3608 - Minimum Time for K Connected Components.

Medium

You are given an integer n and an undirected graph with n nodes labeled from 0 to n - 1. This is represented by a 2D array edges, where <code>edgesi = u_i, v_i, time_i</code> indicates an undirected edge between nodes <code>u_i</code> and <code>v_i</code> that can be removed at <code>time_i</code>.

You are also given an integer k.

Initially, the graph may be connected or disconnected. Your task is to find the minimum time t such that after removing all edges with time <= t, the graph contains at least k connected components.

Return the minimum time t.

A connected component is a subgraph of a graph in which there exists a path between any two vertices, and no vertex of the subgraph shares an edge with a vertex outside of the subgraph.

Example 1:

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

Output: 3

Explanation:

• Initially, there is one connected component {0, 1}.

• At time = 1 or 2, the graph remains unchanged.

• At time = 3, edge [0, 1] is removed, resulting in k = 2 connected components {0}, {1}. Thus, the answer is 3.

Example 2:

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

Output: 4

Explanation:

• Initially, there is one connected component {0, 1, 2}.

• At time = 2, edge [0, 1] is removed, resulting in two connected components {0}, {1, 2}.

• At time = 4, edge [1, 2] is removed, resulting in k = 3 connected components {0}, {1}, {2}. Thus, the answer is 4.

Example 3:

Input: n = 3, edges = [0,2,5], k = 2

Output: 0

Explanation:

• Since there are already k = 2 disconnected components {1}, {0, 2}, no edge removal is needed. Thus, the answer is 0.

Constraints:

• <code>1 <= n <= 10⁵</code>

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

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

• <code>0 <= u_i, v_i< n</code>

• <code>u_i != v_i</code>

• <code>1 <= time_i<= 10⁹</code>

• 1 <= k <= n

• There are no duplicate edges.