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

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            "title": "Number of Sets of K Non-Overlapping Line Segments",
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            "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>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>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>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",
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            "hints": [
                "Try to use dynamic programming where the current index and remaining number of line segments to form can describe any intermediate state.",
                "To make the computation of each state in constant time, we could add another flag to the state that indicates whether or not we are in the middle of placing a line (placed start point but no endpoint)."
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