1786 - Number of Restricted Paths From First to Last Node.

Medium

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 <code>edgesi = u_i, v_i, weight_i</code> denotes that there is an edge between nodes <code>u_i</code> and <code>v_i</code> with weight equal to <code>weight_i</code>.

A path from node start to node end is a sequence of nodes <code>z₀, z₁, z₂, ..., z_k</code> such that <code>z₀ = start</code> and <code>z_k = end</code> and there is an edge between <code>z_i</code> and <code>z_i+1</code> 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 <code>distanceToLastNode(z_i) > distanceToLastNode(z_i+1)</code> 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 <code>10⁹ + 7</code>.

Example 1:

Input: n = 5, edges = \[\[1,2,3],1,3,3,2,3,1,1,4,2,5,2,2,3,5,1,5,4,10]

Output: 3

Explanation: Each circle contains the node number in black and its distanceToLastNode value in blue. The three restricted paths are:

• 1 --> 2 --> 5

• 1 --> 2 --> 3 --> 5

• 1 --> 3 --> 5

Example 2:

Input: 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]

Output: 1

Explanation: Each circle contains the node number in black and its distanceToLastNode value in blue. The only restricted path is 1 --> 3 --> 7.

Constraints:

• <code>1 <= n <= 2 * 10⁴</code>

• <code>n - 1 <= edges.length <= 4 * 10⁴</code>

• edges[i].length == 3

• <code>1 <= u_i, v_i<= n</code>

• <code>u_i != v_i</code>

• <code>1 <= weight_i<= 10⁵</code>

• There is at most one edge between any two nodes.

• There is at least one path between any two nodes.