((static_cast<int64_t>(inv_fact.back()) * inv.back()) % MOD);
}
return (((static_cast<int64_t>(fact[n]) * inv_fact[n - k]) % MOD) * inv_fact[k]) % MOD;
};
while (size(s) - 1 >= size(cnt)) { // cached
cnt.emplace_back(cnt[__builtin_popcount(size(cnt))] + 1);
}
int result = 0;
for (int i = 0, curr = 0; i < size(s); ++i) {
if (s[i] != '1') {
continue;
}
for (int c = 0; c <= size(s) - (i + 1); ++c) {
if (cnt[curr + c] < k) {
result = (result + nCr(size(s) - (i + 1), c)) % MOD;
}
}
++curr;
}
return (((result - 1) % MOD) + MOD) % MOD;
}
};
Beginner Explanation
What is Count K-Reducible Numbers Less Than N?
Count K-Reducible Numbers Less Than N (LeetCode #3352) is a Hard problem that primarily trains dynamic programming.
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 dynamic programming.
- Only then translate the idea into code.
Why this problem matters
Hard problems force you to combine patterns and prove complexity carefully — interview gold. Official solution notes mention: DP, Combinatorics.
AlgoForge explanations are original teaching notes. Always open the official problem statement on LeetCode for constraints and examples.
Interview Walkthrough
Interview approach for Count K-Reducible Numbers Less Than N
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 dynamic programming.
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^2)) and space (O(n)) 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^2) time and O(n) space.
Pattern focus: dynamic programming
Use the pattern as a checklist:
- dynamic programming — 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^2) |
| Space | O(n) |
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 Count K-Reducible Numbers Less Than N
- Skipping edge cases — empty collections, single-element inputs, max constraints.
- Wrong invariant for dynamic programming — 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 dynamic programming:
- 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: dynamic programming.
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 Count K-Reducible Numbers Less Than N in a second language (cpp, python).
- Drill 3–5 more problems tagged dynamic programming.
- 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 dynamic programming 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
Count K-Reducible Numbers Less Than N (#3352) — Hard. Pattern: dynamic programming. Complexity: O(n^2) time / O(n) space. Re-derive the invariant before coding.
FAQs
What is the time complexity of Count K-Reducible Numbers Less Than N?+
The reference solutions aim for O(n^2) time and O(n) space. Always re-derive complexity from the code you write in the interview.
What pattern does Count K-Reducible Numbers Less Than N use?+
It primarily maps to dynamic programming, within the broader topic of dynamic programming.
Is Count K-Reducible Numbers Less Than N good for interviews?+
Yes — as a Hard 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/count-k-reducible-numbers-less-than-n/