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find-the-safest-path-in-a-grid.cpp
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find-the-safest-path-in-a-grid.cpp
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// 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;
}
};