leetcode-problemset/leetcode-cn/originData/number-of-sets-of-k-non-overlapping-line-segments.json

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            "categoryTitle": "Algorithms",
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            "title": "Number of Sets of K Non-Overlapping Line Segments",
            "titleSlug": "number-of-sets-of-k-non-overlapping-line-segments",
            "content": "<p>Given <code>n</code> points on a 1-D plane, where the <code>i<sup>th</sup></code> point (from <code>0</code> to <code>n-1</code>) is at <code>x = i</code>, find the number of ways we can draw <strong>exactly</strong> <code>k</code> <strong>non-overlapping</strong> line segments such that each segment covers two or more points. The endpoints of each segment must have <strong>integral coordinates</strong>. The <code>k</code> line segments <strong>do not</strong> have to cover all <code>n</code> points, and they are <strong>allowed</strong> to share endpoints.</p>\n\n<p>Return <em>the number of ways we can draw </em><code>k</code><em> non-overlapping line segments</em><em>.</em> Since this number can be huge, return it <strong>modulo</strong> <code>10<sup>9</sup> + 7</code>.</p>\n\n<p>&nbsp;</p>\n<p><strong class=\"example\">Example 1:</strong></p>\n<img alt=\"\" src=\"https://assets.leetcode.com/uploads/2020/09/07/ex1.png\" style=\"width: 179px; height: 222px;\" />\n<pre>\n<strong>Input:</strong> n = 4, k = 2\n<strong>Output:</strong> 5\n<strong>Explanation:</strong> The two line segments are shown in red and blue.\nThe image above shows the 5 different ways {(0,2),(2,3)}, {(0,1),(1,3)}, {(0,1),(2,3)}, {(1,2),(2,3)}, {(0,1),(1,2)}.\n</pre>\n\n<p><strong class=\"example\">Example 2:</strong></p>\n\n<pre>\n<strong>Input:</strong> n = 3, k = 1\n<strong>Output:</strong> 3\n<strong>Explanation:</strong> The 3 ways are {(0,1)}, {(0,2)}, {(1,2)}.\n</pre>\n\n<p><strong class=\"example\">Example 3:</strong></p>\n\n<pre>\n<strong>Input:</strong> n = 30, k = 7\n<strong>Output:</strong> 796297179\n<strong>Explanation:</strong> The total number of possible ways to draw 7 line segments is 3796297200. Taking this number modulo 10<sup>9</sup> + 7 gives us 796297179.\n</pre>\n\n<p>&nbsp;</p>\n<p><strong>Constraints:</strong></p>\n\n<ul>\n\t<li><code>2 &lt;= n &lt;= 1000</code></li>\n\t<li><code>1 &lt;= k &lt;= n-1</code></li>\n</ul>\n",
            "translatedTitle": "大小为 K 的不重叠线段的数目",
            "translatedContent": "<p>给你一维空间的 <code>n</code> 个点，其中第 <code>i</code> 个点（编号从 <code>0</code> 到 <code>n-1</code>）位于 <code>x = i</code> 处，请你找到 <strong>恰好</strong> <code>k</code> <strong>个不重叠</strong> 线段且每个线段至少覆盖两个点的方案数。线段的两个端点必须都是 <strong>整数坐标</strong> 。这 <code>k</code> 个线段不需要全部覆盖全部 <code>n</code> 个点，且它们的端点 <strong>可以 </strong>重合。</p>\n\n<p>请你返回 <code>k</code> 个不重叠线段的方案数。由于答案可能很大，请将结果对 <code>10<sup>9</sup> + 7</code> <strong>取余</strong> 后返回。</p>\n\n<p> </p>\n\n<p><strong>示例 1：</strong></p>\n<img alt=\"\" src=\"https://assets.leetcode-cn.com/aliyun-lc-upload/uploads/2020/10/17/ex1.png\" style=\"width: 179px; height: 222px;\" />\n<pre>\n<b>输入：</b>n = 4, k = 2\n<b>输出：</b>5\n<strong>解释：\n</strong>如图所示，两个线段分别用红色和蓝色标出。\n上图展示了 5 种不同的方案 {(0,2),(2,3)}，{(0,1),(1,3)}，{(0,1),(2,3)}，{(1,2),(2,3)}，{(0,1),(1,2)} 。</pre>\n\n<p><strong>示例 2：</strong></p>\n\n<pre>\n<b>输入：</b>n = 3, k = 1\n<b>输出：</b>3\n<strong>解释：</strong>总共有 3 种不同的方案 {(0,1)}, {(0,2)}, {(1,2)} 。\n</pre>\n\n<p><strong>示例 3：</strong></p>\n\n<pre>\n<b>输入：</b>n = 30, k = 7\n<b>输出：</b>796297179\n<strong>解释：</strong>画 7 条线段的总方案数为 3796297200 种。将这个数对 10<sup>9</sup> + 7 取余得到 796297179 。\n</pre>\n\n<p><strong>示例 4：</strong></p>\n\n<pre>\n<b>输入：</b>n = 5, k = 3\n<b>输出：</b>7\n</pre>\n\n<p><strong>示例 5：</strong></p>\n\n<pre>\n<b>输入：</b>n = 3, k = 2\n<b>输出：</b>1</pre>\n\n<p> </p>\n\n<p><strong>提示：</strong></p>\n\n<ul>\n\t<li><code>2 <= n <= 1000</code></li>\n\t<li><code>1 <= k <= n-1</code></li>\n</ul>\n",
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                "Try to use dynamic programming where the current index and remaining number of line segments to form can describe any intermediate state.",
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