leetcode-problemset/leetcode-cn/originData/maximize-value-of-function-in-a-ball-passing-game.json

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        "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": "<p>You are given a <strong>0-indexed</strong> integer array <code>receiver</code> of length <code>n</code> and an integer <code>k</code>.</p>\n\n<p>There are <code>n</code> players having a <strong>unique id</strong> in the range <code>[0, n - 1]</code> who will play a ball passing game, and <code>receiver[i]</code> is the id of the player who receives passes from the player with id <code>i</code>. Players can pass to themselves, <strong>i.e.</strong> <code>receiver[i]</code> may be equal to <code>i</code>.</p>\n\n<p>You must choose one of the <code>n</code> players as the starting player for the game, and the ball will be passed <strong>exactly</strong> <code>k</code> times starting from the chosen player.</p>\n\n<p>For a chosen starting player having id <code>x</code>, we define a function <code>f(x)</code> that denotes the <strong>sum</strong> of <code>x</code> and the <strong>ids</strong> of all players who receive the ball during the <code>k</code> passes, <strong>including repetitions</strong>. In other words, <code>f(x) = x + receiver[x] + receiver[receiver[x]] + ... + receiver<sup>(k)</sup>[x]</code>.</p>\n\n<p>Your task is to choose a starting player having id <code>x</code> that <strong>maximizes</strong> the value of <code>f(x)</code>.</p>\n\n<p>Return <em>an integer denoting the <strong>maximum</strong> value of the function.</em></p>\n\n<p><strong>Note:</strong> <code>receiver</code> may contain duplicates.</p>\n\n<p>&nbsp;</p>\n<p><strong class=\"example\">Example 1:</strong></p>\n\n<table border=\"1\" cellspacing=\"3\" style=\"border-collapse: separate; text-align: center;\">\n\t<tbody>\n\t\t<tr>\n\t\t\t<th style=\"padding: 5px; border: 1px solid black;\">Pass Number</th>\n\t\t\t<th style=\"padding: 5px; border: 1px solid black;\">Sender ID</th>\n\t\t\t<th style=\"padding: 5px; border: 1px solid black;\">Receiver ID</th>\n\t\t\t<th style=\"padding: 5px; border: 1px solid black;\">x + Receiver IDs</th>\n\t\t</tr>\n\t\t<tr>\n\t\t\t<td style=\"padding: 5px; border: 1px solid black;\">&nbsp;</td>\n\t\t\t<td style=\"padding: 5px; border: 1px solid black;\">&nbsp;</td>\n\t\t\t<td style=\"padding: 5px; border: 1px solid black;\">&nbsp;</td>\n\t\t\t<td style=\"padding: 5px; border: 1px solid black;\">2</td>\n\t\t</tr>\n\t\t<tr>\n\t\t\t<td style=\"padding: 5px; border: 1px solid black;\">1</td>\n\t\t\t<td style=\"padding: 5px; border: 1px solid black;\">2</td>\n\t\t\t<td style=\"padding: 5px; border: 1px solid black;\">1</td>\n\t\t\t<td style=\"padding: 5px; border: 1px solid black;\">3</td>\n\t\t</tr>\n\t\t<tr>\n\t\t\t<td style=\"padding: 5px; border: 1px solid black;\">2</td>\n\t\t\t<td style=\"padding: 5px; border: 1px solid black;\">1</td>\n\t\t\t<td style=\"padding: 5px; border: 1px solid black;\">0</td>\n\t\t\t<td style=\"padding: 5px; border: 1px solid black;\">3</td>\n\t\t</tr>\n\t\t<tr>\n\t\t\t<td style=\"padding: 5px; border: 1px solid black;\">3</td>\n\t\t\t<td style=\"padding: 5px; border: 1px solid black;\">0</td>\n\t\t\t<td style=\"padding: 5px; border: 1px solid black;\">2</td>\n\t\t\t<td