leetcode-problemset/leetcode-cn/originData/find-the-safest-path-in-a-grid.json

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        "question": {
            "questionId": "2914",
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            "categoryTitle": "Algorithms",
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            "title": "Find the Safest Path in a Grid",
            "titleSlug": "find-the-safest-path-in-a-grid",
            "content": "<p>You are given a <strong>0-indexed</strong> 2D matrix <code>grid</code> of size <code>n x n</code>, where <code>(r, c)</code> represents:</p>\n\n<ul>\n\t<li>A cell containing a thief if <code>grid[r][c] = 1</code></li>\n\t<li>An empty cell if <code>grid[r][c] = 0</code></li>\n</ul>\n\n<p>You are initially positioned at cell <code>(0, 0)</code>. In one move, you can move to any adjacent cell in the grid, including cells containing thieves.</p>\n\n<p>The <strong>safeness factor</strong> of a path on the grid is defined as the <strong>minimum</strong> manhattan distance from any cell in the path to any thief in the grid.</p>\n\n<p>Return <em>the <strong>maximum safeness factor</strong> of all paths leading to cell </em><code>(n - 1, n - 1)</code><em>.</em></p>\n\n<p>An <strong>adjacent</strong> cell of cell <code>(r, c)</code>, is one of the cells <code>(r, c + 1)</code>, <code>(r, c - 1)</code>, <code>(r + 1, c)</code> and <code>(r - 1, c)</code> if it exists.</p>\n\n<p>The <strong>Manhattan distance</strong> between two cells <code>(a, b)</code> and <code>(x, y)</code> is equal to <code>|a - x| + |b - y|</code>, where <code>|val|</code> denotes the absolute value of val.</p>\n\n<p>&nbsp;</p>\n<p><strong class=\"example\">Example 1:</strong></p>\n<img alt=\"\" src=\"https://assets.leetcode.com/uploads/2023/07/02/example1.png\" style=\"width: 362px; height: 242px;\" />\n<pre>\n<strong>Input:</strong> grid = [[1,0,0],[0,0,0],[0,0,1]]\n<strong>Output:</strong> 0\n<strong>Explanation:</strong> All paths from (0, 0) to (n - 1, n - 1) go through the thieves in cells (0, 0) and (n - 1, n - 1).\n</pre>\n\n<p><strong class=\"example\">Example 2:</strong></p>\n<img alt=\"\" src=\"https://assets.leetcode.com/uploads/2023/07/02/example2.png\" style=\"width: 362px; height: 242px;\" />\n<pre>\n<strong>Input:</strong> grid = [[0,0,1],[0,0,0],[0,0,0]]\n<strong>Output:</strong> 2\n<strong>Explanation:</strong> The path depicted in the picture above has a safeness factor of 2 since:\n- The closest cell of the path to the thief at cell (0, 2) is cell (0, 0). The distance between them is | 0 - 0 | + | 0 - 2 | = 2.\nIt can be shown that there are no other paths with a higher safeness factor.\n</pre>\n\n<p><strong class=\"example\">Example 3:</strong></p>\n<img alt=\"\" src=\"https://assets.leetcode.com/uploads/2023/07/02/example3.png\" style=\"width: 362px; height: 242px;\" />\n<pre>\n<strong>Input:</strong> grid = [[0,0,0,1],[0,0,0,0],[0,0,0,0],[1,0,0,0]]\n<strong>Output:</strong> 2\n<strong>Explanation:</strong> The path depicted in the picture above has a safeness factor of 2 since:\n- The closest cell of the path to the thief at cell (0, 3) is cell (1, 2). The distance between them is | 0 - 1 | + | 3 - 2 | = 2.\n- The closest cell of the path to the thief at cell (3, 0) is cell (3, 2). The distance between them is | 3 - 3 | + | 0 - 2 | = 2.\nIt can be shown that there are no other paths with a higher safeness factor.\n</pre>\n\n<p>&nbsp;</p>\n<p><strong>Constraints:</strong></p>\n\n<ul>\n\t<li><code>1 &lt;= grid.length == n &lt;= 400</code></li>\n\t<li><code>grid[i].length == n</code></li>\n\t<li><code>grid[i][j]</code> is either <code>0</code> or <code>1</code>.</li>\n\t<li>There is at least one thief in the <code>grid</code>.</li>\n</ul>\n",
