mirror of
https://gitee.com/coder-xiaomo/leetcode-problemset
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214 lines
35 KiB
JSON
214 lines
35 KiB
JSON
{
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"questionId": "3919",
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"boundTopicId": 3727223,
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"title": "Network Recovery Pathways",
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"content": "<p data-end=\"502\" data-start=\"75\">You are given a directed acyclic graph of <code>n</code> nodes numbered from 0 to <code>n − 1</code>. This is represented by a 2D array <code data-end=\"201\" data-start=\"194\">edges</code> of length<font face=\"monospace\"> <code>m</code></font>, where <code data-end=\"255\" data-start=\"227\">edges[i] = [u<sub>i</sub>, v<sub>i</sub>, cost<sub>i</sub>]</code> indicates a one‑way communication from node <code data-end=\"304\" data-start=\"300\">u<sub>i</sub></code> to node <code data-end=\"317\" data-start=\"313\">v<sub>i</sub></code> with a recovery cost of <code data-end=\"349\" data-start=\"342\">cost<sub>i</sub></code>.</p>\n\n<p data-end=\"502\" data-start=\"75\">Some nodes may be offline. You are given a boolean array <code data-end=\"416\" data-start=\"408\">online</code> where <code data-end=\"441\" data-start=\"423\">online[i] = true</code> means node <code data-end=\"456\" data-start=\"453\">i</code> is online. Nodes 0 and <code>n − 1</code> are always online.</p>\n\n<p data-end=\"547\" data-start=\"504\">A path from 0 to <code>n − 1</code> is <strong data-end=\"541\" data-start=\"532\">valid</strong> if:</p>\n\n<ul>\n\t<li>All intermediate nodes on the path are online.</li>\n\t<li data-end=\"676\" data-start=\"605\">The total recovery cost of all edges on the path does not exceed <code>k</code>.</li>\n</ul>\n\n<p data-end=\"771\" data-start=\"653\">For each valid path, define its <strong data-end=\"694\" data-start=\"685\">score</strong> as the minimum edge‑cost along that path.</p>\n\n<p data-end=\"913\" data-start=\"847\">Return the <strong>maximum</strong> path score (i.e., the largest <strong>minimum</strong>-edge cost) among all valid paths. If no valid path exists, return -1.</p>\n\n<p> </p>\n<p><strong class=\"example\">Example 1:</strong></p>\n\n<div class=\"example-block\">\n<p><strong>Input:</strong> <span class=\"example-io\">edges = [[0,1,5],[1,3,10],[0,2,3],[2,3,4]], online = [true,true,true,true], k = 10</span></p>\n\n<p><strong>Output:</strong> <span class=\"example-io\">3</span></p>\n\n<p><strong>Explanation:</strong></p>\n\n<p><img alt=\"\" src=\"https://assets.leetcode.com/uploads/2025/06/06/graph-10.png\" style=\"width: 239px; height: 267px;\" /></p>\n\n<ul data-end=\"551\" data-start=\"146\">\n\t<li data-end=\"462\" data-start=\"146\">\n\t<p data-end=\"206\" data-start=\"148\">The graph has two possible routes from node 0 to node 3:</p>\n\n\t<ol data-end=\"462\" data-start=\"209\">\n\t\t<li data-end=\"315\" data-start=\"209\">\n\t\t<p data-end=\"228\" data-start=\"212\">Path <code>0 → 1 → 3</code></p>\n\n\t\t<ul data-end=\"315\" data-start=\"234\">\n\t\t\t<li data-end=\"315\" data-start=\"234\">\n\t\t\t<p data-end=\"315\" data-start=\"236\">Total cost = <code>5 + 10 = 15</code>, which exceeds k (<code>15 > 10</code>), so this path is invalid.