{ "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 an integer array receiver of length n and an integer k. n players are playing a ball-passing game.

\n\n

You choose the starting player, i. The game proceeds as follows: player i passes the ball to player receiver[i], who then passes it to receiver[receiver[i]], and so on, for k passes in total. The game's score is the sum of the indices of the players who touched the ball, including repetitions, i.e. i + receiver[i] + receiver[receiver[i]] + ... + receiver(k)[i].

\n\n

Return the maximum possible score.

\n\n

Notes:

\n\n\n\n

 

\n

Example 1:

\n\n
\n

Input: receiver = [2,0,1], k = 4

\n\n

Output: 6

\n\n

Explanation:

\n\n

Starting with player i = 2 the initial score is 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
PassSender IndexReceiver IndexScore
1213
2103
3025
4216
\n
\n\n

Example 2:

\n\n
\n

Input: receiver = [1,1,1,2,3], k = 3

\n\n

Output: 10

\n\n

Explanation:

\n\n

Starting with player i = 4 the initial score is 4:

\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\n
PassSender IndexReceiver IndexScore
1437
2329
32110
\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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