            "translatedTitle": "找出最安全路径",
            "translatedContent": "<p>给你一个下标从 <strong>0</strong> 开始、大小为 <code>n x n</code> 的二维矩阵 <code>grid</code> ，其中 <code>(r, c)</code> 表示：</p>\n\n<ul>\n\t<li>如果 <code>grid[r][c] = 1</code> ，则表示一个存在小偷的单元格</li>\n\t<li>如果 <code>grid[r][c] = 0</code> ，则表示一个空单元格</li>\n</ul>\n\n<p>你最开始位于单元格 <code>(0, 0)</code> 。在一步移动中，你可以移动到矩阵中的任一相邻单元格，包括存在小偷的单元格。</p>\n\n<p>矩阵中路径的 <strong>安全系数</strong> 定义为：从路径中任一单元格到矩阵中任一小偷所在单元格的 <strong>最小</strong> 曼哈顿距离。</p>\n\n<p>返回所有通向单元格<em> </em><code>(n - 1, n - 1)</code> 的路径中的 <strong>最大安全系数</strong> 。</p>\n\n<p>单元格 <code>(r, c)</code> 的某个 <strong>相邻</strong> 单元格，是指在矩阵中存在的 <code>(r, c + 1)</code>、<code>(r, c - 1)</code>、<code>(r + 1, c)</code> 和 <code>(r - 1, c)</code> 之一。</p>\n\n<p>两个单元格 <code>(a, b)</code> 和 <code>(x, y)</code> 之间的 <strong>曼哈顿距离</strong> 等于 <code>| a - x | + | b - y |</code> ，其中 <code>|val|</code> 表示 <code>val</code> 的绝对值。</p>\n\n<p>&nbsp;</p>\n\n<p><strong>示例 1：</strong></p>\n<img alt=\"\" src=\"https://assets.leetcode.com/uploads/2023/07/02/example1.png\" style=\"width: 362px; height: 242px;\" />\n<pre>\n<strong>输入：</strong>grid = [[1,0,0],[0,0,0],[0,0,1]]\n<strong>输出：</strong>0\n<strong>解释：</strong>从 (0, 0) 到 (n - 1, n - 1) 的每条路径都经过存在小偷的单元格 (0, 0) 和 (n - 1, n - 1) 。\n</pre>\n\n<p><strong>示例 2：</strong></p>\n<img alt=\"\" src=\"https://assets.leetcode.com/uploads/2023/07/02/example2.png\" style=\"width: 362px; height: 242px;\" />\n<pre>\n<strong>输入：</strong>grid = [[0,0,1],[0,0,0],[0,0,0]]\n<strong>输出：</strong>2\n<strong>解释：</strong>\n上图所示路径的安全系数为 2：\n- 该路径上距离小偷所在单元格（0，2）最近的单元格是（0，0）。它们之间的曼哈顿距离为 | 0 - 0 | + | 0 - 2 | = 2 。\n可以证明，不存在安全系数更高的其他路径。\n</pre>\n\n<p><strong>示例 3：</strong></p>\n<img alt=\"\" src=\"https://assets.leetcode.com/uploads/2023/07/02/example3.png\" style=\"width: 362px; height: 242px;\" />\n<pre>\n<strong>输入：</strong>grid = [[0,0,0,1],[0,0,0,0],[0,0,0,0],[1,0,0,0]]\n<strong>输出：</strong>2\n<strong>解释：</strong>\n上图所示路径的安全系数为 2：\n- 该路径上距离小偷所在单元格（0，3）最近的单元格是（1，2）。它们之间的曼哈顿距离为 | 0 - 1 | + | 3 - 2 | = 2 。\n- 该路径上距离小偷所在单元格（3，0）最近的单元格是（3，2）。它们之间的曼哈顿距离为 | 3 - 3 | + | 0 - 2 | = 2 。\n可以证明，不存在安全系数更高的其他路径。</pre>\n\n<p>&nbsp;</p>\n\n<p><strong>提示：</strong></p>\n\n<ul>\n\t<li><code>1 &lt;= grid.length == n &lt;= 400</code></li>\n\t<li><code>grid[i].length == n</code></li>\n\t<li><code>grid[i][j]</code> 为 <code>0</code> 或 <code>1</code></li>\n\t<li><code>grid</code> 至少存在一个小偷</li>\n</ul>\n",
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                "Consider using both BFS and binary search together.",
                "Launch a BFS starting from all the cells containing thieves to calculate d[x][y] which is the smallest Manhattan distance from (x, y) to the nearest grid that contains thieves.",
                "To check if the bottom-right cell of the grid can be reached **through a path of safeness factor v**, eliminate all cells (x, y) such that grid[x][y]  < v. if (0, 0) and (n - 1, n - 1) are still connected, there exists a path between (0, 0) and (n - 1, n - 1) of safeness factor v.",
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