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70 lines
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70 lines
4.1 KiB
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<p>There is an undirected tree with <code>n</code> nodes labeled from <code>0</code> to <code>n - 1</code>, rooted at node <code>0</code>. You are given a 2D integer array <code>edges</code> of length <code>n - 1</code> where <code>edges[i] = [a<sub>i</sub>, b<sub>i</sub>]</code> indicates that there is an edge between nodes <code>a<sub>i</sub></code> and <code>b<sub>i</sub></code> in the tree.</p>
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<p>At every node <code>i</code>, there is a gate. You are also given an array of even integers <code>amount</code>, where <code>amount[i]</code> represents:</p>
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<ul>
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<li>the price needed to open the gate at node <code>i</code>, if <code>amount[i]</code> is negative, or,</li>
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<li>the cash reward obtained on opening the gate at node <code>i</code>, otherwise.</li>
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</ul>
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<p>The game goes on as follows:</p>
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<ul>
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<li>Initially, Alice is at node <code>0</code> and Bob is at node <code>bob</code>.</li>
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<li>At every second, Alice and Bob <b>each</b> move to an adjacent node. Alice moves towards some <strong>leaf node</strong>, while Bob moves towards node <code>0</code>.</li>
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<li>For <strong>every</strong> node along their path, Alice and Bob either spend money to open the gate at that node, or accept the reward. Note that:
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<ul>
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<li>If the gate is <strong>already open</strong>, no price will be required, nor will there be any cash reward.</li>
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<li>If Alice and Bob reach the node <strong>simultaneously</strong>, they share the price/reward for opening the gate there. In other words, if the price to open the gate is <code>c</code>, then both Alice and Bob pay <code>c / 2</code> each. Similarly, if the reward at the gate is <code>c</code>, both of them receive <code>c / 2</code> each.</li>
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</ul>
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</li>
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<li>If Alice reaches a leaf node, she stops moving. Similarly, if Bob reaches node <code>0</code>, he stops moving. Note that these events are <strong>independent</strong> of each other.</li>
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</ul>
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<p>Return<em> the <strong>maximum</strong> net income Alice can have if she travels towards the optimal leaf node.</em></p>
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<p> </p>
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<p><strong class="example">Example 1:</strong></p>
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<img alt="" src="https://assets.leetcode.com/uploads/2022/10/29/eg1.png" style="width: 275px; height: 275px;" />
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<pre>
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<strong>Input:</strong> edges = [[0,1],[1,2],[1,3],[3,4]], bob = 3, amount = [-2,4,2,-4,6]
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<strong>Output:</strong> 6
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<strong>Explanation:</strong>
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The above diagram represents the given tree. The game goes as follows:
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- Alice is initially on node 0, Bob on node 3. They open the gates of their respective nodes.
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Alice's net income is now -2.
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- Both Alice and Bob move to node 1.
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Since they reach here simultaneously, they open the gate together and share the reward.
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Alice's net income becomes -2 + (4 / 2) = 0.
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- Alice moves on to node 3. Since Bob already opened its gate, Alice's income remains unchanged.
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Bob moves on to node 0, and stops moving.
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- Alice moves on to node 4 and opens the gate there. Her net income becomes 0 + 6 = 6.
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Now, neither Alice nor Bob can make any further moves, and the game ends.
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It is not possible for Alice to get a higher net income.
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</pre>
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<p><strong class="example">Example 2:</strong></p>
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<img alt="" src="https://assets.leetcode.com/uploads/2022/10/29/eg2.png" style="width: 250px; height: 78px;" />
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<pre>
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<strong>Input:</strong> edges = [[0,1]], bob = 1, amount = [-7280,2350]
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<strong>Output:</strong> -7280
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<strong>Explanation:</strong>
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Alice follows the path 0->1 whereas Bob follows the path 1->0.
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Thus, Alice opens the gate at node 0 only. Hence, her net income is -7280.
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</pre>
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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>2 <= n <= 10<sup>5</sup></code></li>
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<li><code>edges.length == n - 1</code></li>
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<li><code>edges[i].length == 2</code></li>
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<li><code>0 <= a<sub>i</sub>, b<sub>i</sub> < n</code></li>
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<li><code>a<sub>i</sub> != b<sub>i</sub></code></li>
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<li><code>edges</code> represents a valid tree.</li>
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<li><code>1 <= bob < n</code></li>
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<li><code>amount.length == n</code></li>
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<li><code>amount[i]</code> is an <strong>even</strong> integer in the range <code>[-10<sup>4</sup>, 10<sup>4</sup>]</code>.</li>
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</ul>
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