</p>\n\t\t\t</li>\n\t\t</ul>\n\t\t</li>\n\t\t<li data-end=\"462\" data-start=\"318\">\n\t\t<p data-end=\"337\" data-start=\"321\">Path <code>0 → 2 → 3</code></p>\n\n\t\t<ul data-end=\"462\" data-start=\"343\">\n\t\t\t<li data-end=\"397\" data-start=\"343\">\n\t\t\t<p data-end=\"397\" data-start=\"345\">Total cost = <code>3 + 4 = 7 <= k</code>, so this path is valid.</p>\n\t\t\t</li>\n\t\t\t<li data-end=\"462\" data-start=\"403\">\n\t\t\t<p data-end=\"462\" data-start=\"405\">The minimum edge‐cost along this path is <code>min(3, 4) = 3</code>.</p>\n\t\t\t</li>\n\t\t</ul>\n\t\t</li>\n\t</ol>\n\t</li>\n\t<li data-end=\"551\" data-start=\"463\">\n\t<p data-end=\"551\" data-start=\"465\">There are no other valid paths. Hence, the maximum among all valid path‐scores is 3.</p>\n\t</li>\n</ul>\n</div>\n\n<p><strong class=\"example\">Example 2:</strong></p>\n\n<div class=\"example-block\">\n<p><strong>Input:</strong> <span class=\"example-io\">edges = [[0,1,7],[1,4,5],[0,2,6],[2,3,6],[3,4,2],[2,4,6]], online = [true,true,true,false,true], k = 12</span></p>\n\n<p><strong>Output:</strong> <span class=\"example-io\">6</span></p>\n\n<p><strong>Explanation:</strong></p>\n\n<p><img alt=\"\" src=\"https://assets.leetcode.com/uploads/2025/06/06/graph-11.png\" style=\"width: 343px; height: 194px;\" /></p>\n\n<ul>\n\t<li data-end=\"790\" data-start=\"726\">\n\t<p data-end=\"790\" data-start=\"728\">Node 3 is offline, so any path passing through 3 is invalid.</p>\n\t</li>\n\t<li data-end=\"1231\" data-start=\"791\">\n\t<p data-end=\"837\" data-start=\"793\">Consider the remaining routes from 0 to 4:</p>\n\n\t<ol data-end=\"1231\" data-start=\"840\">\n\t\t<li data-end=\"985\" data-start=\"840\">\n\t\t<p data-end=\"859\" data-start=\"843\">Path <code>0 → 1 → 4</code></p>\n\n\t\t<ul data-end=\"985\" data-start=\"865\">\n\t\t\t<li data-end=\"920\" data-start=\"865\">\n\t\t\t<p data-end=\"920\" data-start=\"867\">Total cost = <code>7 + 5 = 12 <= k</code>, so this path is valid.</p>\n\t\t\t</li>\n\t\t\t<li data-end=\"985\" data-start=\"926\">\n\t\t\t<p data-end=\"985\" data-start=\"928\">The minimum edge‐cost along this path is <code>min(7, 5) = 5</code>.</p>\n\t\t\t</li>\n\t\t</ul>\n\t\t</li>\n\t\t<li data-end=\"1083\" data-start=\"988\">\n\t\t<p data-end=\"1011\" data-start=\"991\">Path <code>0 → 2 → 3 → 4</code></p>\n\n\t\t<ul data-end=\"1083\" data-start=\"1017\">\n\t\t\t<li data-end=\"1083\" data-start=\"1017\">\n\t\t\t<p data-end=\"1083\" data-start=\"1019\">Node 3 is offline, so this path is invalid regardless of cost.</p>\n\t\t\t</li>\n\t\t</ul>\n\t\t</li>\n\t\t<li data-end=\"1231\" data-start=\"1086\">\n\t\t<p data-end=\"1105\" data-start=\"1089\">Path <code>0 → 2 → 4</code></p>\n\n\t\t<ul data-end=\"1231\" data-start=\"1111\">\n\t\t\t<li data-end=\"1166\" data-start=\"1111\">\n\t\t\t<p data-end=\"1166\" data-start=\"1113\">Total cost = <code>6 + 6 = 12 <= k</code>, so this path is valid.</p>\n\t\t\t</li>\n\t\t\t<li data-end=\"1231\" data-start=\"1172\">\n\t\t\t<p data-end=\"1231\" data-start=\"1174\">The minimum edge‐cost along this path is <code>min(6, 6) = 6</code>.