{ "data": { "question": { "questionId": "1912", "questionFrontendId": "1786", "categoryTitle": "Algorithms", "boundTopicId": 635606, "title": "Number of Restricted Paths From First to Last Node", "titleSlug": "number-of-restricted-paths-from-first-to-last-node", "content": "
There is an undirected weighted connected graph. You are given a positive integer n
which denotes that the graph has n
nodes labeled from 1
to n
, and an array edges
where each edges[i] = [ui, vi, weighti]
denotes that there is an edge between nodes ui
and vi
with weight equal to weighti
.
A path from node start
to node end
is a sequence of nodes [z0, z1, z2, ..., zk]
such that z0 = start
and zk = end
and there is an edge between zi
and zi+1
where 0 <= i <= k-1
.
The distance of a path is the sum of the weights on the edges of the path. Let distanceToLastNode(x)
denote the shortest distance of a path between node n
and node x
. A restricted path is a path that also satisfies that distanceToLastNode(zi) > distanceToLastNode(zi+1)
where 0 <= i <= k-1
.
Return the number of restricted paths from node 1
to node n
. Since that number may be too large, return it modulo 109 + 7
.
\n
Example 1:
\n\n\nInput: n = 5, edges = [[1,2,3],[1,3,3],[2,3,1],[1,4,2],[5,2,2],[3,5,1],[5,4,10]]\nOutput: 3\nExplanation: Each circle contains the node number in black and its distanceToLastNode value in blue.
The three restricted paths are:\n1) 1 --> 2 --> 5\n2) 1 --> 2 --> 3 --> 5\n3) 1 --> 3 --> 5\n
\n\nExample 2:
\n\n\nInput: 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]]\nOutput: 1\nExplanation: Each circle contains the node number in black and its distanceToLastNode value in blue.
The only restricted path is 1 --> 3 --> 7.\n
\n\n\n
Constraints:
\n\n1 <= n <= 2 * 104
n - 1 <= edges.length <= 4 * 104
edges[i].length == 3
1 <= ui, vi <= n
ui != vi
1 <= weighti <= 105
现有一个加权无向连通图。给你一个正整数 n
,表示图中有 n
个节点,并按从 1
到 n
给节点编号;另给你一个数组 edges
,其中每个 edges[i] = [ui, vi, weighti]
表示存在一条位于节点 ui
和 vi
之间的边,这条边的权重为 weighti
。
从节点 start
出发到节点 end
的路径是一个形如 [z0, z1, z2, ..., zk]
的节点序列,满足 z0 = start
、zk = end
且在所有符合 0 <= i <= k-1
的节点 zi
和 zi+1
之间存在一条边。
路径的距离定义为这条路径上所有边的权重总和。用 distanceToLastNode(x)
表示节点 n
和 x
之间路径的最短距离。受限路径 为满足 distanceToLastNode(zi) > distanceToLastNode(zi+1)
的一条路径,其中 0 <= i <= k-1
。
返回从节点 1
出发到节点 n
的 受限路径数 。由于数字可能很大,请返回对 109 + 7
取余 的结果。
\n\n
示例 1:
\n\n\n输入: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输出:3\n解释:每个圆包含黑色的节点编号和蓝色的 distanceToLastNode 值。三条受限路径分别是:\n1) 1 --> 2 --> 5\n2) 1 --> 2 --> 3 --> 5\n3) 1 --> 3 --> 5\n\n\n
示例 2:
\n\n\n输入: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输出:1\n解释:每个圆包含黑色的节点编号和蓝色的 distanceToLastNode 值。唯一一条受限路径是:1 --> 3 --> 7 。\n\n
\n\n
提示:
\n\n1 <= n <= 2 * 104
n - 1 <= edges.length <= 4 * 104
edges[i].length == 3
1 <= ui, vi <= n
ui != vi
1 <= weighti <= 105
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