leetcode-problemset/leetcode-cn/originData/minimum-cost-to-make-at-least-one-valid-path-in-a-grid.json

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            "questionId": "1485",
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            "categoryTitle": "Algorithms",
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            "title": "Minimum Cost to Make at Least One Valid Path in a Grid",
            "titleSlug": "minimum-cost-to-make-at-least-one-valid-path-in-a-grid",
            "content": "<p>Given an <code>m x n</code> grid. Each cell of the grid has a sign pointing to the next cell you should visit if you are currently in this cell. The sign of <code>grid[i][j]</code> can be:</p>\n\n<ul>\n\t<li><code>1</code> which means go to the cell to the right. (i.e go from <code>grid[i][j]</code> to <code>grid[i][j + 1]</code>)</li>\n\t<li><code>2</code> which means go to the cell to the left. (i.e go from <code>grid[i][j]</code> to <code>grid[i][j - 1]</code>)</li>\n\t<li><code>3</code> which means go to the lower cell. (i.e go from <code>grid[i][j]</code> to <code>grid[i + 1][j]</code>)</li>\n\t<li><code>4</code> which means go to the upper cell. (i.e go from <code>grid[i][j]</code> to <code>grid[i - 1][j]</code>)</li>\n</ul>\n\n<p>Notice that there could be some signs on the cells of the grid that point outside the grid.</p>\n\n<p>You will initially start at the upper left cell <code>(0, 0)</code>. A valid path in the grid is a path that starts from the upper left cell <code>(0, 0)</code> and ends at the bottom-right cell <code>(m - 1, n - 1)</code> following the signs on the grid. The valid path does not have to be the shortest.</p>\n\n<p>You can modify the sign on a cell with <code>cost = 1</code>. You can modify the sign on a cell <strong>one time only</strong>.</p>\n\n<p>Return <em>the minimum cost to make the grid have at least one valid path</em>.</p>\n\n<p>&nbsp;</p>\n<p><strong class=\"example\">Example 1:</strong></p>\n<img alt=\"\" src=\"https://assets.leetcode.com/uploads/2020/02/13/grid1.png\" style=\"width: 400px; height: 390px;\" />\n<pre>\n<strong>Input:</strong> grid = [[1,1,1,1],[2,2,2,2],[1,1,1,1],[2,2,2,2]]\n<strong>Output:</strong> 3\n<strong>Explanation:</strong> You will start at point (0, 0).\nThe path to (3, 3) is as follows. (0, 0) --&gt; (0, 1) --&gt; (0, 2) --&gt; (0, 3) change the arrow to down with cost = 1 --&gt; (1, 3) --&gt; (1, 2) --&gt; (1, 1) --&gt; (1, 0) change the arrow to down with cost = 1 --&gt; (2, 0) --&gt; (2, 1) --&gt; (2, 2) --&gt; (2, 3) change the arrow to down with cost = 1 --&gt; (3, 3)\nThe total cost = 3.\n</pre>\n\n<p><strong class=\"example\">Example 2:</strong></p>\n<img alt=\"\" src=\"https://assets.leetcode.com/uploads/2020/02/13/grid2.png\" style=\"width: 350px; height: 341px;\" />\n<pre>\n<strong>Input:</strong> grid = [[1,1,3],[3,2,2],[1,1,4]]\n<strong>Output:</strong> 0\n<strong>Explanation:</strong> You can follow the path from (0, 0) to (2, 2).\n</pre>\n\n<p><strong class=\"example\">Example 3:</strong></p>\n<img alt=\"\" src=\"https://assets.leetcode.com/uploads/2020/02/13/grid3.png\" style=\"width: 200px; height: 192px;\" />\n<pre>\n<strong>Input:</strong> grid = [[1,2],[4,3]]\n<strong>Output:</strong> 1\n</pre>\n\n<p>&nbsp;</p>\n<p><strong>Constraints:</strong></p>\n\n<ul>\n\t<li><code>m == grid.length</code></li>\n\t<li><code>n == grid[i].length</code></li>\n\t<li><code>1 &lt;= m, n &lt;= 100</code></li>\n\t<li><code>1 &lt;= grid[i][j] &lt;= 4</code></li>\n</ul>\n",
            "translatedTitle": "使网格图至少有一条有效路径的最小代价",