style=\"padding: 5px; border: 1px solid black;\">5</td>\n\t\t</tr>\n\t\t<tr>\n\t\t\t<td style=\"padding: 5px; border: 1px solid black;\">4</td>\n\t\t\t<td style=\"padding: 5px; border: 1px solid black;\">2</td>\n\t\t\t<td style=\"padding: 5px; border: 1px solid black;\">1</td>\n\t\t\t<td style=\"padding: 5px; border: 1px solid black;\">6</td>\n\t\t</tr>\n\t</tbody>\n</table>\n\n<pre>\n<strong>Input:</strong> receiver = [2,0,1], k = 4\n<strong>Output:</strong> 6\n<strong>Explanation:</strong> 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</pre>\n\n<p><strong class=\"example\">Example 2:</strong></p>\n\n<table border=\"1\" cellspacing=\"3\" style=\"border-collapse: separate; text-align: center;\">\n\t<tbody>\n\t\t<tr>\n\t\t\t<th style=\"padding: 5px; border: 1px solid black;\">Pass Number</th>\n\t\t\t<th style=\"padding: 5px; border: 1px solid black;\">Sender ID</th>\n\t\t\t<th style=\"padding: 5px; border: 1px solid black;\">Receiver ID</th>\n\t\t\t<th style=\"padding: 5px; border: 1px solid black;\">x + Receiver IDs</th>\n\t\t</tr>\n\t\t<tr>\n\t\t\t<td style=\"padding: 5px; border: 1px solid black;\">&nbsp;</td>\n\t\t\t<td style=\"padding: 5px; border: 1px solid black;\">&nbsp;</td>\n\t\t\t<td style=\"padding: 5px; border: 1px solid black;\">&nbsp;</td>\n\t\t\t<td style=\"padding: 5px; border: 1px solid black;\">4</td>\n\t\t</tr>\n\t\t<tr>\n\t\t\t<td style=\"padding: 5px; border: 1px solid black;\">1</td>\n\t\t\t<td style=\"padding: 5px; border: 1px solid black;\">4</td>\n\t\t\t<td style=\"padding: 5px; border: 1px solid black;\">3</td>\n\t\t\t<td style=\"padding: 5px; border: 1px solid black;\">7</td>\n\t\t</tr>\n\t\t<tr>\n\t\t\t<td style=\"padding: 5px; border: 1px solid black;\">2</td>\n\t\t\t<td style=\"padding: 5px; border: 1px solid black;\">3</td>\n\t\t\t<td style=\"padding: 5px; border: 1px solid black;\">2</td>\n\t\t\t<td style=\"padding: 5px; border: 1px solid black;\">9</td>\n\t\t</tr>\n\t\t<tr>\n\t\t\t<td style=\"padding: 5px; border: 1px solid black;\">3</td>\n\t\t\t<td style=\"padding: 5px; border: 1px solid black;\">2</td>\n\t\t\t<td style=\"padding: 5px; border: 1px solid black;\">1</td>\n\t\t\t<td style=\"padding: 5px; border: 1px solid black;\">10</td>\n\t\t</tr>\n\t</tbody>\n</table>\n\n<pre>\n<strong>Input:</strong> receiver = [1,1,1,2,3], k = 3\n<strong>Output:</strong> 10\n<strong>Explanation:</strong> 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</pre>\n\n<p>&nbsp;</p>\n<p><strong>Constraints:</strong></p>\n\n<ul>\n\t<li><code>1 &lt;= receiver.length == n &lt;= 10<sup>5</sup></code></li>\n\t<li><code>0 &lt;= receiver[i] &lt;= n - 1</code></li>\n\t<li><code>1 &lt;= k &lt;= 10<sup>10</sup></code></li>\n</ul>\n",
            "translatedTitle": "在传球游戏中最大化函数值",
            "translatedContent": "<p>给你一个长度为 <code>n</code>&nbsp;下标从 <strong>0</strong>&nbsp;开始的整数数组&nbsp;<code>receiver</code>&nbsp;和一个整数&nbsp;<code>k</code>&nbsp;。</p>\n\n<p>总共有&nbsp;<code>n</code>&nbsp;名玩家，玩家 <strong>编号</strong>&nbsp;互不相同，且为&nbsp;<code>[0, n - 1]</code>&nbsp;中的整数。这些玩家玩一个传球游戏，<code>receiver[i]</code>&nbsp;表示编号为 <code>i</code>&nbsp;的玩家会传球给编号为 <code>receiver[i]</code>&nbsp;的玩家。玩家可以传球给自己，也就是说&nbsp;<code>receiver[i]</code>&nbsp;可能等于&nbsp;<code>i</code>&nbsp;。