leetcode-problemset/leetcode-cn/originData/minimum-rectangles-to-cover-points.json

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        "question": {
            "questionId": "3390",
            "questionFrontendId": "3111",
            "categoryTitle": "Algorithms",
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            "title": "Minimum Rectangles to Cover Points",
            "titleSlug": "minimum-rectangles-to-cover-points",
            "content": "<p>You are given a 2D integer array <code>points</code>, where <code>points[i] = [x<sub>i</sub>, y<sub>i</sub>]</code>. You are also given an integer <code>w</code>. Your task is to <strong>cover</strong> <strong>all</strong> the given points with rectangles.</p>\n\n<p>Each rectangle has its lower end at some point <code>(x<sub>1</sub>, 0)</code> and its upper end at some point <code>(x<sub>2</sub>, y<sub>2</sub>)</code>, where <code>x<sub>1</sub> &lt;= x<sub>2</sub></code>, <code>y<sub>2</sub> &gt;= 0</code>, and the condition <code>x<sub>2</sub> - x<sub>1</sub> &lt;= w</code> <strong>must</strong> be satisfied for each rectangle.</p>\n\n<p>A point is considered covered by a rectangle if it lies within or on the boundary of the rectangle.</p>\n\n<p>Return an integer denoting the <strong>minimum</strong> number of rectangles needed so that each point is covered by <strong>at least one</strong> rectangle<em>.</em></p>\n\n<p><strong>Note:</strong> A point may be covered by more than one rectangle.</p>\n\n<p>&nbsp;</p>\n<p><strong class=\"example\">Example 1:</strong></p>\n\n<p><img alt=\"\" src=\"https://assets.leetcode.com/uploads/2024/03/04/screenshot-from-2024-03-04-20-33-05.png\" style=\"width: 205px; height: 300px;\" /></p>\n\n<div class=\"example-block\" style=\"\n    border-color: var(--border-tertiary);\n    border-left-width: 2px;\n    color: var(--text-secondary);\n    font-size: .875rem;\n    margin-bottom: 1rem;\n    margin-top: 1rem;\n    overflow: visible;\n    padding-left: 1rem;\n\">\n<p><strong>Input:</strong> <span class=\"example-io\" style=\"\n    font-family: Menlo,sans-serif;\n    font-size: 0.85rem;\n\">points = [[2,1],[1,0],[1,4],[1,8],[3,5],[4,6]], w = 1</span></p>\n\n<p><strong>Output:</strong> <span class=\"example-io\" style=\"\n    font-family: Menlo,sans-serif;\n    font-size: 0.85rem;\n\">2</span></p>\n\n<p><strong>Explanation: </strong></p>\n\n<p>The image above shows one possible placement of rectangles to cover the points:</p>\n\n<ul>\n\t<li>A rectangle with a lower end at <code>(1, 0)</code> and its upper end at <code>(2, 8)</code></li>\n\t<li>A rectangle with a lower end at <code>(3, 0)</code> and its upper end at <code>(4, 8)</code></li>\n</ul>\n</div>\n\n<p><strong class=\"example\">Example 2:</strong></p>\n\n<p><img alt=\"\" src=\"https://assets.leetcode.com/uploads/2024/03/04/screenshot-from-2024-03-04-18-59-12.png\" style=\"width: 260px; height: 250px;\" /></p>\n\n<div class=\"example-block\" style=\"\n    border-color: var(--border-tertiary);\n    border-left-width: 2px;\n    color: var(--text-secondary);\n    font-size: .875rem;\n    margin-bottom: 1rem;\n    margin-top: 1rem;\n    overflow: visible;\n    padding-left: 1rem;\n\">\n<p><strong>Input:</strong> <span class=\"example-io\" style=\"\n    font-family: Menlo,sans-serif;\n    font-size: 0.85rem;\n\">points = [[0,0],[1,1],[2,2],[3,3],[4,4],[5,5],[6,6]], w = 2</span></p>\n\n<p><strong>Output:</strong> <span class=\"example-io\" style=\"\n    font-family: Menlo,sans-serif;\n    font-size: 0.85rem;\n\">3</span></p>\n\n<p><strong>Explanation: </strong></p>\n\n<p>The image above shows one possible placement of rectangles to cover the points:</p>\n\n<ul>\n\t<li>A rectangle with a lower end at <code>(0, 0)</code> and its upper end at <code>(2, 2)</code></li>\n\t<li>A rectangle with a lower end at <code>(3, 0)</code> and its upper end at <code>(5, 5)</code></li>\n\t<li>A rectangle with a lower end at <code>(6, 0)</code> and its upper end at <code>(6, 6)</code></li>\n</ul>\n</div>\n\n<p><strong class=\"example\">Example 3:</strong></p>\n\n<p><img alt=\"\" src=\"https://assets.leetcode.com/uploads/2024/03/04/screenshot-from-2024-03-04-20-24-03.png\" style=\"height: 150px; width: 127px;\" /></p>\n\n<div class=\"example-block\" style=\"\n    border-color: var(--border-tertiary);\n    border-left-width: 2px;\n    color: var(--text-secondary);\n    font-size: .875rem;\n    margin-bottom: 1rem;\n    margin-top: 1rem;\n    overflow: visible;\n    padding-left: 1rem;\n\">\n<p><strong>Input:</strong> <span class=\"example-io\" style=\"\n    font-family: Menlo,sans-serif;\n    font-size: 0.85rem;\n\">points = [[2,3],[1,2]], w = 0</span></p>\n\n<p><strong>Output:</strong> <span class=\"example-io\" style=\"\n    font-family: Menlo,sans-serif;\n    font-size: 0.85rem;\n\">2</span></p>\n\n<p><strong>Explanation: </strong></p>\n\n<p>The image above shows one possible placement of rectangles to cover the points:</p>\n\n<ul>\n\t<li>A rectangle with a lower end at <code>(1, 0)</code> and its upper end at <code>(1, 2)</code></li>\n\t<li>A rectangle with a lower end at <code>(2, 0)</code> and its upper end at <code>(2, 3)</code></li>\n</ul>\n</div>\n\n<p>&nbsp;</p>\n<p><strong>Constraints:</strong></p>\n\n<ul>\n\t<li><code>1 &lt;= points.length &lt;= 10<sup>5</sup></code></li>\n\t<li><code>points[i].length == 2</code></li>\n\t<li><code>0 &lt;= x<sub>i</sub> == points[i][0] &lt;= 10<sup>9</sup></code></li>\n\t<li><code>0 &lt;= y<sub>i</sub> == points[i][1] &lt;= 10<sup>9</sup></code></li>\n\t<li><code>0 &lt;= w &lt;= 10<sup>9</sup></code></li>\n\t<li>All pairs <code>(x<sub>i</sub>, y<sub>i</sub>)</code> are distinct.</li>\n</ul>\n",
            "translatedTitle": "覆盖所有点的最少矩形数目",
            "translatedContent": "<p>给你一个二维整数数组&nbsp;<code>point</code>&nbsp;，其中&nbsp;<code>points[i] = [x<sub>i</sub>, y<sub>i</sub>]</code>&nbsp;表示二维平面内的一个点。同时给你一个整数&nbsp;<code>w</code>&nbsp;。你需要用矩形&nbsp;<strong>覆盖所有</strong>&nbsp;点。</p>\n\n<p>每个矩形的左下角在某个点&nbsp;<code>(x<sub>1</sub>, 0)</code>&nbsp;处，且右上角在某个点&nbsp;<code>(x<sub>2</sub>, y<sub>2</sub>)</code>&nbsp;处，其中&nbsp;<code>x<sub>1</sub> &lt;= x<sub>2</sub></code>&nbsp;且&nbsp;<code>y<sub>2</sub> &gt;= 0</code>&nbsp;，同时对于每个矩形都&nbsp;<strong>必须</strong>&nbsp;满足&nbsp;<code>x<sub>2</sub> - x<sub>1</sub> &lt;= w</code>&nbsp;。</p>\n\n<p>如果一个点在矩形内或者在边上，我们说这个点被矩形覆盖了。</p>\n\n<p>请你在确保每个点都 <strong>至少</strong>&nbsp;被一个矩形覆盖的前提下，<strong>最少</strong>&nbsp;需要多少个矩形。</p>\n\n<p><strong>注意：</strong>一个点可以被多个矩形覆盖。</p>\n\n<p>&nbsp;</p>\n\n<p><strong class=\"example\">示例 1：</strong></p>\n\n<p><img alt=\"\" src=\"https://assets.leetcode.com/uploads/2024/03/04/screenshot-from-2024-03-04-20-33-05.png\" style=\"width: 205px; height: 300px;\" /></p>\n\n<div class=\"example-block\" style=\"border-color: var(--border-tertiary); border-left-width: 2px; color: var(--text-secondary); margin-bottom: 1rem; margin-top: 1rem; overflow: visible; padding-left: 1rem;\">\n<p style=\"\"><span class=\"example-io\" style=\"font-size: 8.75px;\"><b>输入：</b></span><span class=\"example-io\" style=\"font-size: 0.85rem; font-family: Menlo, sans-serif;\">points = [[2,1],[1,0],[1,4],[1,8],[3,5],[4,6]], w = 1</span></p>\n\n<p style=\"\"><span class=\"example-io\" style=\"font-size: 8.75px;\"><b>输出：</b></span><span class=\"example-io\" style=\"font-size: 0.85rem; font-family: Menlo, sans-serif;\">2</span></p>\n\n<p style=\"font-size: 0.875rem;\"><strong>解释：</strong></p>\n\n<p style=\"font-size: 0.875rem;\">上图展示了一种可行的矩形放置方案：</p>\n\n<ul style=\"font-size: 0.875rem;\">\n\t<li>一个矩形的左下角在&nbsp;<code>(1, 0)</code>&nbsp;，右上角在&nbsp;<code>(2, 8)</code>&nbsp;。