3515 - Shortest Path in a Weighted Tree.

Hard

You are given an integer n and an undirected, weighted tree rooted at node 1 with n nodes numbered from 1 to n. This is represented by a 2D array edges of length n - 1, where <code>edgesi = u_i, v_i, w_i</code> indicates an undirected edge from node <code>u_i</code> to <code>v_i</code> with weight <code>w_i</code>.

You are also given a 2D integer array queries of length q, where each queries[i] is either:

• [1, u, v, w'] – Update the weight of the edge between nodes u and v to w', where (u, v) is guaranteed to be an edge present in edges.

• [2, x] – Compute the shortest path distance from the root node 1 to node x.

Return an integer array answer, where answer[i] is the shortest path distance from node 1 to x for the <code>i^th</code> query of [2, x].

Example 1:

Input: n = 2, edges = [1,2,7], queries = [2,2,1,1,2,4,2,2]

Output: 7,4

Explanation:

• Query [2,2]: The shortest path from root node 1 to node 2 is 7.

• Query [1,1,2,4]: The weight of edge (1,2) changes from 7 to 4.

• Query [2,2]: The shortest path from root node 1 to node 2 is 4.

Example 2:

Input: n = 3, edges = [1,2,2,1,3,4], queries = [2,1,2,3,1,1,3,7,2,2,2,3]

Output: 0,4,2,7

Explanation:

• Query [2,1]: The shortest path from root node 1 to node 1 is 0.

• Query [2,3]: The shortest path from root node 1 to node 3 is 4.

• Query [1,1,3,7]: The weight of edge (1,3) changes from 4 to 7.

• Query [2,2]: The shortest path from root node 1 to node 2 is 2.

• Query [2,3]: The shortest path from root node 1 to node 3 is 7.

Example 3:

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

Output: 8,3,2,5

Explanation:

• Query [2,4]: The shortest path from root node 1 to node 4 consists of edges (1,2), (2,3), and (3,4) with weights 2 + 1 + 5 = 8.

• Query [2,3]: The shortest path from root node 1 to node 3 consists of edges (1,2) and (2,3) with weights 2 + 1 = 3.

• Query [1,2,3,3]: The weight of edge (2,3) changes from 1 to 3.

• Query [2,2]: The shortest path from root node 1 to node 2 is 2.

• Query [2,3]: The shortest path from root node 1 to node 3 consists of edges (1,2) and (2,3) with updated weights 2 + 3 = 5.

Constraints:

• <code>1 <= n <= 10⁵</code>

• edges.length == n - 1

• <code>edgesi == u_i, v_i, w_i</code>

• <code>1 <= u_i, v_i<= n</code>

• <code>1 <= w_i<= 10⁴</code>

• The input is generated such that edges represents a valid tree.

• <code>1 <= queries.length == q <= 10⁵</code>

• queries[i].length == 2 or 4