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137 lines
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137 lines
7.3 KiB
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<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>
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<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>
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<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>
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<ul>
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<li>All intermediate nodes on the path are online.</li>
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<li data-end="676" data-start="605">The total recovery cost of all edges on the path does not exceed <code>k</code>.</li>
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</ul>
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<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>
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<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>
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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">edges = [[0,1,5],[1,3,10],[0,2,3],[2,3,4]], online = [true,true,true,true], k = 10</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 alt="" src="https://assets.leetcode.com/uploads/2025/06/06/graph-10.png" style="width: 239px; height: 267px;" /></p>
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<ul data-end="551" data-start="146">
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<li data-end="462" data-start="146">
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<p data-end="206" data-start="148">The graph has two possible routes from node 0 to node 3:</p>
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<ol data-end="462" data-start="209">
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<li data-end="315" data-start="209">
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<p data-end="228" data-start="212">Path <code>0 → 1 → 3</code></p>
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<ul data-end="315" data-start="234">
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<li data-end="315" data-start="234">
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<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>
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</li>
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</ul>
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</li>
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<li data-end="462" data-start="318">
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<p data-end="337" data-start="321">Path <code>0 → 2 → 3</code></p>
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<ul data-end="462" data-start="343">
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<li data-end="397" data-start="343">
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<p data-end="397" data-start="345">Total cost = <code>3 + 4 = 7 <= k</code>, so this path is valid.</p>
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</li>
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<li data-end="462" data-start="403">
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<p data-end="462" data-start="405">The minimum edge‐cost along this path is <code>min(3, 4) = 3</code>.</p>
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</li>
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</ul>
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</li>
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</ol>
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</li>
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<li data-end="551" data-start="463">
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<p data-end="551" data-start="465">There are no other valid paths. Hence, the maximum among all valid path‐scores is 3.</p>
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</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">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>
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<p><strong>Output:</strong> <span class="example-io">6</span></p>
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<p><strong>Explanation:</strong></p>
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<p><img alt="" src="https://assets.leetcode.com/uploads/2025/06/06/graph-11.png" style="width: 343px; height: 194px;" /></p>
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<ul>
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<li data-end="790" data-start="726">
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<p data-end="790" data-start="728">Node 3 is offline, so any path passing through 3 is invalid.</p>
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</li>
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<li data-end="1231" data-start="791">
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<p data-end="837" data-start="793">Consider the remaining routes from 0 to 4:</p>
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<ol data-end="1231" data-start="840">
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<li data-end="985" data-start="840">
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<p data-end="859" data-start="843">Path <code>0 → 1 → 4</code></p>
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<ul data-end="985" data-start="865">
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<li data-end="920" data-start="865">
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<p data-end="920" data-start="867">Total cost = <code>7 + 5 = 12 <= k</code>, so this path is valid.</p>
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</li>
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<li data-end="985" data-start="926">
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<p data-end="985" data-start="928">The minimum edge‐cost along this path is <code>min(7, 5) = 5</code>.</p>
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</li>
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</ul>
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</li>
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<li data-end="1083" data-start="988">
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<p data-end="1011" data-start="991">Path <code>0 → 2 → 3 → 4</code></p>
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<ul data-end="1083" data-start="1017">
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<li data-end="1083" data-start="1017">
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<p data-end="1083" data-start="1019">Node 3 is offline, so this path is invalid regardless of cost.</p>
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</li>
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</ul>
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</li>
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<li data-end="1231" data-start="1086">
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<p data-end="1105" data-start="1089">Path <code>0 → 2 → 4</code></p>
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<ul data-end="1231" data-start="1111">
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<li data-end="1166" data-start="1111">
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<p data-end="1166" data-start="1113">Total cost = <code>6 + 6 = 12 <= k</code>, so this path is valid.</p>
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</li>
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<li data-end="1231" data-start="1172">
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<p data-end="1231" data-start="1174">The minimum edge‐cost along this path is <code>min(6, 6) = 6</code>.</p>
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</li>
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</ul>
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</li>
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</ol>
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</li>
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<li data-end="1314" data-is-last-node="" data-start="1232">
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<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>
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</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 data-end="42" data-start="20"><code data-end="40" data-start="20">n == online.length</code></li>
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<li data-end="63" data-start="45"><code data-end="61" data-start="45">2 <= n <= 5 * 10<sup>4</sup></code></li>
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<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>
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<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>
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<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>
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<li data-end="166" data-start="154"><code data-end="164" data-start="154">u<sub>i</sub> != v<sub>i</sub></code></li>
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<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>
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<li data-end="213" data-start="194"><code data-end="211" data-start="194">0 <= k <= 5 * 10<sup>13</sup></code></li>
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<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>
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<li data-end="362" data-is-last-node="" data-start="312">The given graph is a directed acyclic graph.</li>
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</ul>
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