mirror of
https://gitee.com/coder-xiaomo/leetcode-problemset
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193 lines
19 KiB
JSON
193 lines
19 KiB
JSON
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"title": "Number of Restricted Paths From First to Last Node",
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"content": "<p>There is an undirected weighted connected graph. You are given a positive integer <code>n</code> which denotes that the graph has <code>n</code> nodes labeled from <code>1</code> to <code>n</code>, and an array <code>edges</code> where each <code>edges[i] = [u<sub>i</sub>, v<sub>i</sub>, weight<sub>i</sub>]</code> denotes that there is an edge between nodes <code>u<sub>i</sub></code> and <code>v<sub>i</sub></code> with weight equal to <code>weight<sub>i</sub></code>.</p>\n\n<p>A path from node <code>start</code> to node <code>end</code> is a sequence of nodes <code>[z<sub>0</sub>, z<sub>1</sub>,<sub> </sub>z<sub>2</sub>, ..., z<sub>k</sub>]</code> such that <code>z<sub>0 </sub>= start</code> and <code>z<sub>k</sub> = end</code> and there is an edge between <code>z<sub>i</sub></code> and <code>z<sub>i+1</sub></code> where <code>0 <= i <= k-1</code>.</p>\n\n<p>The distance of a path is the sum of the weights on the edges of the path. Let <code>distanceToLastNode(x)</code> denote the shortest distance of a path between node <code>n</code> and node <code>x</code>. A <strong>restricted path</strong> is a path that also satisfies that <code>distanceToLastNode(z<sub>i</sub>) > distanceToLastNode(z<sub>i+1</sub>)</code> where <code>0 <= i <= k-1</code>.</p>\n\n<p>Return <em>the number of restricted paths from node</em> <code>1</code> <em>to node</em> <code>n</code>. Since that number may be too large, 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/2021/02/17/restricted_paths_ex1.png\" style=\"width: 351px; height: 341px;\" />\n<pre>\n<strong>Input:</strong> n = 5, edges = [[1,2,3],[1,3,3],[2,3,1],[1,4,2],[5,2,2],[3,5,1],[5,4,10]]\n<strong>Output:</strong> 3\n<strong>Explanation:</strong> Each circle contains the node number in black and its <code>distanceToLastNode value in blue. </code>The three restricted paths are:\n1) 1 --> 2 --> 5\n2) 1 --> 2 --> 3 --> 5\n3) 1 --> 3 --> 5\n</pre>\n\n<p><strong class=\"example\">Example 2:</strong></p>\n<img alt=\"\" src=\"https://assets.leetcode.com/uploads/2021/02/17/restricted_paths_ex22.png\" style=\"width: 356px; height: 401px;\" />\n<pre>\n<strong>Input:</strong> n = 7, edges = [[1,3,1],[4,1,2],[7,3,4],[2,5,3],[5,6,1],[6,7,2],[7,5,3],[2,6,4]]\n<strong>Output:</strong> 1\n<strong>Explanation:</strong> Each circle contains the node number in black and its <code>distanceToLastNode value in blue. </code>The only restricted path is 1 --> 3 --> 7.\n</pre>\n\n<p> </p>\n<p><strong>Constraints:</strong></p>\n\n<ul>\n\t<li><code>1 <= n <= 2 * 10<sup>4</sup></code></li>\n\t<li><code>n - 1 <= edges.length <= 4 * 10<sup>4</sup></code></li>\n\t<li><code>edges[i].length == 3</code></li>\n\t<li><code>1 <= u<sub>i</sub>, v<sub>i</sub> <= n</code></li>\n\t<li><code>u<sub>i </sub>!= v<sub>i</sub></code></li>\n\t<li><code>1 <= weight<sub>i</sub> <= 10<sup>5</sup></code></li>\n\t<li>There is at most one edge between any two nodes.</li>\n\t<li>There is at least one path between any two nodes.</li>\n</ul>\n",
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"Run a Dijkstra from node numbered n to compute distance from the last node.",
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"Consider all edges [u, v] one by one and direct them such that distance of u to n > distance of v to n. If both u and v are at the same distance from n, discard this edge.",
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"Now this problem reduces to computing the number of paths from 1 to n in a DAG, a standard DP problem."
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