{ "data": { "question": { "questionId": "3826", "questionFrontendId": "3530", "categoryTitle": "Algorithms", "boundTopicId": 3662345, "title": "Maximum Profit from Valid Topological Order in DAG", "titleSlug": "maximum-profit-from-valid-topological-order-in-dag", "content": "

You are given a Directed Acyclic Graph (DAG) with n nodes labeled from 0 to n - 1, represented by a 2D array edges, where edges[i] = [ui, vi] indicates a directed edge from node ui to vi. Each node has an associated score given in an array score, where score[i] represents the score of node i.

\n\n

You must process the nodes in a valid topological order. Each node is assigned a 1-based position in the processing order.

\n\n

The profit is calculated by summing up the product of each node's score and its position in the ordering.

\n\n

Return the maximum possible profit achievable with an optimal topological order.

\n\n

A topological order of a DAG is a linear ordering of its nodes such that for every directed edge u → v, node u comes before v in the ordering.

\n\n

 

\n

Example 1:

\n\n
\n

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

\n\n

Output: 8

\n\n

Explanation:

\n\n

\n\n

Node 1 depends on node 0, so a valid order is [0, 1].

\n\n\n\t\n\t\t\n\t\t\t\n\t\t\t\n\t\t\t\n\t\t\t\n\t\t\t\n\t\t\n\t\n\t\n\t\t\n\t\t\t\n\t\t\t\n\t\t\t\n\t\t\t\n\t\t\t\n\t\t\n\t\t\n\t\t\t\n\t\t\t\n\t\t\t\n\t\t\t\n\t\t\t\n\t\t\n\t\n
NodeProcessing OrderScoreMultiplierProfit Calculation
01st212 × 1 = 2
12nd323 × 2 = 6
\n\n

The maximum total profit achievable over all valid topological orders is 2 + 6 = 8.

\n
\n\n

Example 2:

\n\n
\n

Input: n = 3, edges = [[0,1],[0,2]], score = [1,6,3]

\n\n

Output: 25

\n\n

Explanation:

\n\n

\"\"

\n\n

Nodes 1 and 2 depend on node 0, so the most optimal valid order is [0, 2, 1].

\n\n\n\t\n\t\t\n\t\t\t\n\t\t\t\n\t\t\t\n\t\t\t\n\t\t\t\n\t\t\n\t\n\t\n\t\t\n\t\t\t\n\t\t\t\n\t\t\t\n\t\t\t\n\t\t\t\n\t\t\n\t\t\n\t\t\t\n\t\t\t\n\t\t\t\n\t\t\t\n\t\t\t\n\t\t\n\t\t\n\t\t\t\n\t\t\t\n\t\t\t\n\t\t\t\n\t\t\t\n\t\t\n\t\n
NodeProcessing OrderScoreMultiplierProfit Calculation
01st111 × 1 = 1
22nd323 × 2 = 6
13rd636 × 3 = 18
\n\n

The maximum total profit achievable over all valid topological orders is 1 + 6 + 18 = 25.

\n
\n\n

 

\n

Constraints:

\n\n\n", "translatedTitle": "有向无环图中合法拓扑排序的最大利润", "translatedContent": "

给你一个由 n 个节点组成的有向无环图（DAG），节点编号从 0 到 n - 1，通过二维数组 edges 表示，其中 edges[i] = [ui, vi] 表示一条从节点 ui 指向节点 vi 的有向边。每个节点都有一个对应的 得分 ，由数组 score 给出，其中 score[i] 表示节点 i 的得分。

\n\n

你需要以 有效的拓扑排序 顺序处理这些节点。每个节点在处理顺序中被分配一个编号从 1 开始的位置。

\n\n

将每个节点的得分乘以其在拓扑排序中的位置，然后求和，得到的值称为 利润。

\n\n

请返回在所有合法拓扑排序中可获得的 最大利润 。

\n\n

拓扑排序 是一个对 DAG 中所有节点的线性排序，使得每条有向边 u → v 中，节点 u 都出现在 v 之前。

\n\n

 

\n\n

示例 1：

\n\n
\n

输入： n = 2, edges = [[0,1]], score = [2,3]

\n\n

输出： 8

\n\n

解释：

\n\n

\n\n

节点 1 依赖于节点 0，因此一个合法顺序是 [0, 1]。

\n\n\n\t\n\t\t\n\t\t\t\n\t\t\t\n\t\t\t\n\t\t\t\n\t\t\t\n\t\t\n\t\n\t\n\t\t\n\t\t\t\n\t\t\t\n\t\t\t\n\t\t\t\n\t\t\t\n\t\t\n\t\t\n\t\t\t\n\t\t\t\n\t\t\t\n\t\t\t\n\t\t\t\n\t\t\n\t\n
节点处理顺序得分乘数利润计算
0第 1 个212 × 1 = 2
1第 2 个323 × 2 = 6
\n\n

所有合法拓扑排序中可获得的最大总利润是 2 + 6 = 8。

\n
\n\n

示例 2：

\n\n
\n

输入： n = 3, edges = [[0,1],[0,2]], score = [1,6,3]

\n\n

输出： 25

\n\n

解释：

\n\n

\"\"

\n\n

节点 1 和 2 都依赖于节点 0，因此最优的合法顺序是 [0, 2, 1]。

\n\n\n\t\n\t\t\n\t\t\t\n\t\t\t\n\t\t\t\n\t\t\t\n\t\t\t\n\t\t\n\t\n\t\n\t\t\n\t\t\t\n\t\t\t\n\t\t\t\n\t\t\t\n\t\t\t\n\t\t\n\t\t\n\t\t\t\n\t\t\t\n\t\t\t\n\t\t\t\n\t\t\t\n\t\t\n\t\t\n\t\t\t\n\t\t\t\n\t\t\t\n\t\t\t\n\t\t\t\n\t\t\n\t\n
节点处理顺序得分乘数利润计算
0第 1 个111 × 1 = 1
2第 2 个323 × 2 = 6
1第 3 个636 × 3 = 18
\n\n

所有合法拓扑排序中可获得的最大总利润是 1 + 6 + 18 = 25。

\n
\n\n

 

\n\n

提示：

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