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leetcode-cn/problem (Chinese)/使子序列的和等于目标的最少操作次数 [minimum-operations-to-form-subsequence-with-target-sum].html Normal file
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<p>给你一个下标从 <strong>0</strong>&nbsp;开始的数组&nbsp;<code>nums</code>&nbsp;，它包含 <strong>非负</strong>&nbsp;整数，且全部为 <code>2</code>&nbsp;的幂，同时给你一个整数&nbsp;<code>target</code>&nbsp;。</p>
<p>一次操作中，你必须对数组做以下修改：</p>
<ul>
<li>选择数组中一个元素&nbsp;<code>nums[i]</code>&nbsp;，满足&nbsp;<code>nums[i] &gt; 1</code>&nbsp;。</li>
<li>将&nbsp;<code>nums[i]</code>&nbsp;从数组中删除。</li>
<li>在 <code>nums</code>&nbsp;的 <strong>末尾</strong>&nbsp;添加 <strong>两个</strong>&nbsp;数，值都为&nbsp;<code>nums[i] / 2</code>&nbsp;。</li>
</ul>
<p>你的目标是让 <code>nums</code>&nbsp;的一个 <strong>子序列</strong>&nbsp;的元素和等于&nbsp;<code>target</code>&nbsp;，请你返回达成这一目标的 <strong>最少操作次数</strong>&nbsp;。如果无法得到这样的子序列，请你返回 <code>-1</code>&nbsp;。</p>
<p>数组中一个 <strong>子序列</strong>&nbsp;是通过删除原数组中一些元素，并且不改变剩余元素顺序得到的剩余数组。</p>
<p>&nbsp;</p>
<p><strong class="example">示例 1：</strong></p>
<pre>
<b>输入：</b>nums = [1,2,8], target = 7
<b>输出：</b>1
<b>解释：</b>第一次操作中，我们选择元素 nums[2] 。数组变为 nums = [1,2,4,4] 。
这时候，nums 包含子序列 [1,2,4] ，和为 7 。
无法通过更少的操作得到和为 7 的子序列。
</pre>
<p><strong class="example">示例 2：</strong></p>
<pre>
<b>输入：</b>nums = [1,32,1,2], target = 12
<b>输出：</b>2
<b>解释：</b>第一次操作中，我们选择元素 nums[1] 。数组变为 nums = [1,1,2,16,16] 。
第二次操作中，我们选择元素 nums[3] 。数组变为 nums = [1,1,2,16,8,8] 。
这时候，nums 包含子序列 [1,1,2,8] ，和为 12 。
无法通过更少的操作得到和为 12 的子序列。</pre>
<p><strong class="example">示例 3：</strong></p>
<pre>
<b>输入：</b>nums = [1,32,1], target = 35
<b>输出：</b>-1
<b>解释：</b>无法得到和为 35 的子序列。
</pre>
<p>&nbsp;</p>
<p><strong>提示：</strong></p>
<ul>
<li><code>1 &lt;= nums.length &lt;= 1000</code></li>
<li><code>1 &lt;= nums[i] &lt;= 2<sup>30</sup></code></li>
<li><code>nums</code>&nbsp;只包含非负整数，且均为 2 的幂。</li>
<li><code>1 &lt;= target &lt; 2<sup>31</sup></code></li>
</ul>

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leetcode-cn/problem (Chinese)/在传球游戏中最大化函数值 [maximize-value-of-function-in-a-ball-passing-game].html Normal file
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<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>
<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>
<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>
<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>
<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>
<p>请你返回函数的 <strong>最大值</strong>&nbsp;。</p>
<p><strong>注意：</strong><code>receiver</code>&nbsp;可能含有重复元素。</p>
<p>&nbsp;</p>
<p><strong class="example">示例 1：</strong></p>
<table border="1" cellspacing="3" style="border-collapse: separate; text-align: center;">
<tbody>
<tr>
<th style="padding: 5px; border: 1px solid black;">传递次数</th>
<th style="padding: 5px; border: 1px solid black;">传球者编号</th>
<th style="padding: 5px; border: 1px solid black;">接球者编号</th>
<th style="padding: 5px; border: 1px solid black;">x + 所有接球者编号</th>
</tr>
<tr>
<td style="padding: 5px; border: 1px solid black;">&nbsp;</td>
<td style="padding: 5px; border: 1px solid black;">&nbsp;</td>
<td style="padding: 5px; border: 1px solid black;">&nbsp;</td>
<td style="padding: 5px; border: 1px solid black;">2</td>
</tr>
<tr>
<td style="padding: 5px; border: 1px solid black;">1</td>
<td style="padding: 5px; border: 1px solid black;">2</td>
<td style="padding: 5px; border: 1px solid black;">1</td>
<td style="padding: 5px; border: 1px solid black;">3</td>
</tr>
