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77 lines
3.8 KiB
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77 lines
3.8 KiB
HTML
<p>You are given an integer <code>n</code> and an undirected graph with <code>n</code> nodes labeled from 0 to <code>n - 1</code>. This is represented by a 2D array <code>edges</code>, where <code>edges[i] = [u<sub>i</sub>, v<sub>i</sub>, time<sub>i</sub>]</code> indicates an undirected edge between nodes <code>u<sub>i</sub></code> and <code>v<sub>i</sub></code> that can be removed at <code>time<sub>i</sub></code>.</p>
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<p>You are also given an integer <code>k</code>.</p>
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<p>Initially, the graph may be connected or disconnected. Your task is to find the <strong>minimum</strong> time <code>t</code> such that after removing all edges with <code>time <= t</code>, the graph contains <strong>at least</strong> <code>k</code> connected components.</p>
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<p>Return the <strong>minimum</strong> time <code>t</code>.</p>
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<p>A <strong>connected component</strong> 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.</p>
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<p> </p>
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<p><strong class="example">Example 1:</strong></p>
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<div class="example-block">
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<p><strong>Input:</strong> <span class="example-io">n = 2, edges = [[0,1,3]], k = 2</span></p>
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<p><strong>Output:</strong> <span class="example-io">3</span></p>
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<p><strong>Explanation:</strong></p>
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<p><img src="https://assets.leetcode.com/uploads/2025/05/31/screenshot-2025-06-01-at-022724.png" style="width: 230px; height: 85px;" /></p>
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<ul>
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<li>Initially, there is one connected component <code>{0, 1}</code>.</li>
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<li>At <code>time = 1</code> or <code>2</code>, the graph remains unchanged.</li>
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<li>At <code>time = 3</code>, edge <code>[0, 1]</code> is removed, resulting in <code>k = 2</code> connected components <code>{0}</code>, <code>{1}</code>. Thus, the answer is 3.</li>
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</ul>
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</div>
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<p><strong class="example">Example 2:</strong></p>
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<div class="example-block">
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<p><strong>Input:</strong> <span class="example-io">n = 3, edges = [[0,1,2],[1,2,4]], k = 3</span></p>
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<p><strong>Output:</strong> <span class="example-io">4</span></p>
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<p><strong>Explanation:</strong></p>
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<p><img src="https://assets.leetcode.com/uploads/2025/05/31/screenshot-2025-06-01-at-022812.png" style="width: 180px; height: 164px;" /></p>
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<ul>
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<li>Initially, there is one connected component <code>{0, 1, 2}</code>.</li>
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<li>At <code>time = 2</code>, edge <code>[0, 1]</code> is removed, resulting in two connected components <code>{0}</code>, <code>{1, 2}</code>.</li>
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<li>At <code>time = 4</code>, edge <code>[1, 2]</code> is removed, resulting in <code>k = 3</code> connected components <code>{0}</code>, <code>{1}</code>, <code>{2}</code>. Thus, the answer is 4.</li>
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</ul>
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</div>
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<p><strong class="example">Example 3:</strong></p>
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<div class="example-block">
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<p><strong>Input:</strong> <span class="example-io">n = 3, edges = [[0,2,5]], k = 2</span></p>
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<p><strong>Output:</strong> <span class="example-io">0</span></p>
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<p><strong>Explanation:</strong></p>
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<p><img src="https://assets.leetcode.com/uploads/2025/05/31/screenshot-2025-06-01-at-022930.png" style="width: 180px; height: 155px;" /></p>
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<ul>
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<li>Since there are already <code>k = 2</code> disconnected components <code>{1}</code>, <code>{0, 2}</code>, no edge removal is needed. Thus, the answer is 0.</li>
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</ul>
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</div>
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<p> </p>
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<p><strong>Constraints:</strong></p>
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<ul>
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<li><code>1 <= n <= 10<sup>5</sup></code></li>
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<li><code>0 <= edges.length <= 10<sup>5</sup></code></li>
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<li><code>edges[i] = [u<sub>i</sub>, v<sub>i</sub>, time<sub>i</sub>]</code></li>
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<li><code>0 <= u<sub>i</sub>, v<sub>i</sub> < n</code></li>
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<li><code>u<sub>i</sub> != v<sub>i</sub></code></li>
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<li><code>1 <= time<sub>i</sub> <= 10<sup>9</sup></code></li>
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<li><code>1 <= k <= n</code></li>
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<li>There are no duplicate edges.</li>
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</ul>
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