</p>\n\t\t\t</li>\n\t\t</ul>\n\t\t</li>\n\t</ol>\n\t</li>\n\t<li data-end=\"1314\" data-is-last-node=\"\" data-start=\"1232\">\n\t<p data-end=\"1314\" data-is-last-node=\"\" data-start=\"1234\">Among the two valid paths, their scores are 5 and 6. Therefore, the answer is 6.</p>\n\t</li>\n</ul>\n</div>\n\n<p> </p>\n<p><strong>Constraints:</strong></p>\n\n<ul>\n\t<li data-end=\"42\" data-start=\"20\"><code data-end=\"40\" data-start=\"20\">n == online.length</code></li>\n\t<li data-end=\"63\" data-start=\"45\"><code data-end=\"61\" data-start=\"45\">2 <= n <= 5 * 10<sup>4</sup></code></li>\n\t<li data-end=\"102\" data-start=\"66\"><code data-end=\"100\" data-start=\"66\">0 <= m == edges.length <= </code><code>min(10<sup>5</sup>, n * (n - 1) / 2)</code></li>\n\t<li data-end=\"102\" data-start=\"66\"><code data-end=\"127\" data-start=\"105\">edges[i] = [u<sub>i</sub>, v<sub>i</sub>, cost<sub>i</sub>]</code></li>\n\t<li data-end=\"151\" data-start=\"132\"><code data-end=\"149\" data-start=\"132\">0 <= u<sub>i</sub>, v<sub>i</sub> < n</code></li>\n\t<li data-end=\"166\" data-start=\"154\"><code data-end=\"164\" data-start=\"154\">u<sub>i</sub> != v<sub>i</sub></code></li>\n\t<li data-end=\"191\" data-start=\"169\"><code data-end=\"189\" data-start=\"169\">0 <= cost<sub>i</sub> <= 10<sup>9</sup></code></li>\n\t<li data-end=\"213\" data-start=\"194\"><code data-end=\"211\" data-start=\"194\">0 <= k <= 5 * 10<sup>13</sup></code></li>\n\t<li data-end=\"309\" data-start=\"216\"><code data-end=\"227\" data-start=\"216\">online[i]</code> is either <code data-end=\"244\" data-is-only-node=\"\" data-start=\"238\">true</code> or <code data-end=\"255\" data-start=\"248\">false</code>, and both <code data-end=\"277\" data-start=\"266\">online[0]</code> and <code data-end=\"295\" data-start=\"282\">online[n − 1]</code> are <code data-end=\"306\" data-start=\"300\">true</code>.</li>\n\t<li data-end=\"362\" data-is-last-node=\"\" data-start=\"312\">The given graph is a directed acyclic graph.</li>\n</ul>\n",
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"translatedTitle": "恢复网络路径",
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"translatedContent": "<p>给你一个包含 <code>n</code> 个节点（编号从 0 到 <code>n - 1</code>）的有向无环图。图由长度为 <code>m</code> 的二维数组 <code>edges</code> 表示，其中 <code>edges[i] = [u<sub>i</sub>, v<sub>i</sub>, cost<sub>i</sub>]</code> 表示从节点 <code>u<sub>i</sub></code> 到节点 <code>v<sub>i</sub></code> 的单向通信，恢复成本为 <code>cost<sub>i</sub></code>。</p>\n\n<p>一些节点可能处于离线状态。给定一个布尔数组 <code>online</code>，其中 <code>online[i] = true</code> 表示节点 <code>i</code> 在线。节点 0 和 <code>n - 1</code> 始终在线。</p>\n\n<p>从 0 到 <code>n - 1</code> 的路径如果满足以下条件，那么它是 <strong>有效 </strong>的：</p>\n\n<ul>\n\t<li>路径上的所有中间节点都在线。</li>\n\t<li>路径上所有边的总恢复成本不超过 <code>k</code>。</li>\n</ul>\n\n<p>对于每条有效路径，其 <strong>分数 </strong>定义为该路径上的最小边成本。</p>\n\n<p>返回所有有效路径中的 <strong>最大 </strong>路径分数（即最大 <strong>最小 </strong>边成本）。如果没有有效路径，则返回 -1。