# https://leetcode.com/problems/find-minimum-time-to-reach-last-room-i

import heapq
from typing import List, Tuple

class Solution:
    def minTimeToReach(self, moveTime: List[List[int]]) -> int:
        n, m = len(moveTime), len(moveTime[0])
        dirs: List[Tuple[int, int]] = [(-1, 0), (1, 0), (0, -1), (0, 1)]
        min_time = [[float('inf')] * m for _ in range(n)]
        min_time[0][0] = 0
        heap: List[Tuple[int, int, int]] = [(0, 0, 0)]

        while heap:
            curr_time, y, x = heapq.heappop(heap)
            if (y, x) == (n - 1, m - 1):
                return curr_time
            if curr_time > min_time[y][x]:
                continue
            for dy, dx in dirs:
                ny, nx = y + dy, x + dx
                if 0 <= ny < n and 0 <= nx < m:
                    start_time = max(curr_time, moveTime[ny][nx])
                    arrival_time = start_time + 1
                    if arrival_time < min_time[ny][nx]:
                        min_time[ny][nx] = arrival_time
                        heapq.heappush(heap, (arrival_time, ny, nx))
        return -1