</li>\n\t<li>一个矩形的左下角在&nbsp;<code>(3, 0)</code>&nbsp;，右上角在&nbsp;<code>(4, 8)</code>&nbsp;。</li>\n</ul>\n</div>\n\n<p><strong class=\"example\">示例 2：</strong></p>\n\n<p><img alt=\"\" src=\"https://assets.leetcode.com/uploads/2024/03/04/screenshot-from-2024-03-04-18-59-12.png\" style=\"width: 260px; height: 250px;\" /></p>\n\n<div class=\"example-block\" style=\"border-color: var(--border-tertiary); border-left-width: 2px; color: var(--text-secondary); margin-bottom: 1rem; margin-top: 1rem; overflow: visible; padding-left: 1rem;\">\n<p style=\"\"><span class=\"example-io\" style=\"font-size: 8.75px;\"><b>输入：</b></span><span class=\"example-io\" style=\"font-size: 0.85rem; font-family: Menlo, sans-serif;\">points = [[0,0],[1,1],[2,2],[3,3],[4,4],[5,5],[6,6]], w = 2</span></p>\n\n<p style=\"\"><span class=\"example-io\" style=\"font-size: 8.75px;\"><b>输出：</b></span><span class=\"example-io\" style=\"font-size: 0.85rem; font-family: Menlo, sans-serif;\">3</span></p>\n\n<p style=\"font-size: 0.875rem;\"><b>解释：</b></p>\n\n<p style=\"font-size: 0.875rem;\">上图展示了一种可行的矩形放置方案：</p>\n\n<ul style=\"font-size: 0.875rem;\">\n\t<li>一个矩形的左下角在&nbsp;<code>(0, 0)</code>&nbsp;，右上角在&nbsp;<code>(2, 2)</code>&nbsp;。</li>\n\t<li>一个矩形的左下角在&nbsp;<code>(3, 0)</code>&nbsp;，右上角在&nbsp;<code>(5, 5)</code>&nbsp;。</li>\n\t<li>一个矩形的左下角在&nbsp;<code>(6, 0)</code>&nbsp;，右上角在&nbsp;<code>(6, 6)</code>&nbsp;。</li>\n</ul>\n</div>\n\n<p><strong class=\"example\">示例 3：</strong></p>\n\n<p><img alt=\"\" src=\"https://assets.leetcode.com/uploads/2024/03/04/screenshot-from-2024-03-04-20-24-03.png\" style=\"height: 150px; width: 127px;\" /></p>\n\n<div class=\"example-block\" style=\"border-color: var(--border-tertiary); border-left-width: 2px; color: var(--text-secondary); margin-bottom: 1rem; margin-top: 1rem; overflow: visible; padding-left: 1rem;\">\n<p style=\"\"><span class=\"example-io\" style=\"font-size: 8.75px;\"><b>输入：</b></span><span class=\"example-io\" style=\"font-size: 0.85rem; font-family: Menlo, sans-serif;\">points = [[2,3],[1,2]], w = 0</span></p>\n\n<p style=\"\"><span class=\"example-io\" style=\"font-size: 8.75px;\"><b>输出：</b></span><span class=\"example-io\" style=\"font-size: 0.85rem; font-family: Menlo, sans-serif;\">2</span></p>\n\n<p style=\"font-size: 0.875rem;\"><strong>解释：</strong></p>\n\n<p style=\"font-size: 0.875rem;\">上图展示了一种可行的矩形放置方案：</p>\n\n<ul style=\"font-size: 0.875rem;\">\n\t<li>一个矩形的左下角在&nbsp;<code>(1, 0)</code>&nbsp;，右上角在&nbsp;<code>(1, 2)</code>&nbsp;。</li>\n\t<li>一个矩形的左下角在&nbsp;<code>(2, 0)</code>&nbsp;，右上角在&nbsp;<code>(2, 3)</code>&nbsp;。</li>\n</ul>\n</div>\n\n<p>&nbsp;</p>\n\n<p><strong>提示：</strong></p>\n\n<ul>\n\t<li><code>1 &lt;= points.length &lt;= 10<sup>5</sup></code></li>\n\t<li><code>points[i].length == 2</code></li>\n\t<li><code>0 &lt;= x<sub>i</sub> == points[i][0] &lt;= 10<sup>9</sup></code></li>\n\t<li><code>0 &lt;= y<sub>i</sub> == points[i][1] &lt;= 10<sup>9</sup></code></li>\n\t<li><code>0 &lt;= w &lt;= 10<sup>9</sup></code></li>\n\t<li>所有点坐标&nbsp;<code>(x<sub>i</sub>, y<sub>i</sub>)</code>&nbsp;互不相同。</li>\n</ul>\n",
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