#2812Medium~35 min
Find the Safest Path in a Grid
Time O(n^2) · Space O(n^2) · Official statement on LeetCode
Solutions
// Time: O(n^2)
// Space: O(n^2)
// bfs, bucket sort, union find
class Solution {
public:
int maximumSafenessFactor(vector<vector<int>>& grid) {
static const vector<pair<int, int>> DIRECTIONS = {{1, 0}, {0, 1}, {-1, 0}, {0, -1}};
const auto& bfs = [&]() {
vector<vector<int>> dist(size(grid), vector<int>(size(grid[0]), -1));
vector<pair<int, int>> q;
for (int r = 0; r < size(grid); ++r) {
for (int c = 0; c < size(grid[0]); ++c) {
if (grid[r][c]) {
dist[r][c] = 0;
q.emplace_back(r, c);
}
}
}
int d = 0;
for (int d = 0; !empty(q); ++d) {
vector<pair<int, int>> new_q;
for (const auto& [r, c] : q) {
for (const auto& [dr, dc] : DIRECTIONS) {
const int nr = r + dr, nc = c + dc;
if (!(0 <= nr && nr < size(dist) && 0 <= nc && nc < size(dist[0]) && dist[nr][nc] == -1)) {
continue;
}
dist[nr][nc] = d + 1;
new_q.emplace_back(nr, nc);
}
}
q = move(new_q);
}
return dist;
};
auto dist = bfs();
vector<vector<pair<int, int>>> buckets((size(grid) - 1) + (size(grid[0]) - 1) + 1);
for (int r = 0; r < size(grid); ++r) {
for (int c = 0; c < size(grid[0]); ++c) {
buckets[dist[r][c]].emplace_back(r, c);
}
}
vector<vector<bool>> lookup(size(grid), vector<bool>(size(grid[0])));
UnionFind uf(size(grid) * size(grid[0]));
int d = size(buckets) - 1;
for (; d >= 0; --d) {
for (const auto& [r, c] : buckets[d]) {
for (const auto& [dr, dc] : DIRECTIONS) {
const int nr = r + dr, nc = c + dc;
if (!(0 <= nr && nr < size(dist) && 0 <= nc && nc < size(dist[0]) && lookup[nr][nc])) {
continue;
}
uf.union_set(nr * size(grid[0]) + nc, r * size(grid[0]) + c);
}
lookup[r][c] = true;
}
if (uf.find_set(0 * size(grid[0]) + 0) == uf.find_set((size(grid) - 1) * size(grid[0]) + (size(grid[0]) - 1))) {
break;
}
}
return d;
}
private:
class UnionFind {
public:
UnionFind(int n)
: set_(n)
, rank_(n) {
iota(begin(set_), end(set_), 0);
}
int find_set(int x) {
if (set_[x] != x) {
set_[x] = find_set(set_[x]); // Path compression.
}
return set_[x];
}
bool union_set(int x, int y) {
x = find_set(x), y = find_set(y);
if (x == y) {
return false;
}
if (rank_[x] > rank_[y]) {
swap(x, y);
}
set_[x] = y; // Union by rank.
if (rank_[x] == rank_[y]) {
++rank_[y];
}
return true;
}
private:
vector<int> set_;
vector<int> rank_;
};
};
// Time: O(n^2 * logn)
// Space: O(n^2)
// bfs, dijkstra's algorithm
class Solution2 {
public:
int maximumSafenessFactor(vector<vector<int>>& grid) {
static const vector<pair<int, int>> DIRECTIONS = {{1, 0}, {0, 1}, {-1, 0}, {0, -1}};
const auto& bfs = [&]() {
vector<vector<int>> dist(size(grid), vector<int>(size(grid[0]), -1));
vector<pair<int, int>> q;
for (int r = 0; r < size(grid); ++r) {
for (int c = 0; c < size(grid[0]); ++c) {
if (grid[r][c]) {
dist[r][c] = 0;
q.emplace_back(r, c);
}
}
}
int d = 0;
for (int d = 0; !empty(q); ++d) {
vector<pair<int, int>> new_q;
for (const auto& [r, c] : q) {
for (const auto& [dr, dc] : DIRECTIONS) {
const int nr = r + dr, nc = c + dc;
if (!(0 <= nr && nr < size(dist) && 0 <= nc && nc < size(dist[0]) && dist[nr][nc] == -1)) {
continue;
}
dist[nr][nc] = d + 1;
new_q.emplace_back(nr, nc);
}
}
q = move(new_q);
}
return dist;
};
auto dist = bfs();
const auto& dijkstra = [&](const auto& start, const auto& target) {
priority_queue<pair<int, pair<int, int>>> max_heap;
max_heap.emplace(dist[start.first][start.second], start);
dist[start.first][start.second] = -1;
while (!empty(max_heap)) {
auto [curr, u] = max_heap.top(); max_heap.pop();
if (u == target) {
return curr;
}
for (const auto& [dr, dc] : DIRECTIONS) {
const int nr = u.first + dr, nc = u.second + dc;
if (!(0 <= nr && nr < size(dist) && 0 <= nc && nc < size(dist[0]) && dist[nr][nc] != -1)) {
continue;
}
max_heap.emplace(min(curr, dist[nr][nc]), pair(nr, nc));
dist[nr][nc] = -1;
}
}
return -1;
};
return dijkstra(pair(0, 0), pair(static_cast<int>(size(dist) - 1), static_cast<int>(size(dist[0]) - 1)));
}
};
// Time: O(n^2 * logn)
// Space: O(n^2)
// bfs, binary search
class Solution3 {
public:
int maximumSafenessFactor(vector<vector<int>>& grid) {
static const vector<pair<int, int>> DIRECTIONS = {{1, 0}, {0, 1}, {-1, 0}, {0, -1}};
const auto& bfs = [&]() {
vector<vector<int>> dist(size(grid), vector<int>(size(grid[0]), -1));
vector<pair<int, int>> q;
for (int r = 0; r < size(grid); ++r) {
for (int c = 0; c < size(grid[0]); ++c) {
if (grid[r][c]) {
dist[r][c] = 0;
q.emplace_back(r, c);
}
}
}
int d = 0;
for (int d = 0; !empty(q); ++d) {
