Balanced K-Factor Decomposition
Time precompute: O(rlogr) runtime: O(k * (logn)^(k - 1)) · Space O(rlogr) · Official statement on LeetCode
Solutions
// Time: precompute: O(rlogr)
// runtime: O(k * (logn)^(k - 1))
// Space: O(rlogr)
// backtracking, number theory
const auto& factors = [](int n) {
vector<vector<int>> result(n + 1);
for (int i = 1; i <= n; ++i) {
for (int j = i; j <= n; j += i) {
result[j].emplace_back(i);
}
}
return result;
};
const int MAX_N = 1e5;
const auto& FACTORS = factors(MAX_N);
class Solution {
public:
vector<int> minDifference(int n, int k) {
vector<int> result, curr;
function<void (int)> backtracking = [&](int remain) {
const int start = !empty(curr) ? curr.back() : 1;
if (size(curr) == k - 1) {
if (remain >= start) {
curr.emplace_back(remain);
if (empty(result) || result.back() - result[0] > curr.back() - curr[0]) {
result = curr;
}
curr.pop_back();
}
return;
}
const auto& factors = FACTORS[remain];
for (auto it = lower_bound(cbegin(factors), cend(factors), start); it != cend(factors); ++it) {
curr.emplace_back(*it);
backtracking(remain / *it);
curr.pop_back();
}
};
backtracking(n);
return result;
}
};
// Time: O(k * (n^(1/2) * n^(1/4) * n^(1/8) * n^(1/6) + n^(1/2) * n^(1/4) * n^(1/8) + n^(1/2) * n^(1/4) + n^(1/2))) <= O(k^2 * n)
// Space: O(k)
// backtracking, number theory
class Solution2 {
public:
vector<int> minDifference(int n, int k) {
vector<int> result, curr;
function<void (int)> backtracking = [&](int remain) {
const int start = !empty(curr) ? curr.back() : 1;
if (size(curr) == k - 1) {
if (remain >= start) {
curr.emplace_back(remain);
if (empty(result) || result.back() - result[0] > curr.back() - curr[0]) {
result = curr;
}
curr.pop_back();
}
return;
}
for (int i = 1; i * i <= remain; ++i) {
if (remain % i) {
continue;
}
const int j = remain / i;
if (i >= start) {
curr.emplace_back(i);
backtracking(j);
curr.pop_back();
}
if (j == i) {
continue;
}
if (j >= start) {
curr.emplace_back(j);
backtracking(i);
curr.pop_back();
}
}
};
backtracking(n);
return result;
}
};
// Time: O(2^(k-1) * k * n)
// Space: O(k)
// backtracking, number theory
class Solution3 {
public:
vector<int> minDifference(int n, int k) {
vector<int> result, curr;
function<void (int)> backtracking = [&](int remain) {
if (size(curr) == k - 1) {
curr.emplace_back(remain);
if (empty(result) || ranges::max(result) - ranges::min(result) > ranges::max(curr) - ranges::min(curr)) {
result = curr;
}
curr.pop_back();
return;
}
for (int i = 1; i * i <= remain; ++i) {
if (remain % i) {
continue;
}
const int j = remain / i;
curr.emplace_back(i);
backtracking(j);
curr.pop_back();
if (j == i) {
continue;
}
curr.emplace_back(j);
backtracking(i);
curr.pop_back();
}
};
backtracking(n);
return result;
}
};
Beginner Explanation
What is Balanced K-Factor Decomposition?
Balanced K-Factor Decomposition (LeetCode #3669) is a Medium problem that primarily trains backtracking.
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
It sits in the sweet spot of interview difficulty: multiple valid approaches, clear trade-offs. Official solution notes mention: Backtracking, Number Theory.
AlgoForge explanations are original teaching notes. Always open the official problem statement on LeetCode for constraints and examples.
Interview Walkthrough
Interview approach for Balanced K-Factor Decomposition
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 (precompute: O(rlogr) runtime: O(k * (logn)^(k - 1))) and space (O(rlogr)) 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 precompute: O(rlogr) runtime: O(k * (logn)^(k - 1)) time and O(rlogr) 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 | precompute: O(rlogr) runtime: O(k * (logn)^(k - 1)) |
| Space | O(rlogr) |
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 Balanced K-Factor Decomposition
- 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: backtracking.
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 Balanced K-Factor Decomposition in a second language (cpp, python).
- Drill 3–5 more problems tagged backtracking.
- 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
Balanced K-Factor Decomposition (#3669) — Medium. Pattern: dfs backtracking. Complexity: precompute: O(rlogr) runtime: O(k * (logn)^(k - 1)) time / O(rlogr) space. Re-derive the invariant before coding.
FAQs
What is the time complexity of Balanced K-Factor Decomposition?+
The reference solutions aim for precompute: O(rlogr) runtime: O(k * (logn)^(k - 1)) time and O(rlogr) space. Always re-derive complexity from the code you write in the interview.
What pattern does Balanced K-Factor Decomposition use?+
It primarily maps to dfs backtracking, within the broader topic of backtracking.
Is Balanced K-Factor Decomposition 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/balanced-k-factor-decomposition/