{ "data": { "question": { "questionId": "3032", "questionFrontendId": "2836", "categoryTitle": "Algorithms", "boundTopicId": 2404057, "title": "Maximize Value of Function in a Ball Passing Game", "titleSlug": "maximize-value-of-function-in-a-ball-passing-game", "content": "

You are given a 0-indexed integer array receiver of length n and an integer k.

\n\n

There are n players having a unique id in the range [0, n - 1] who will play a ball passing game, and receiver[i] is the id of the player who receives passes from the player with id i. Players can pass to themselves, i.e. receiver[i] may be equal to i.

\n\n

You must choose one of the n players as the starting player for the game, and the ball will be passed exactly k times starting from the chosen player.

\n\n

For a chosen starting player having id x, we define a function f(x) that denotes the sum of x and the ids of all players who receive the ball during the k passes, including repetitions. In other words, f(x) = x + receiver[x] + receiver[receiver[x]] + ... + receiver(k)[x].

\n\n

Your task is to choose a starting player having id x that maximizes the value of f(x).

\n\n

Return an integer denoting the maximum value of the function.

\n\n

Note: receiver may contain duplicates.

\n\n

 

\n

Example 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\n\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\n\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\n\t\t\n\t\t\t\n\t\t\t\n\t\t\t\n\t\t\t\n\t\t\n\t\n
Pass NumberSender IDReceiver IDx + Receiver IDs
   2
1213
2103
3025
4216
\n\n
\nInput: receiver = [2,0,1], k = 4\nOutput: 6\nExplanation: The table above shows a simulation of the game starting with the player having id x = 2. \nFrom the table, f(2) is equal to 6. \nIt can be shown that 6 is the maximum achievable value of the function. \nHence, the output is 6. \n
\n\n

Example 2:

\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\n\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\n\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\n\t\n
Pass NumberSender IDReceiver IDx + Receiver IDs
   4
1437
2329
32110
\n\n
\nInput: receiver = [1,1,1,2,3], k = 3\nOutput: 10\nExplanation: The table above shows a simulation of the game starting with the player having id x = 4. \nFrom the table, f(4) is equal to 10. \nIt can be shown that 10 is the maximum achievable value of the function. \nHence, the output is 10. \n
\n\n

 

\n

Constraints:

\n\n\n", "translatedTitle": "在传球游戏中最大化函数值", "translatedContent": "

给你一个长度为 n 下标从 0 开始的整数数组 receiver 和一个整数 k 。

\n\n

总共有 n 名玩家，玩家 编号 互不相同，且为 [0, n - 1] 中的整数。这些玩家玩一个传球游戏，receiver[i] 表示编号为 i 的玩家会传球给编号为 receiver[i] 的玩家。玩家可以传球给自己，也就是说 receiver[i] 可能等于 i 。

\n\n

你需要从 n 名玩家中选择一名玩家作为游戏开始时唯一手中有球的玩家，球会被传 恰好 k 次。

\n\n

如果选择编号为 x 的玩家作为开始玩家，定义函数 f(x) 表示从编号为 x 的玩家开始，k 次传球内所有接触过球玩家的编号之 和 ，如果有玩家多次触球，则 累加多次 。换句话说， f(x) = x + receiver[x] + receiver[receiver[x]] + ... + receiver(k)[x] 。

\n\n

你的任务时选择开始玩家 x ，目的是 最大化 f(x) 。

\n\n

请你返回函数的 最大值 。

\n\n

注意：receiver 可能含有重复元素。

\n\n

 

\n\n

示例 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\n\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\n\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\n\t\t\n\t\t\t\n\t\t\t\n\t\t\t\n\t\t\t\n\t\t\n\t\n
传递次数传球者编号接球者编号x + 所有接球者编号
   2
1213
2103
3025
4216
\n\n

 

\n\n
\n输入：receiver = [2,0,1], k = 4\n输出：6\n解释：上表展示了从编号为 x = 2 开始的游戏过程。\n从表中可知，f(2) 等于 6 。\n6 是能得到最大的函数值。\n所以输出为 6 。\n
\n\n

示例 2：

\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\n\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\n\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\n\t\n
传递次数传球者编号接球者编号x + 所有接球者编号
   4
1437
2329
32110
\n\n

 

\n\n
\n输入：receiver = [1,1,1,2,3], k = 3\n输出：10\n解释：上表展示了从编号为 x = 4 开始的游戏过程。\n从表中可知，f(4) 等于 10 。\n10 是能得到最大的函数值。\n所以输出为 10 。\n
\n\n

 

\n\n

提示：

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We can solve the problem using binary lifting.
", "
For each player with id x and for every i in the range [0, ceil(log2k)], we can determine the last receiver's id and compute the sum of player ids who receive the ball after 2i passes, starting from x.
", "
Let last_receiver[x][i] = the last receiver's id after 2i passes, and sum[x][i] = the sum of player ids who receive the ball after 2i passes. For all x in the range [0, n - 1], last_receiver[x][0] = receiver[x], and sum[x][0] = receiver[x].
", "
Then for i in range [1, ceil(log2k)], last_receiver[x][i] = last_receiver[last_receiver[x][i - 1]][i - 1] and sum[x][i] = sum[x][i - 1] + sum[last_receiver[x][i - 1]][i - 1], for all x in the range [0, n - 1].
", "
Starting from each player id x, we can now go through the powers of 2 in the binary representation of k and make jumps corresponding to each power, using the pre-computed values, to compute f(x).
", "
The answer is the maximum f(x) from each player id.
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