</p>\n\n<p> </p>\n\n<p><strong class=\"example\">示例 1:</strong></p>\n\n<div class=\"example-block\">\n<p><strong>输入:</strong> <span class=\"example-io\">edges = [[0,1,5],[1,3,10],[0,2,3],[2,3,4]], online = [true,true,true,true], k = 10</span></p>\n\n<p><strong>输出:</strong> <span class=\"example-io\">3</span></p>\n\n<p><strong>解释:</strong></p>\n\n<p><img alt=\"\" src=\"https://assets.leetcode.com/uploads/2025/06/06/graph-10.png\" style=\"width: 239px; height: 267px;\" /></p>\n\n<ul>\n\t<li>\n\t<p>图中有两条从节点 0 到节点 3 的可能路线：</p>\n\n\t<ol>\n\t\t<li>\n\t\t<p>路径 <code>0 → 1 → 3</code></p>\n\n\t\t<ul>\n\t\t\t<li>\n\t\t\t<p>总成本 = <code>5 + 10 = 15</code>，超过了 k (<code>15 > 10</code>)，因此此路径无效。</p>\n\t\t\t</li>\n\t\t</ul>\n\t\t</li>\n\t\t<li>\n\t\t<p>路径 <code>0 → 2 → 3</code></p>\n\n\t\t<ul>\n\t\t\t<li>\n\t\t\t<p>总成本 = <code>3 + 4 = 7 <= k</code>，因此此路径有效。</p>\n\t\t\t</li>\n\t\t\t<li>\n\t\t\t<p>此路径上的最小边成本为 <code>min(3, 4) = 3</code>。</p>\n\t\t\t</li>\n\t\t</ul>\n\t\t</li>\n\t</ol>\n\t</li>\n\t<li>\n\t<p>没有其他有效路径。因此，所有有效路径分数中的最大值为 3。</p>\n\t</li>\n</ul>\n</div>\n\n<p><strong class=\"example\">示例 2:</strong></p>\n\n<div class=\"example-block\">\n<p><strong>输入:</strong> <span class=\"example-io\">edges = [[0,1,7],[1,4,5],[0,2,6],[2,3,6],[3,4,2],[2,4,6]], online = [true,true,true,false,true], k = 12</span></p>\n\n<p><strong>输出:</strong> <span class=\"example-io\">6</span></p>\n\n<p><strong>解释:</strong></p>\n\n<p><img alt=\"\" src=\"https://assets.leetcode.com/uploads/2025/06/06/graph-11.png\" style=\"width: 343px; height: 194px;\" /></p>\n\n<ul>\n\t<li>\n\t<p>节点 3 离线，因此任何通过 3 的路径都是无效的。</p>\n\t</li>\n\t<li>\n\t<p>考虑从 0 到 4 的其余路线：</p>\n\n\t<ol>\n\t\t<li>\n\t\t<p>路径 <code>0 → 1 → 4</code></p>\n\n\t\t<ul>\n\t\t\t<li>\n\t\t\t<p>总成本 = <code>7 + 5 = 12 <= k</code>，因此此路径有效。</p>\n\t\t\t</li>\n\t\t\t<li>\n\t\t\t<p>此路径上的最小边成本为 <code>min(7, 5) = 5</code>。</p>\n\t\t\t</li>\n\t\t</ul>\n\t\t</li>\n\t\t<li>\n\t\t<p>路径 <code>0 → 2 → 3 → 4</code></p>\n\n\t\t<ul>\n\t\t\t<li>\n\t\t\t<p>节点 3 离线，因此无论成本多少，此路径无效。</p>\n\t\t\t</li>\n\t\t</ul>\n\t\t</li>\n\t\t<li>\n\t\t<p>路径 <code>0 → 2 → 4</code></p>\n\n\t\t<ul>\n\t\t\t<li>\n\t\t\t<p>总成本 = <code>6 + 6 = 12 <= k</code>，因此此路径有效。</p>\n\t\t\t</li>\n\t\t\t<li>\n\t\t\t<p>此路径上的最小边成本为 <code>min(6, 6) = 6</code>。</p>\n\t\t\t</li>\n\t\t</ul>\n\t\t</li>\n\t</ol>\n\t</li>\n\t<li>\n\t<p>在两条有效路径中，它们的分数分别为 5 和 6。因此，答案是 6。</p>\n\t</li>\n</ul>\n</div>\n\n<p> </p>\n\n<p><strong>提示:</strong></p>\n\n<ul>\n\t<li><code>n == online.length</code></li>\n\t<li><code>2 <= n <= 5 * 10<sup>4</sup></code></li>\n\t<li><code>0 <= m == edges.length <= min(10<sup>5</sup>, n * (n - 1) / 2)</code></li>\n\t<li><code>edges[i] = [u<sub>i</sub>, v<sub>i</sub>, cost<sub>i</sub>]</code></li>\n\t<li><code>0 <= u<sub>i</sub>, v<sub>i</sub> < n</code></li>\n\t<li><code>u<sub>i</sub> != v<sub>i</sub></code></li>\n\t<li><code>0 <= cost<sub>i</sub> <= 10<sup>9</sup></code></li>\n\t<li><code>0 <= k <= 5 * 10<sup>13</sup></code></li>\n\t<li><code>online[i]</code> 是 <code>true</code> 或 <code>false</code>，且 <code>online[0]</code> 和 <code>online[n - 1]</code> 均为 <code>true</code>。</li>\n\t<li>给定的图是一个有向无环图。</li>\n</ul>\n",
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|
||
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"Implement the check function using either <code>Dijkstra</code> or DP (via topological sorting, since the graph is a DAG)."
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