K-th Smallest in Lexicographical Order
Time O(logn) · Space O(logn) · Official statement on LeetCode
Solutions
// Time: O(logn)
// Space: O(logn)
class Solution {
public:
int findKthNumber(int n, int k) {
int result = 0;
vector<int> cnts(10);
for (int i = 1; i <= 9; ++i) {
cnts[i] = cnts[i - 1] * 10 + 1;
}
vector<int> nums;
for (int i = n; i > 0; i /= 10) {
nums.emplace_back(i % 10);
}
int total = n;
int target = 0;
for (int i = nums.size() - 1; i >= 0 && k; --i) {
target = target * 10 + nums[i];
const auto start = i == nums.size() - 1 ? 1 : 0;
for (int j = start; j <= 9; ++j) {
const auto candidate = result * 10 + j;
int num;
if (candidate < target) {
num = cnts[i + 1];
} else if (candidate > target) {
num = cnts[i];
} else {
num = total - cnts[i + 1] * (j - start) - cnts[i] * (9 - j);
}
if (k > num) {
k -= num;
} else {
result = candidate;
--k;
total = num - 1;
break;
}
}
}
return result;
}
};
// Time: O(logn * logn)
// Space: O(logn)
class Solution2 {
public:
int findKthNumber(int n, int k) {
int result = 0;
int index = 0;
findKthNumberHelper(n, k, 0, &index, &result);
return result;
}
private:
bool findKthNumberHelper(int n, int k, int cur, int *index, int *result) {
if (cur) {
++(*index);
if (*index == k) {
*result = cur;
return true;
}
}
for (int i = (cur == 0 ? 1 : 0); i <= 9; ++i, cur /= 10) {
cur = cur * 10 + i;
int cnt = count(n, cur);
if (k > cnt + *index) {
*index += cnt;
continue;
}
if (cur <= n && findKthNumberHelper(n, k, cur, index, result)) {
return true;
}
}
return false;
}
int count(int n, long long prefix) { // Time: O(logn)
int result = 0;
int number = 1;
while (prefix <= n) {
result += number;
prefix *= 10;
number *= 10;
}
result -= max(number / 10 - (n - prefix / 10 + 1), static_cast<long long>(0));
return result;
}
};
Beginner Explanation
What is K-th Smallest in Lexicographical Order?
K-th Smallest in Lexicographical Order (LeetCode #440) is a Hard problem that primarily trains depth first search.
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 dfs backtracking.
- Only then translate the idea into code.
Why this problem matters
Hard problems force you to combine patterns and prove complexity carefully — interview gold.
AlgoForge explanations are original teaching notes. Always open the official problem statement on LeetCode for constraints and examples.
Interview Walkthrough
Interview approach for K-th Smallest in Lexicographical Order
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 dfs backtracking.
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(logn)) and space (O(logn)) 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(logn) time and O(logn) space.
Pattern focus: dfs backtracking
Use the pattern as a checklist:
- dfs backtracking — 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(logn) |
| Space | O(logn) |
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 K-th Smallest in Lexicographical Order
- Skipping edge cases — empty collections, single-element inputs, max constraints.
- Wrong invariant for dfs backtracking — 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 dfs backtracking:
- 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: depth first search.
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 K-th Smallest in Lexicographical Order in a second language (cpp, python).
- Drill 3–5 more problems tagged depth first search.
- 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 dfs backtracking 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
K-th Smallest in Lexicographical Order (#440) — Hard. Pattern: dfs backtracking. Complexity: O(logn) time / O(logn) space. Re-derive the invariant before coding.
FAQs
What is the time complexity of K-th Smallest in Lexicographical Order?+
The reference solutions aim for O(logn) time and O(logn) space. Always re-derive complexity from the code you write in the interview.
What pattern does K-th Smallest in Lexicographical Order use?+
It primarily maps to dfs backtracking, within the broader topic of depth first search.
Is K-th Smallest in Lexicographical Order 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/k-th-smallest-in-lexicographical-order/