Tiling a Rectangle with the Fewest Squares
Time O(n^2 * m^2 * m^(n * m)) · Space O(n * m) · Official statement on LeetCode
Solutions
// Time: O(n^2 * m^2 * m^(n * m)), given m < n
// Space: O(n * m)
class Solution {
public:
int tilingRectangle(int n, int m) {
if (m > n) {
return tilingRectangle(m, n);
}
vector<vector<int>> board(n, vector<int>(m));
int result = numeric_limits<int>::max();
backtracking(&board, 0, &result);
return result;
}
private:
pair<int, int> find_next(const vector<vector<int>>& board) {
for (int i = 0; i < board.size(); ++i) {
for (int j = 0; j < board[0].size(); ++j) {
if (!board[i][j]) {
return {i, j};
}
}
}
return {-1, -1};
}
int find_max_length(const vector<vector<int>>& board, int i, int j) {
int max_length = 1;
while (i + max_length - 1 < board.size() &&
j + max_length - 1 < board[0].size()) {
for (int r = i; r < i + max_length - 1; ++r) {
if (board[r][j + max_length - 1]) {
return max_length - 1;
}
}
for (int c = j; c < j + max_length; ++c) {
if (board[i + max_length - 1][c]) {
return max_length - 1;
}
}
++max_length;
}
return max_length - 1;
}
void fill(vector<vector<int>> *board,
int i, int j, int length, int val) {
for (int r = i; r < i + length; ++r) {
for (int c = j; c < j + length; ++c) {
(*board)[r][c] = val;
}
}
}
void backtracking(vector<vector<int>> *board,
int count, int *result) {
if (count >= *result) { // pruning
return;
}
const auto& [i, j] = find_next(*board);
if (i == -1 && j == -1) { // finished
*result = min(*result, count);
return;
}
const auto& max_length = find_max_length(*board, i, j);
for (int k = max_length; k >= 1; --k) {
fill(board, i, j, k, 1);
backtracking(board, count + 1, result);
fill(board, i, j, k, 0);
}
}
};
Beginner Explanation
What is Tiling a Rectangle with the Fewest Squares?
Tiling a Rectangle with the Fewest Squares (LeetCode #1240) is a Hard 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
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 Tiling a Rectangle with the Fewest Squares
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(n^2 * m^2 * m^(n * m))) and space (O(n * m)) 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 * m^2 * m^(n * m)) time and O(n * m) 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(n^2 * m^2 * m^(n * m)) |
| Space | O(n * m) |
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 Tiling a Rectangle with the Fewest Squares
- 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 Tiling a Rectangle with the Fewest Squares 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
Tiling a Rectangle with the Fewest Squares (#1240) — Hard. Pattern: dfs backtracking. Complexity: O(n^2 * m^2 * m^(n * m)) time / O(n * m) space. Re-derive the invariant before coding.
FAQs
What is the time complexity of Tiling a Rectangle with the Fewest Squares?+
The reference solutions aim for O(n^2 * m^2 * m^(n * m)) time and O(n * m) space. Always re-derive complexity from the code you write in the interview.
What pattern does Tiling a Rectangle with the Fewest Squares use?+
It primarily maps to dfs backtracking, within the broader topic of backtracking.
Is Tiling a Rectangle with the Fewest Squares 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/tiling-a-rectangle-with-the-fewest-squares/