</p>\n\n<p>你需要从 <code>n</code>&nbsp;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 <strong>恰好</strong>&nbsp;<code>k</code>&nbsp;次。</p>\n\n<p>如果选择编号为 <code>x</code>&nbsp;的玩家作为开始玩家，定义函数&nbsp;<code>f(x)</code>&nbsp;表示从编号为&nbsp;<code>x</code>&nbsp;的玩家开始，<code>k</code>&nbsp;次传球内所有接触过球玩家的编号之&nbsp;<strong>和</strong>&nbsp;，如果有玩家多次触球，则 <strong>累加多次</strong>&nbsp;。换句话说，&nbsp;<code>f(x) = x + receiver[x] + receiver[receiver[x]] + ... + receiver<sup>(k)</sup>[x]</code>&nbsp;。</p>\n\n<p>你的任务时选择开始玩家 <code>x</code>&nbsp;，目的是<strong>&nbsp;最大化</strong>&nbsp;<code>f(x)</code>&nbsp;。</p>\n\n<p>请你返回函数的 <strong>最大值</strong>&nbsp;。</p>\n\n<p><strong>注意：</strong><code>receiver</code>&nbsp;可能含有重复元素。</p>\n\n<p>&nbsp;</p>\n\n<p><strong class=\"example\">示例 1：</strong></p>\n\n<table border=\"1\" cellspacing=\"3\" style=\"border-collapse: separate; text-align: center;\">\n\t<tbody>\n\t\t<tr>\n\t\t\t<th style=\"padding: 5px; border: 1px solid black;\">传递次数</th>\n\t\t\t<th style=\"padding: 5px; border: 1px solid black;\">传球者编号</th>\n\t\t\t<th style=\"padding: 5px; border: 1px solid black;\">接球者编号</th>\n\t\t\t<th style=\"padding: 5px; border: 1px solid black;\">x + 所有接球者编号</th>\n\t\t</tr>\n\t\t<tr>\n\t\t\t<td style=\"padding: 5px; border: 1px solid black;\">&nbsp;</td>\n\t\t\t<td style=\"padding: 5px; border: 1px solid black;\">&nbsp;</td>\n\t\t\t<td style=\"padding: 5px; border: 1px solid black;\">&nbsp;</td>\n\t\t\t<td style=\"padding: 5px; border: 1px solid black;\">2</td>\n\t\t</tr>\n\t\t<tr>\n\t\t\t<td style=\"padding: 5px; border: 1px solid black;\">1</td>\n\t\t\t<td style=\"padding: 5px; border: 1px solid black;\">2</td>\n\t\t\t<td style=\"padding: 5px; border: 1px solid black;\">1</td>\n\t\t\t<td style=\"padding: 5px; border: 1px solid black;\">3</td>\n\t\t</tr>\n\t\t<tr>\n\t\t\t<td style=\"padding: 5px; border: 1px solid black;\">2</td>\n\t\t\t<td style=\"padding: 5px; border: 1px solid black;\">1</td>\n\t\t\t<td style=\"padding: 5px; border: 1px solid black;\">0</td>\n\t\t\t<td style=\"padding: 5px; border: 1px solid black;\">3</td>\n\t\t</tr>\n\t\t<tr>\n\t\t\t<td style=\"padding: 5px; border: 1px solid black;\">3</td>\n\t\t\t<td style=\"padding: 5px; border: 1px solid black;\">0</td>\n\t\t\t<td style=\"padding: 5px; border: 1px solid black;\">2</td>\n\t\t\t<td style=\"padding: 5px; border: 1px solid black;\">5</td>\n\t\t</tr>\n\t\t<tr>\n\t\t\t<td style=\"padding: 5px; border: 1px solid black;\">4</td>\n\t\t\t<td style=\"padding: 5px; border: 1px solid black;\">2</td>\n\t\t\t<td style=\"padding: 5px; border: 1px solid black;\">1</td>\n\t\t\t<td style=\"padding: 5px; border: 1px solid black;\">6</td>\n\t\t</tr>\n\t</tbody>\n</table>\n\n<p>&nbsp;</p>\n\n<pre>\n<b>输入：</b>receiver = [2,0,1], k = 4\n<b>输出：</b>6\n<b>解释：</b>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</pre>\n\n<p><strong class=\"example\">示例 2：</strong></p>\n\n<table border=\"1\" cellspacing=\"3\" style=\"border-collapse: separate; text-align: center;\">\n\t<tbody>\n\t\t<tr>\n\t\t\t<th style=\"padding: 5px; border: 1px solid black;\">传递次数</th>\n\t\t\t<th style=\"padding: 5px; border: 1px solid black;\">传球者编号</th>\n\t\t\t<th style=\"padding: 5px; border: 1px solid black;\">接球者编号</th>\n\t\t\t<th style=\"padding: 5px; border: 1px solid black;\">x + 所有接球者编号</th>\n\t\t</tr>\n\t\t<tr>\n\t\t\t<td style=\"padding: 