vector<pair<int, int>> new_q;
for (const auto& [r, c] : q) {
for (const auto& [dr, dc] : DIRECTIONS) {
const int nr = r + dr, nc = c + dc;
if (!(0 <= nr && nr < size(dist) && 0 <= nc && nc < size(dist[0]) && dist[nr][nc] == -1)) {
continue;
}
dist[nr][nc] = d + 1;
new_q.emplace_back(nr, nc);
}
}
q = move(new_q);
}
return dist;
};
const auto& dist = bfs();
const auto& check = [&](int x) {
vector<vector<int>> lookup(size(dist), vector<int>(size(dist[0])));
vector<pair<int, int>> q = {{0, 0}};
lookup[0][0] = 1;
while (!empty(q)) {
vector<pair<int, int>> new_q;
for (const auto& [r, c] : q) {
for (const auto& [dr, dc] : DIRECTIONS) {
const int nr = r + dr, nc = c + dc;
if (!(0 <= nr && nr < size(dist) && 0 <= nc && nc < size(dist[0]) && dist[nr][nc] >= x && !lookup[nr][nc])) {
continue;
}
lookup[nr][nc] = 1;
new_q.emplace_back(nr, nc);
}
}
q = move(new_q);
}
return lookup.back().back();
};
int left = 0, right = dist[0][0];
while (left <= right) {
const auto& mid = left + (right - left) / 2;
if (!check(mid)) {
right = mid - 1;
} else {
left = mid + 1;
}
}
return right;
}
};
Beginner Explanation
What is Find the Safest Path in a Grid?
Find the Safest Path in a Grid (LeetCode #2812) is a Medium problem that primarily trains breadth first search.
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 queue bfs, bucket sort, union find, dijkstras algorithm, and binary search.
- 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: BFS, Bucket Sort, Union Find, Dijkstra's Algorithm.
AlgoForge explanations are original teaching notes. Always open the official problem statement on LeetCode for constraints and examples.
Interview Walkthrough
Interview approach for Find the Safest Path in a Grid
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 queue bfs, bucket sort, union find, dijkstras algorithm, and binary search.
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)) and space (O(n^2)) 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) time and O(n^2) space.
Pattern focus: queue bfs, bucket sort, union find, dijkstras algorithm, and binary search
Use the pattern as a checklist:
- queue bfs — confirm the invariant holds after each step
- bucket sort — confirm the invariant holds after each step
- union find — confirm the invariant holds after each step
- dijkstras algorithm — confirm the invariant holds after each step
- binary search — 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) |
| Space | O(n^2) |
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 Find the Safest Path in a Grid
- Skipping edge cases — empty collections, single-element inputs, max constraints.
- Wrong invariant for queue bfs, bucket sort, union find, dijkstras algorithm, and binary search — 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 queue bfs, bucket sort, union find, dijkstras algorithm, and binary search:
- 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: breadth first search.
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 Find the Safest Path in a Grid in a second language (cpp, python).
- Drill 3–5 more problems tagged breadth first search.
- Teach the solution out loud in under 5 minutes.
- Add this problem to your revision calendar in 3 days and 14 days.
Visualization
Conceptual diagram for Find the Safest Path in a Grid: show input structure (breadth first search), highlight the moving parts of the queue bfs, bucket sort, union find, dijkstras algorithm, and binary search approach, and annotate each step with the maintained invariant and complexity.
Study checklist
- Read the official problem statement on LeetCode
- Solve on paper / whiteboard first
- Implement the queue bfs, bucket sort, union find, dijkstras algorithm, and binary search 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
Find the Safest Path in a Grid (#2812) — Medium. Pattern: queue bfs, bucket sort, union find, dijkstras algorithm, and binary search. Complexity: O(n^2) time / O(n^2) space. Re-derive the invariant before coding.
FAQs
What is the time complexity of Find the Safest Path in a Grid?+
The reference solutions aim for O(n^2) time and O(n^2) space. Always re-derive complexity from the code you write in the interview.
What pattern does Find the Safest Path in a Grid use?+
It primarily maps to queue bfs, bucket sort, union find, dijkstras algorithm, and binary search, within the broader topic of breadth first search.
Is Find the Safest Path in a Grid 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/find-the-safest-path-in-a-grid/