Sum of k-Mirror Numbers
Time O(10^6) · Space O(1) · Official statement on LeetCode
Solutions
// Time: O(10^6), the most times of finding x is 665502 (k = 7, n = 30)
// Space: O(1)
class Solution {
public:
long long kMirror(int k, int n) {
const int base1 = k, base2 = 10; // (10, k) is slower
int64_t result = 0;
vector<int> prefix_num(2, 1), total(2, base1);
uint8_t odd = 1;
while (n--) {
int64_t x;
do {
x = mirror(prefix_num[odd], base1, odd);
if (++prefix_num[odd] == total[odd]) {
total[odd] *= base1;
odd ^= 1;
}
} while (x != reverse(x, base2));
result += x;
}
return result;
}
private:
int64_t mirror(int n, int base, bool odd) {
int64_t result = n;
if (odd) {
n /= base;
}
for (; n; n /= base) {
result = result * base + (n % base);
}
return result;
}
int64_t reverse(int64_t n, int base) {
int64_t result = 0;
for (; n; n /= base) {
result = result * base + n % base;
}
return result;
}
};
// Time: O(10^6), the most times of finding x is 665502 (k = 7, n = 30)
// Space: O(1)
class Solution2 {
public:
long long kMirror(int k, int n) {
string s = "0";
int64_t result = 0;
while (n--) {
int64_t x;
do {
x = next_num_in_base_k(k, &s);
} while (!is_mirror(to_string(x)));
result += x;
}
return result;
}
private:
int64_t next_num_in_base_k(int k, string *s) {
int result = 0;
for (int i = size(*s) / 2; i < size(*s); ++i) {
if ((*s)[i] + 1 - k < '0') {
(*s)[i] = (*s)[size(*s) - 1 - i] = (*s)[i] + 1;
break;
}
(*s)[i] = (*s)[size(*s) - 1 - i] = '0';
}
if ((*s)[0] == '0') {
s->back() = '1';
s->insert(begin(*s), '1');
}
return to_int_from_base_k(*s, k);
}
int64_t to_int_from_base_k(const string& s, int k) {
int64_t result = 0;
for (int64_t i = size(s) - 1, base = 1; i >= 0; --i, base *= k) {
result += (s[i] - '0') * base;
}
return result;
}
bool is_mirror(const string& s) {
int left = 0, right = size(s) - 1;
while (left < right) {
if (s[left++] != s[right--]) {
return false;
}
}
return true;
}
};
Beginner Explanation
What is Sum of k-Mirror Numbers?
Sum of k-Mirror Numbers (LeetCode #2081) is a Hard problem that primarily trains string.
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
Hard problems force you to combine patterns and prove complexity carefully — interview gold. Official solution notes mention: String, Palindrome, Brute Force.
AlgoForge explanations are original teaching notes. Always open the official problem statement on LeetCode for constraints and examples.
Interview Walkthrough
Interview approach for Sum of k-Mirror Numbers
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(10^6)) 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(10^6) 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(10^6) |
| 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 Sum of k-Mirror Numbers
- 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: string.
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 Sum of k-Mirror Numbers in a second language (cpp, python).
- Drill 3–5 more problems tagged string.
- 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
Sum of k-Mirror Numbers (#2081) — Hard. Pattern: general problem-solving. Complexity: O(10^6) time / O(1) space. Re-derive the invariant before coding.
FAQs
What is the time complexity of Sum of k-Mirror Numbers?+
The reference solutions aim for O(10^6) time and O(1) space. Always re-derive complexity from the code you write in the interview.
What pattern does Sum of k-Mirror Numbers use?+
It primarily maps to general problem-solving, within the broader topic of string.
Is Sum of k-Mirror Numbers 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/sum-of-k-mirror-numbers/