778. Swim in Rising Water

778. Swim in Rising Water

Difficulty: Hard

Topics: Array, Binary Search, Depth-First Search, Breadth-First Search, Union Find, Heap (Priority Queue), Matrix, Weekly Contest 70

You are given an n x n integer matrix grid where each value grid[i][j] represents the elevation at that point (i, j).

It starts raining, and water gradually rises over time. At time t, the water level is t, meaning any cell with elevation less than equal to t is submerged or reachable.

You can swim from a square to another 4-directionally adjacent square if and only if the elevation of both squares individually are at most t. You can swim infinite distances in zero time. Of course, you must stay within the boundaries of the grid during your swim.

Return the minimum time until you can reach the bottom right square (n - 1, n - 1) if you start at the top left square (0, 0).

Example 1:

• Input: grid = [[0,2],[1,3]]
• Output: 3
• Explanation:
  • At time 0, you are in grid location (0, 0).
  • You cannot go anywhere else because 4-directionally adjacent neighbors have a higher elevation than t = 0.
  • You cannot reach point (1, 1) until time 3.
  • When the depth of water is 3, we can swim anywhere inside the grid.

Example 2:

• Input: grid = [[0,1,2,3,4],[24,23,22,21,5],[12,13,14,15,16],[11,17,18,19,20],[10,9,8,7,6]]
• Output: 16
• Explanation:
  • The final route is shown.
  • We need to wait until time 16 so that (0, 0) and (4, 4) are connected.

Constraints:

• n == grid.length
• n == grid[i].length
• 1 <= n <= 50
• 0 <= grid[i][j] < n²
• Each value grid[i][j] is unique.

Hint:

1. Use either Dijkstra's, or binary search for the best time T for which you can reach the end if you only step on squares at most T.

Solution:

We need to determine the minimum time required to swim from the top-left corner (0, 0) to the bottom-right corner (n-1, n-1) of an n x n grid, where each cell has a unique elevation. The water level rises over time, and at time t, any cell with an elevation ≤ t is reachable. We can move to adjacent cells (up, down, left, right) only if both the current and target cells have elevations ≤ t.

Approach

1. Binary Search on Time: We perform a binary search on the possible time values (from 0 to the maximum elevation in the grid) to find the minimum time t such that there exists a path from (0, 0) to (n-1, n-1) where all cells in the path have elevations ≤ t.
2. Breadth-First Search (BFS) for Path Checking: For each candidate time t during the binary search, we use BFS to check if a path exists from the start to the end where all cells in the path have elevations ≤ t. We mark visited cells to avoid cycles.

Let's implement this solution in PHP: 778. Swim in Rising Water

<?php
/**
 * @param Integer[][] $grid
 * @return Integer
 */
function swimInWater($grid) {
    ...
    ...
    ...
    /**
     * go to ./solution.php
     */
}

/**
 * @param $grid
 * @param $time
 * @param $n
 * @return bool
 */
function canReach($grid, $time, $n) {
    ...
    ...
    ...
    /**
     * go to ./solution.php
     */
}

// Test cases
$grid1 = [[0,2],[1,3]];
echo swimInWater($grid1) . PHP_EOL; // 3

$grid2 = [
    [0,1,2,3,4],
    [24,23,22,21,5],
    [12,13,14,15,16],
    [11,17,18,19,20],
    [10,9,8,7,6]
];
echo swimInWater($grid2) . PHP_EOL; // 16
?>

Explanation:

1. Binary Search Setup: We initialize left to 0 and right to the maximum possible elevation (n² - 1). This range covers all possible time values from the start to the maximum elevation in the grid.
2. Path Existence Check: For each midpoint mid in the binary search, we check if a path exists from (0, 0) to (n-1, n-1) using BFS, considering only cells with elevations ≤ mid.
3. BFS Implementation: The BFS starts from (0, 0) and explores all reachable cells. If the destination (n-1, n-1) is reached, the candidate time mid is feasible, and we adjust the binary search range accordingly.
4. Result: The binary search converges to the minimum time t such that there exists a valid path from the start to the end under the water level t.

This approach efficiently narrows down the possible time values using binary search and checks path feasibility using BFS, ensuring optimal performance even for the upper constraint limits.

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