3650 - Minimum Cost Path with Edge Reversals.

Medium

You are given a directed, weighted graph with n nodes labeled from 0 to n - 1, and an array edges where <code>edgesi = u_i, v_i, w_i</code> represents a directed edge from node <code>u_i</code> to node <code>v_i</code> with cost <code>w_i</code>.

Each node <code>u_i</code> has a switch that can be used at most once: when you arrive at <code>u_i</code> and have not yet used its switch, you may activate it on one of its incoming edges <code>v_i → u_i</code> reverse that edge to <code>u_i → v_i</code> and immediately traverse it.

The reversal is only valid for that single move, and using a reversed edge costs <code>2 * w_i</code>.

Return the minimum total cost to travel from node 0 to node n - 1. If it is not possible, return -1.

Example 1:

Input: n = 4, edges = [0,1,3,3,1,1,2,3,4,0,2,2]

Output: 5

Explanation:

• Use the path 0 → 1 (cost 3).

• At node 1 reverse the original edge 3 → 1 into 1 → 3 and traverse it at cost 2 * 1 = 2.

• Total cost is 3 + 2 = 5.

Example 2:

Input: n = 4, edges = [0,2,1,2,1,1,1,3,1,2,3,3]

Output: 3

Explanation:

• No reversal is needed. Take the path 0 → 2 (cost 1), then 2 → 1 (cost 1), then 1 → 3 (cost 1).

• Total cost is 1 + 1 + 1 = 3.

Constraints:

• <code>2 <= n <= 5 * 10⁴</code>

• <code>1 <= edges.length <= 10⁵</code>

• <code>edgesi = u_i, v_i, w_i</code>

• <code>0 <= u_i, v_i<= n - 1</code>

• <code>1 <= w_i<= 1000</code>