Largest Magic Square
Time O(max(m, n) * min(m, n)^3) · Space O(m * n) · Official statement on LeetCode
Solutions
// Time: O(max(m, n) * min(m, n)^3)
// Space: O(m + n)
class Solution {
public:
int largestMagicSquare(vector<vector<int>>& grid) {
vector<vector<int>> prefix_row(size(grid), vector<int>(size(grid[0]) + 1));
vector<vector<int>> prefix_col(size(grid[0]), vector<int>(size(grid) + 1));
for (int i = 0; i < size(grid); ++i) {
for (int j = 0; j < size(grid[0]); ++j) {
prefix_row[i][j + 1] = prefix_row[i][j] + grid[i][j];
prefix_col[j][i + 1] = prefix_col[j][i] + grid[i][j];
}
}
for (int l = min(size(grid), size(grid[0])); l >= 1; --l) {
for (int i = 0; i + l - 1 < size(grid); ++i) {
for (int j = 0; j + l - 1 < size(grid[0]); ++j) {
if (check(grid, prefix_row, prefix_col, l, i, j)) {
return l;
}
}
}
}
return 1;
}
private:
bool check(const vector<vector<int>>& grid,
const vector<vector<int>>& prefix_row,
const vector<vector<int>>& prefix_col,
int l, int i, int j) {
int diag = 0, anti_diag = 0;
for (int d = 0; d < l; ++d) {
diag += grid[i + d][j + d];
anti_diag += grid[i + d][j + l - 1 - d];
}
if (diag != anti_diag) {
return false;
}
for (int ni = i; ni < i + l; ++ni) {
if (diag != get_sum(prefix_row[ni], j, j + l - 1)) {
return false;
}
}
for (int nj = j; nj < j + l; ++nj) {
if (diag != get_sum(prefix_col[nj], i, i + l - 1)) {
return false;
}
}
return true;
}
int get_sum(const vector<int>& prefix, int a, int b) {
return prefix[b + 1] - prefix[a];
}
};
Beginner Explanation
What is Largest Magic Square?
Largest Magic Square (LeetCode #1895) is a Medium problem that primarily trains array.
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
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 Largest Magic Square
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(max(m, n) * min(m, n)^3)) 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(max(m, n) * min(m, n)^3) time and O(m * n) 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(max(m, n) * min(m, n)^3) |
| 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 Largest Magic Square
- 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: array.
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 Largest Magic Square in a second language (cpp, python).
- Drill 3–5 more problems tagged array.
- 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
Largest Magic Square (#1895) — Medium. Pattern: general problem-solving. Complexity: O(max(m, n) * min(m, n)^3) time / O(m * n) space. Re-derive the invariant before coding.
FAQs
What is the time complexity of Largest Magic Square?+
The reference solutions aim for O(max(m, n) * min(m, n)^3) time and O(m * n) space. Always re-derive complexity from the code you write in the interview.
What pattern does Largest Magic Square use?+
It primarily maps to general problem-solving, within the broader topic of array.
Is Largest Magic Square 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/largest-magic-square/