Problem: You are given a directed weighted graph with n nodes and 2n - 2 edges. The nodes are labeled from 1 to n, while the edges are labeled from 1 to 2n - 2. The graph's edges can be split into two parts.
- The first n - 1 edges will form a rooted spanning tree, with node 1 as the root. All these edges will point away from the root.
- The last n - 1 edges will be from node i to node 1, for all 2 ≤ i ≤ n.
You are given q queries. There are two types of queries
- 1 i w: Change the weight of the i-th edge to w
- 2 u v: Print the length of the shortest path between nodes u to v
Given these queries, print the shortest path lengths.
Input Format: The first line of input will contain two integers n, q (2 ≤ n, q ≤ 200 000), the number of nodes, and the number of queries, respectively.
The next 2n - 2 integers will contain 3 integers ai, bi, ci, denoting a directed edge from node ai to node bi with weight ci.
The first n - 1 of these lines will describe a rooted spanning tree pointing away from node 1, while the last n - 1 of these lines will have bi = 1.
More specifically,
- The edges (a1, b1), (a2, b2), ... (an - 1, bn - 1) will describe a rooted spanning tree pointing away from node 1.
- bj = 1 for n ≤ j ≤ 2n - 2.
- an, an + 1, ..., a2n - 2 will be distinct and between 2 and n.
The next q lines will contain 3 integers, describing a query in the format described in the statement.
All edge weights will be between 1 and 106.
Output Format: For each type 2 query, print the length of the shortest path in its own line.