Rank Transform of a Matrix
Time O(m * n * log(m * n)) · Space O(m * n) · Official statement on LeetCode
Solutions
// Time: O(m * n * log(m * n) + m * n * α(m * n)) = O(m * n * log(m * n))
// Space: O(m * n)
class Solution {
public:
vector<vector<int>> matrixRankTransform(vector<vector<int>>& matrix) {
map<int, vector<pair<int, int>>> lookup;
for (int i = 0; i < size(matrix); ++i) {
for (int j = 0; j < size(matrix[0]); ++j) {
lookup[matrix[i][j]].emplace_back(i, j);
}
}
vector<int> rank(size(matrix) + size(matrix[0]));
for (const auto& [x, pairs] : lookup) {
vector<int> new_rank(rank);
const auto& cb = [&new_rank](int x, int y, int z) {
new_rank[x] = max(new_rank[y], new_rank[z]);
};
UnionFind union_find(size(matrix) + size(matrix[0]), cb);
for (const auto& [i, j] : pairs) {
union_find.union_set(i, j + size(matrix));
}
for (const auto& [i, j] : pairs) {
matrix[i][j] = rank[i] = rank[j + size(matrix)] = new_rank[union_find.find_set(i)] + 1;
}
}
return matrix;
}
private:
class UnionFind {
public:
UnionFind(const int n, function<void(int, int, int)> cb)
: set_(n)
, rank_(n)
, count_(n)
, cb_(cb) {
iota(set_.begin(), set_.end(), 0);
}
int find_set(const int x) {
if (set_[x] != x) {
set_[x] = find_set(set_[x]); // Path compression.
}
return set_[x];
}
bool union_set(const int x, const int y) {
int x_root = find_set(x), y_root = find_set(y);
if (x_root == y_root) {
return false;
}
if (rank_[x_root] < rank_[y_root]) { // Union by rank.
set_[x_root] = y_root;
cb_(y_root, x_root, y_root);
} else if (rank_[x_root] > rank_[y_root]) {
set_[y_root] = x_root;
cb_(x_root, x_root, y_root);
} else {
set_[y_root] = x_root;
++rank_[x_root];
cb_(x_root, x_root, y_root);
}
--count_;
return true;
}
int size() const {
return count_;
}
private:
vector<int> set_;
vector<int> rank_;
int count_;
function<void(int, int, int)> cb_;
};
};
Beginner Explanation
What is Rank Transform of a Matrix?
Rank Transform of a Matrix (LeetCode #1632) is a Hard problem that primarily trains greedy.
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 union find.
- 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: Union Find.
AlgoForge explanations are original teaching notes. Always open the official problem statement on LeetCode for constraints and examples.
Interview Walkthrough
Interview approach for Rank Transform of a Matrix
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 union find.
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(m * n * log(m * n))) and space (O(m * 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(m * n * log(m * n)) time and O(m * n) space.
Pattern focus: union find
Use the pattern as a checklist:
- union find — 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(m * n * log(m * n)) |
| Space | O(m * 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 Rank Transform of a Matrix
- Skipping edge cases — empty collections, single-element inputs, max constraints.
- Wrong invariant for union find — 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 union find:
- 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: greedy.
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 Rank Transform of a Matrix in a second language (cpp, python).
- Drill 3–5 more problems tagged greedy.
- 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 union find 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
Rank Transform of a Matrix (#1632) — Hard. Pattern: union find. Complexity: O(m * n * log(m * n)) time / O(m * n) space. Re-derive the invariant before coding.
FAQs
What is the time complexity of Rank Transform of a Matrix?+
The reference solutions aim for O(m * n * log(m * n)) time and O(m * n) space. Always re-derive complexity from the code you write in the interview.
What pattern does Rank Transform of a Matrix use?+
It primarily maps to union find, within the broader topic of greedy.
Is Rank Transform of a Matrix 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/rank-transform-of-a-matrix/