<tr>
<td style="padding: 5px; border: 1px solid black;">2</td>
<td style="padding: 5px; border: 1px solid black;">1</td>
<td style="padding: 5px; border: 1px solid black;">0</td>
<td style="padding: 5px; border: 1px solid black;">3</td>
</tr>
<tr>
<td style="padding: 5px; border: 1px solid black;">3</td>
<td style="padding: 5px; border: 1px solid black;">0</td>
<td style="padding: 5px; border: 1px solid black;">2</td>
<td style="padding: 5px; border: 1px solid black;">5</td>
</tr>
<tr>
<td style="padding: 5px; border: 1px solid black;">4</td>
<td style="padding: 5px; border: 1px solid black;">2</td>
<td style="padding: 5px; border: 1px solid black;">1</td>
<td style="padding: 5px; border: 1px solid black;">6</td>
</tr>
</tbody>
</table>
<p>&nbsp;</p>
<pre>
<b>输入：</b>receiver = [2,0,1], k = 4
<b>输出：</b>6
<b>解释：</b>上表展示了从编号为 x = 2 开始的游戏过程。
从表中可知，f(2) 等于 6 。
6 是能得到最大的函数值。
所以输出为 6 。
</pre>
<p><strong class="example">示例 2：</strong></p>
<table border="1" cellspacing="3" style="border-collapse: separate; text-align: center;">
<tbody>
<tr>
<th style="padding: 5px; border: 1px solid black;">传递次数</th>
<th style="padding: 5px; border: 1px solid black;">传球者编号</th>
<th style="padding: 5px; border: 1px solid black;">接球者编号</th>
<th style="padding: 5px; border: 1px solid black;">x + 所有接球者编号</th>
</tr>
<tr>
<td style="padding: 5px; border: 1px solid black;">&nbsp;</td>
<td style="padding: 5px; border: 1px solid black;">&nbsp;</td>
<td style="padding: 5px; border: 1px solid black;">&nbsp;</td>
<td style="padding: 5px; border: 1px solid black;">4</td>
</tr>
<tr>
<td style="padding: 5px; border: 1px solid black;">1</td>
<td style="padding: 5px; border: 1px solid black;">4</td>
<td style="padding: 5px; border: 1px solid black;">3</td>
<td style="padding: 5px; border: 1px solid black;">7</td>
</tr>
<tr>
<td style="padding: 5px; border: 1px solid black;">2</td>
<td style="padding: 5px; border: 1px solid black;">3</td>
<td style="padding: 5px; border: 1px solid black;">2</td>
<td style="padding: 5px; border: 1px solid black;">9</td>
</tr>
<tr>
<td style="padding: 5px; border: 1px solid black;">3</td>
<td style="padding: 5px; border: 1px solid black;">2</td>
<td style="padding: 5px; border: 1px solid black;">1</td>
<td style="padding: 5px; border: 1px solid black;">10</td>
</tr>
</tbody>
</table>
<p>&nbsp;</p>
<pre>
<b>输入：</b>receiver = [1,1,1,2,3], k = 3
<b>输出：</b>10
<b>解释：</b>上表展示了从编号为 x = 4 开始的游戏过程。
从表中可知，f(4) 等于 10 。
10 是能得到最大的函数值。
所以输出为 10 。
</pre>
<p>&nbsp;</p>
<p><strong>提示：</strong></p>
<ul>
<li><code>1 &lt;= receiver.length == n &lt;= 10<sup>5</sup></code></li>
<li><code>0 &lt;= receiver[i] &lt;= n - 1</code></li>
<li><code>1 &lt;= k &lt;= 10<sup>10</sup></code></li>
</ul>

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leetcode-cn/problem (Chinese)/找出美丽数组的最小和 [find-the-minimum-possible-sum-of-a-beautiful-array].html Normal file
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<p>给你两个正整数：<code>n</code> 和 <code>target</code> 。</p>
<p>如果数组 <code>nums</code> 满足下述条件，则称其为 <strong>美丽数组</strong> 。</p>
<ul>
<li><code>nums.length == n</code>.</li>
<li><code>nums</code> 由两两互不相同的正整数组成。</li>
<li>在范围 <code>[0, n-1]</code> 内，<strong>不存在 </strong>两个 <strong>不同</strong> 下标 <code>i</code> 和 <code>j</code> ，使得 <code>nums[i] + nums[j] == target</code> 。</li>
</ul>
<p>返回符合条件的美丽数组所可能具备的 <strong>最小</strong> 和。</p>
<p>&nbsp;</p>
<p><strong>示例 1：</strong></p>
<pre>
<strong>输入：</strong>n = 2, target = 3