5px; border: 1px solid black;\">&nbsp;</td>\n\t\t\t<td style=\"padding: 5px; border: 1px solid black;\">&nbsp;</td>\n\t\t\t<td style=\"padding: 5px; border: 1px solid black;\">&nbsp;</td>\n\t\t\t<td style=\"padding: 5px; border: 1px solid black;\">4</td>\n\t\t</tr>\n\t\t<tr>\n\t\t\t<td style=\"padding: 5px; border: 1px solid black;\">1</td>\n\t\t\t<td style=\"padding: 5px; border: 1px solid black;\">4</td>\n\t\t\t<td style=\"padding: 5px; border: 1px solid black;\">3</td>\n\t\t\t<td style=\"padding: 5px; border: 1px solid black;\">7</td>\n\t\t</tr>\n\t\t<tr>\n\t\t\t<td style=\"padding: 5px; border: 1px solid black;\">2</td>\n\t\t\t<td style=\"padding: 5px; border: 1px solid black;\">3</td>\n\t\t\t<td style=\"padding: 5px; border: 1px solid black;\">2</td>\n\t\t\t<td style=\"padding: 5px; border: 1px solid black;\">9</td>\n\t\t</tr>\n\t\t<tr>\n\t\t\t<td style=\"padding: 5px; border: 1px solid black;\">3</td>\n\t\t\t<td style=\"padding: 5px; border: 1px solid black;\">2</td>\n\t\t\t<td style=\"padding: 5px; border: 1px solid black;\">1</td>\n\t\t\t<td style=\"padding: 5px; border: 1px solid black;\">10</td>\n\t\t</tr>\n\t</tbody>\n</table>\n\n<p>&nbsp;</p>\n\n<pre>\n<b>输入：</b>receiver = [1,1,1,2,3], k = 3\n<b>输出：</b>10\n<b>解释：</b>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</pre>\n\n<p>&nbsp;</p>\n\n<p><strong>提示：</strong></p>\n\n<ul>\n\t<li><code>1 &lt;= receiver.length == n &lt;= 10<sup>5</sup></code></li>\n\t<li><code>0 &lt;= receiver[i] &lt;= n - 1</code></li>\n\t<li><code>1 &lt;= k &lt;= 10<sup>10</sup></code></li>\n</ul>\n",
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                "<div class=\"_1l1MA\">We can solve the problem using binary lifting.</div>",
                "<div class=\"_1l1MA\">For each player with id <code>x</code> and for every <code>i</code> in the range <code>[0, ceil(log<sub>2</sub>k)]</code>, we can determine the last receiver's id and compute the sum of player ids who receive the ball after <code>2<sup>i</sup></code> passes, starting from <code>x</code>.</div>",
                "<div class=\"_1l1MA\">Let <code>last_receiver[x][i] =</code> the last receiver's id after <code>2<sup>i</sup></code> passes, and <code>sum[x][i] =</code> the sum of player ids who receive the ball after <code>2<sup>i</sup></code> passes. For all <code>x</code> in the range <code>[0, n - 1]</code>, <code>last_receiver[x][0] = receiver[x]</code>, and <code>sum[x][0] = receiver[x]</code>.</div>",
                "<div class=\"_1l1MA\">Then for <code>i</code> in range <code>[1, ceil(log<sub>2</sub>k)]</code>, <code>last_receiver[x][i] = last_receiver[last_receiver[x][i - 1]][i - 1]</code> and <code>sum[x][i] = sum[x][i - 1] + sum[last_receiver[x][i - 1]][i - 1]</code>, for all <code>x</code> in the range <code>[0, n - 1]</code>.</div>",
                "<div class=\"_1l1MA\">Starting from each player id <code>x</code>, we can now go through the powers of <code>2</code> in the binary representation of <code>k</code> and make jumps corresponding to each power, using the pre-computed values, to compute <code>f(x)</code>.</div>",
                "<div class=\"_1l1MA\">The answer is the maximum <code>f(x)</code> from each player id.</div>"
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