Kth Smallest Element in a Sorted Matrix
Time O(k * log(min(n, m, k))) · Space O(min(n, m, k)) · Official statement on LeetCode
Solutions
// Time: O(k * log(min(n, m, k))), with n x m matrix
// Space: O(min(n, m, k))
class Solution {
public:
int kthSmallest(vector<vector<int>>& matrix, int k) {
int kth_smallest = 0;
using P = pair<int, pair<int, int>>;
priority_queue<P, vector<P>, greater<P>> q;
auto push = [&matrix, &q](int i, int j) {
if (matrix.size() > matrix[0].size()) {
if (i < matrix[0].size() && j < matrix.size()) {
q.emplace(matrix[j][i], make_pair(i, j));
}
} else {
if (i < matrix.size() && j < matrix[0].size()) {
q.emplace(matrix[i][j], make_pair(i, j));
}
}
};
push(0, 0);
while (!q.empty() && k--) {
auto tmp = q.top(); q.pop();
kth_smallest = tmp.first;
int i, j;
tie(i, j) = tmp.second;
push(i, j + 1);
if (j == 0) {
push(i + 1, 0);
}
}
return kth_smallest;
}
};
Beginner Explanation
What is Kth Smallest Element in a Sorted Matrix?
Kth Smallest Element in a Sorted Matrix (LeetCode #378) is a Medium problem that primarily trains binary heap.
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 heap.
- 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.
AlgoForge explanations are original teaching notes. Always open the official problem statement on LeetCode for constraints and examples.
Interview Walkthrough
Interview approach for Kth Smallest Element in a Sorted 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 heap.
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(k * log(min(n, m, k)))) and space (O(min(n, m, k))) 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(k * log(min(n, m, k))) time and O(min(n, m, k)) space.
Pattern focus: heap
Use the pattern as a checklist:
- heap — 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(k * log(min(n, m, k))) |
| Space | O(min(n, m, k)) |
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 Kth Smallest Element in a Sorted Matrix
- Skipping edge cases — empty collections, single-element inputs, max constraints.
- Wrong invariant for heap — 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 heap:
- 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: binary heap.
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 Kth Smallest Element in a Sorted Matrix in a second language (cpp, python).
- Drill 3–5 more problems tagged binary heap.
- 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 heap 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
Kth Smallest Element in a Sorted Matrix (#378) — Medium. Pattern: heap. Complexity: O(k * log(min(n, m, k))) time / O(min(n, m, k)) space. Re-derive the invariant before coding.
FAQs
What is the time complexity of Kth Smallest Element in a Sorted Matrix?+
The reference solutions aim for O(k * log(min(n, m, k))) time and O(min(n, m, k)) space. Always re-derive complexity from the code you write in the interview.
What pattern does Kth Smallest Element in a Sorted Matrix use?+
It primarily maps to heap, within the broader topic of binary heap.
Is Kth Smallest Element in a Sorted Matrix 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/kth-smallest-element-in-a-sorted-matrix/