            "translatedContent": "<p>给你一个 m x n 的网格图&nbsp;<code>grid</code>&nbsp;。&nbsp;<code>grid</code>&nbsp;中每个格子都有一个数字，对应着从该格子出发下一步走的方向。&nbsp;<code>grid[i][j]</code>&nbsp;中的数字可能为以下几种情况：</p>\n\n<ul>\n\t<li><strong>1</strong>&nbsp;，下一步往右走，也就是你会从 <code>grid[i][j]</code>&nbsp;走到 <code>grid[i][j + 1]</code></li>\n\t<li><strong>2</strong>&nbsp;，下一步往左走，也就是你会从 <code>grid[i][j]</code>&nbsp;走到 <code>grid[i][j - 1]</code></li>\n\t<li><strong>3</strong>&nbsp;，下一步往下走，也就是你会从 <code>grid[i][j]</code>&nbsp;走到 <code>grid[i + 1][j]</code></li>\n\t<li><strong>4</strong>&nbsp;，下一步往上走，也就是你会从 <code>grid[i][j]</code>&nbsp;走到 <code>grid[i - 1][j]</code></li>\n</ul>\n\n<p>注意网格图中可能会有&nbsp;<strong>无效数字</strong>&nbsp;，因为它们可能指向&nbsp;<code>grid</code>&nbsp;以外的区域。</p>\n\n<p>一开始，你会从最左上角的格子&nbsp;<code>(0,0)</code>&nbsp;出发。我们定义一条&nbsp;<strong>有效路径</strong>&nbsp;为从格子&nbsp;<code>(0,0)</code>&nbsp;出发，每一步都顺着数字对应方向走，最终在最右下角的格子&nbsp;<code>(m - 1, n - 1)</code>&nbsp;结束的路径。有效路径&nbsp;<strong>不需要是最短路径</strong>&nbsp;。</p>\n\n<p>你可以花费&nbsp;<code>cost = 1</code>&nbsp;的代价修改一个格子中的数字，但每个格子中的数字&nbsp;<strong>只能修改一次</strong>&nbsp;。</p>\n\n<p>请你返回让网格图至少有一条有效路径的最小代价。</p>\n\n<p>&nbsp;</p>\n\n<p><strong>示例 1：</strong></p>\n\n<p><img alt=\"\" src=\"https://assets.leetcode-cn.com/aliyun-lc-upload/uploads/2020/02/29/grid1.png\" style=\"height: 528px; width: 542px;\"></p>\n\n<pre><strong>输入：</strong>grid = [[1,1,1,1],[2,2,2,2],[1,1,1,1],[2,2,2,2]]\n<strong>输出：</strong>3\n<strong>解释：</strong>你将从点 (0, 0) 出发。\n到达 (3, 3) 的路径为： (0, 0) --&gt; (0, 1) --&gt; (0, 2) --&gt; (0, 3) 花费代价 cost = 1 使方向向下 --&gt; (1, 3) --&gt; (1, 2) --&gt; (1, 1) --&gt; (1, 0) 花费代价 cost = 1 使方向向下 --&gt; (2, 0) --&gt; (2, 1) --&gt; (2, 2) --&gt; (2, 3) 花费代价 cost = 1 使方向向下 --&gt; (3, 3)\n总花费为 cost = 3.\n</pre>\n\n<p><strong>示例 2：</strong></p>\n\n<p><img alt=\"\" src=\"https://assets.leetcode-cn.com/aliyun-lc-upload/uploads/2020/02/29/grid2.png\" style=\"height: 408px; width: 419px;\"></p>\n\n<pre><strong>输入：</strong>grid = [[1,1,3],[3,2,2],[1,1,4]]\n<strong>输出：</strong>0\n<strong>解释：</strong>不修改任何数字你就可以从 (0, 0) 到达 (2, 2) 。\n</pre>\n\n<p><strong>示例 3：</strong></p>\n\n<p><img alt=\"\" src=\"https://assets.leetcode-cn.com/aliyun-lc-upload/uploads/2020/02/29/grid3.png\" style=\"height: 302px; width: 314px;\"></p>\n\n<pre><strong>输入：</strong>grid = [[1,2],[4,3]]\n<strong>输出：</strong>1\n</pre>\n\n<p><strong>示例 4：</strong></p>\n\n<pre><strong>输入：</strong>grid = [[2,2,2],[2,2,2]]\n<strong>输出：</strong>3\n</pre>\n\n<p><strong>示例 5：</strong></p>\n\n<pre><strong>输入：</strong>grid = [[4]]\n<strong>输出：</strong>0\n</pre>\n\n<p>&nbsp;</p>\n\n<p><strong>提示：</strong></p>\n\n<ul>\n\t<li><code>m == grid.length</code></li>\n\t<li><code>n == grid[i].length</code></li>\n\t<li><code>1 &lt;= m, n &lt;= 100</code></li>\n</ul>\n",
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href=\\\"https:\\/\\/github.com\\/emirpasic\\/gods\\/tree\\/v1.18.1\\\" target=\\\"_blank\\\">https:\\/\\/godoc.org\\/github.com\\/emirpasic\\/gods@v1.18.1<\\/a> \\u7b2c\\u4e09\\u65b9\\u5e93\\u3002<\\/p>\"],\"python3\":[\"Python3\",\"<p>\\u7248\\u672c\\uff1a<code>Python 3.10<\\/code><\\/p>\\r\\n\\r\\n<p>\\u4e3a\\u4e86\\u65b9\\u4fbf\\u8d77\\u89c1\\uff0c\\u5927\\u90e8\\u5206\\u5e38\\u7528\\u5e93\\u5df2\\u7ecf\\u88ab\\u81ea\\u52a8 \\u5bfc\\u5165\\uff0c\\u5982<a href=\\\"https:\\/\\/docs.python.org\\/3\\/library\\/array.html\\\" target=\\\"_blank\\\">array<\\/a>, <a href=\\\"https:\\/\\/docs.python.org\\/3\\/library\\/bisect.html\\\" target=\\\"_blank\\\">bisect<\\/a>, <a href=\\\"https:\\/\\/docs.python.org\\/3\\/library\\/collections.html\\\" target=\\\"_blank\\\">collections<\\/a>\\u3002 \\u5982\\u679c\\u60a8\\u9700\\u8981\\u4f7f\\u7528\\u5176\\u4ed6\\u5e93\\u51fd\\u6570\\uff0c\\u8bf7\\u81ea\\u884c\\u5bfc\\u5165\\u3002<\\/p>\\r\\n\\r\\n<p>\\u5982\\u9700\\u4f7f\\u7528 Map\\/TreeMap 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