mirror of
https://gitee.com/coder-xiaomo/leetcode-problemset
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196 lines
19 KiB
JSON
196 lines
19 KiB
JSON
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"questionId": "3112",
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"title": "Count Valid Paths in a Tree",
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"content": "<p>There is an undirected tree with <code>n</code> nodes labeled from <code>1</code> to <code>n</code>. You are given the integer <code>n</code> and a 2D integer array <code>edges</code> of length <code>n - 1</code>, where <code>edges[i] = [u<sub>i</sub>, v<sub>i</sub>]</code> indicates that there is an edge between nodes <code>u<sub>i</sub></code> and <code>v<sub>i</sub></code> in the tree.</p>\n\n<p>Return <em>the <strong>number of valid paths</strong> in the tree</em>.</p>\n\n<p>A path <code>(a, b)</code> is <strong>valid</strong> if there exists <strong>exactly one</strong> prime number among the node labels in the path from <code>a</code> to <code>b</code>.</p>\n\n<p><strong>Note</strong> that:</p>\n\n<ul>\n\t<li>The path <code>(a, b)</code> is a sequence of <strong>distinct</strong> nodes starting with node <code>a</code> and ending with node <code>b</code> such that every two adjacent nodes in the sequence share an edge in the tree.</li>\n\t<li>Path <code>(a, b)</code> and path <code>(b, a)</code> are considered the <strong>same</strong> and counted only <strong>once</strong>.</li>\n</ul>\n\n<p> </p>\n<p><strong class=\"example\">Example 1:</strong></p>\n<img alt=\"\" src=\"https://assets.leetcode.com/uploads/2023/08/27/example1.png\" style=\"width: 440px; height: 357px;\" />\n<pre>\n<strong>Input:</strong> n = 5, edges = [[1,2],[1,3],[2,4],[2,5]]\n<strong>Output:</strong> 4\n<strong>Explanation:</strong> The pairs with exactly one prime number on the path between them are: \n- (1, 2) since the path from 1 to 2 contains prime number 2. \n- (1, 3) since the path from 1 to 3 contains prime number 3.\n- (1, 4) since the path from 1 to 4 contains prime number 2.\n- (2, 4) since the path from 2 to 4 contains prime number 2.\nIt can be shown that there are only 4 valid paths.\n</pre>\n\n<p><strong class=\"example\">Example 2:</strong></p>\n<img alt=\"\" src=\"https://assets.leetcode.com/uploads/2023/08/27/example2.png\" style=\"width: 488px; height: 384px;\" />\n<pre>\n<strong>Input:</strong> n = 6, edges = [[1,2],[1,3],[2,4],[3,5],[3,6]]\n<strong>Output:</strong> 6\n<strong>Explanation:</strong> The pairs with exactly one prime number on the path between them are: \n- (1, 2) since the path from 1 to 2 contains prime number 2.\n- (1, 3) since the path from 1 to 3 contains prime number 3.\n- (1, 4) since the path from 1 to 4 contains prime number 2.\n- (1, 6) since the path from 1 to 6 contains prime number 3.\n- (2, 4) since the path from 2 to 4 contains prime number 2.\n- (3, 6) since the path from 3 to 6 contains prime number 3.\nIt can be shown that there are only 6 valid paths.\n</pre>\n\n<p> </p>\n<p><strong>Constraints:</strong></p>\n\n<ul>\n\t<li><code>1 <= n <= 10<sup>5</sup></code></li>\n\t<li><code>edges.length == n - 1</code></li>\n\t<li><code>edges[i].length == 2</code></li>\n\t<li><code>1 <= u<sub>i</sub>, v<sub>i</sub> <= n</code></li>\n\t<li>The input is generated such that <code>edges</code> represent a valid tree.</li>\n</ul>\n",
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"Use the sieve of Eratosthenes to find all prime numbers in the range <code>[1, n]</code>.****",
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"Root the tree at any node.",
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"Let <code>dp[i][0] = the number of vertical paths starting from i containing no prime nodes </code>, and <code>dp[i][1] = the number of vertical paths starting from i containing one prime node </code>.",
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"If <code>i</code> is not prime, <code>dp[i][0] = sum(dp[child][0]) + 1</code>, and <code>dp[i][1] = sum(dp[child][1])</code> for each <code>child</code> of <code>i</code> in the rooted tree.",
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"If <code>i</code> is prime, <code>dp[i][0] = 0</code>, and <code>dp[i][1] = sum(dp[child][0]) + 1</code> for each <code>child</code> of <code>i</code> in the rooted tree.",
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"For each node <code>i</code>, and using the computed <code>dp</code> matrix, count the number of unordered pairs <code>(a,b)</code> such that <code>lca(a,b) = i</code>, and there exists exactly one prime number on the path from <code>a</code> to <code>b</code>."
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