3108 - Minimum Cost Walk in Weighted Graph.

Hard

There is an undirected weighted graph with n vertices labeled from 0 to n - 1.

You are given the integer n and an array edges, where <code>edgesi = u_i, v_i, w_i</code> indicates that there is an edge between vertices <code>u_i</code> and <code>v_i</code> with a weight of <code>w_i</code>.

A walk on a graph is a sequence of vertices and edges. The walk starts and ends with a vertex, and each edge connects the vertex that comes before it and the vertex that comes after it. It's important to note that a walk may visit the same edge or vertex more than once.

The cost of a walk starting at node u and ending at node v is defined as the bitwise AND of the weights of the edges traversed during the walk. In other words, if the sequence of edge weights encountered during the walk is <code>w₀, w₁, w₂, ..., w_k</code>, then the cost is calculated as <code>w₀& w₁& w₂& ... & w_k</code>, where & denotes the bitwise AND operator.

You are also given a 2D array query, where <code>queryi = s_i, t_i</code>. For each query, you need to find the minimum cost of the walk starting at vertex <code>s_i</code> and ending at vertex <code>t_i</code>. If there exists no such walk, the answer is -1.

Return the array answer, where answer[i] denotes the minimum cost of a walk for query i.

Example 1:

Input: n = 5, edges = [0,1,7,1,3,7,1,2,1], query = [0,3,3,4]

Output: 1,-1

Explanation:

To achieve the cost of 1 in the first query, we need to move on the following edges: 0->1 (weight 7), 1->2 (weight 1), 2->1 (weight 1), 1->3 (weight 7).

In the second query, there is no walk between nodes 3 and 4, so the answer is -1.

Example 2:

Input: n = 3, edges = [0,2,7,0,1,15,1,2,6,1,2,1], query = [1,2]

Output: 0

Explanation:

To achieve the cost of 0 in the first query, we need to move on the following edges: 1->2 (weight 1), 2->1 (weight 6), 1->2 (weight 1).

Constraints:

• <code>2 <= n <= 10⁵</code>

• <code>0 <= edges.length <= 10⁵</code>

• edges[i].length == 3

• <code>0 <= u_i, v_i<= n - 1</code>

• <code>u_i != v_i</code>

• <code>0 <= w_i<= 10⁵</code>

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

• query[i].length == 2

• <code>0 <= s_i, t_i<= n - 1</code>

• <code>s_i != t_i</code>