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1631. Path With Minimum Effort 👍

  • Time: $O(mn\log mn)$
  • Space: $O(mn)$
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struct T {
  int i;
  int j;
  int d;  // the maximum difference of (i, j) and its neighbors
};

class Solution {
 public:
  int minimumEffortPath(vector<vector<int>>& heights) {
    constexpr int dirs[4][2] = {{0, 1}, {1, 0}, {0, -1}, {-1, 0}};
    const int m = heights.size();
    const int n = heights[0].size();
    auto compare = [](const T& a, const T& b) { return a.d > b.d; };
    priority_queue<T, vector<T>, decltype(compare)> minHeap(compare);
    // diff[i][j] := the maximum absolute difference to reach (i, j)
    vector<vector<int>> diff(m, vector<int>(n, INT_MAX));
    vector<vector<bool>> seen(m, vector<bool>(n));

    minHeap.emplace(0, 0, 0);
    diff[0][0] = 0;

    while (!minHeap.empty()) {
      const auto [i, j, d] = minHeap.top();
      minHeap.pop();
      if (i == m - 1 && j == n - 1)
        return d;
      seen[i][j] = true;
      for (const auto& [dx, dy] : dirs) {
        const int x = i + dx;
        const int y = j + dy;
        if (x < 0 || x == m || y < 0 || y == n)
          continue;
        if (seen[x][y])
          continue;
        const int newDiff = abs(heights[i][j] - heights[x][y]);
        const int maxDiff = max(diff[i][j], newDiff);
        if (diff[x][y] > maxDiff) {
          diff[x][y] = maxDiff;
          minHeap.emplace(x, y, maxDiff);
        }
      }
    }

    throw;
  }
};

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class Solution {
  public int minimumEffortPath(int[][] heights) {
    // d := the maximum difference of (i, j) and its neighbors
    record T(int i, int j, int d) {}
    final int[][] dirs = {{0, 1}, {1, 0}, {0, -1}, {-1, 0}};
    final int m = heights.length;
    final int n = heights[0].length;
    Queue<T> minHeap = new PriorityQueue<>((a, b) -> Integer.compare(a.d, b.d));
    // dist[i][j] := the maximum absolute difference to reach (i, j)
    int[][] diff = new int[m][n];
    Arrays.stream(diff).forEach(A -> Arrays.fill(A, Integer.MAX_VALUE));
    boolean[][] seen = new boolean[m][n];

    minHeap.offer(new T(0, 0, 0));
    diff[0][0] = 0;

    while (!minHeap.isEmpty()) {
      final int i = minHeap.peek().i;
      final int j = minHeap.peek().j;
      final int d = minHeap.poll().d;
      if (i == m - 1 && j == n - 1)
        return d;
      seen[i][j] = true;
      for (int[] dir : dirs) {
        final int x = i + dir[0];
        final int y = j + dir[1];
        if (x < 0 || x == m || y < 0 || y == n)
          continue;
        if (seen[x][y])
          continue;
        final int newDiff = Math.abs(heights[i][j] - heights[x][y]);
        final int maxDiff = Math.max(diff[i][j], newDiff);
        if (diff[x][y] > maxDiff) {
          diff[x][y] = maxDiff;
          minHeap.offer(new T(x, y, maxDiff));
        }
      }
    }

    throw new IllegalArgumentException();
  }
}

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class Solution:
  def minimumEffortPath(self, heights: list[list[int]]) -> int:
    dirs = ((0, 1), (1, 0), (0, -1), (-1, 0))
    m = len(heights)
    n = len(heights[0])
    # diff[i][j] := the maximum absolute difference to reach (i, j)
    diff = [[math.inf] * n for _ in range(m)]
    seen = set()

    minHeap = [(0, 0, 0)]  # (d, i, j)
    diff[0][0] = 0

    while minHeap:
      d, i, j = heapq.heappop(minHeap)
      if i == m - 1 and j == n - 1:
        return d
      seen.add((i, j))
      for dx, dy in dirs:
        x = i + dx
        y = j + dy
        if x < 0 or x == m or y < 0 or y == n:
          continue
        if (x, y) in seen:
          continue
        newDiff = abs(heights[i][j] - heights[x][y])
        maxDiff = max(diff[i][j], newDiff)
        if diff[x][y] > maxDiff:
          diff[x][y] = maxDiff
          heapq.heappush(minHeap, (diff[x][y], x, y))