mirror of
https://gitee.com/coder-xiaomo/leetcode-problemset
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174 lines
16 KiB
JSON
174 lines
16 KiB
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> </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> </p>\n<p><strong>Constraints:</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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"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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