<strong>输出：</strong>4
<strong>解释：</strong>nums = [1,3] 是美丽数组。
- nums 的长度为 n = 2 。
- nums 由两两互不相同的正整数组成。
- 不存在两个不同下标 i 和 j ，使得 nums[i] + nums[j] == 3 。
可以证明 4 是符合条件的美丽数组所可能具备的最小和。</pre>
<p><strong>示例 2：</strong></p>
<pre>
<strong>输入：</strong>n = 3, target = 3
<strong>输出：</strong>8
<strong>解释：</strong>
nums = [1,3,4] 是美丽数组。
- nums 的长度为 n = 3 。
- nums 由两两互不相同的正整数组成。
- 不存在两个不同下标 i 和 j ，使得 nums[i] + nums[j] == 3 。
可以证明 8 是符合条件的美丽数组所可能具备的最小和。</pre>
<p><strong>示例 3：</strong></p>
<pre>
<strong>输入：</strong>n = 1, target = 1
<strong>输出：</strong>1
<strong>解释：</strong>nums = [1] 是美丽数组。
</pre>
<p>&nbsp;</p>
<p><strong>提示：</strong></p>
<ul>
<li><code>1 &lt;= n &lt;= 10<sup>5</sup></code></li>
<li><code>1 &lt;= target &lt;= 10<sup>5</sup></code></li>
</ul>

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leetcode-cn/problem (Chinese)/距离原点最远的点 [furthest-point-from-origin].html Normal file
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<p>给你一个长度为 <code>n</code> 的字符串 <code>moves</code> ，该字符串仅由字符 <code>'L'</code>、<code>'R'</code> 和 <code>'_'</code> 组成。字符串表示你在一条原点为 <code>0</code> 的数轴上的若干次移动。</p>
<p>你的初始位置就在原点（<code>0</code>），第 <code>i</code> 次移动过程中，你可以根据对应字符选择移动方向：</p>
<ul>
<li>如果 <code>moves[i] = 'L'</code> 或 <code>moves[i] = '_'</code> ，可以选择向左移动一个单位距离</li>
<li>如果 <code>moves[i] = 'R'</code> 或 <code>moves[i] = '_'</code> ，可以选择向右移动一个单位距离</li>
</ul>
<p>移动 <code>n</code> 次之后，请你找出可以到达的距离原点 <strong>最远</strong> 的点，并返回 <strong>从原点到这一点的距离</strong> 。</p>
<p>&nbsp;</p>
<p><strong>示例 1：</strong></p>
<pre>
<strong>输入：</strong>moves = "L_RL__R"
<strong>输出：</strong>3
<strong>解释：</strong>可以到达的距离原点 0 最远的点是 -3 ，移动的序列为 "LLRLLLR" 。
</pre>
<p><strong>示例 2：</strong></p>
<pre>
<strong>输入：</strong>moves = "_R__LL_"
<strong>输出：</strong>5
<strong>解释：</strong>可以到达的距离原点 0 最远的点是 -5 ，移动的序列为 "LRLLLLL" 。
</pre>
<p><strong>示例 3：</strong></p>
<pre>
<strong>输入：</strong>moves = "_______"
<strong>输出：</strong>7
<strong>解释：</strong>可以到达的距离原点 0 最远的点是 7 ，移动的序列为 "RRRRRRR" 。
</pre>
<p>&nbsp;</p>
<p><strong>提示：</strong></p>
<ul>
<li><code>1 &lt;= moves.length == n &lt;= 50</code></li>
<li><code>moves</code> 仅由字符 <code>'L'</code>、<code>'R'</code> 和 <code>'_'</code> 组成</li>
</ul>

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leetcode-cn/problem (English)/使子序列的和等于目标的最少操作次数(English) [minimum-operations-to-form-subsequence-with-target-sum].html Normal file
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<p>You are given a <strong>0-indexed</strong> array <code>nums</code> consisting of <strong>non-negative</strong> powers of <code>2</code>, and an integer <code>target</code>.</p>
<p>In one operation, you must apply the following changes to the array:</p>
<ul>
<li>Choose any element of the array <code>nums[i]</code> such that <code>nums[i] &gt; 1</code>.</li>
<li>Remove <code>nums[i]</code> from the array.</li>
<li>Add <strong>two</strong> occurrences of <code>nums[i] / 2</code> to the <strong>end</strong> of <code>nums</code>.</li>
</ul>
<p>Return the <em><strong>minimum number of operations</strong> you need to perform so that </em><code>nums</code><em> contains a <strong>subsequence</strong> whose elements sum to</em> <code>target</code>. If it is impossible to obtain such a subsequence, return <code>-1</code>.</p>
<p>A <strong>subsequence</strong> is an array that can be derived from another array by deleting some or no elements without changing the order of the remaining elements.</p>
<p>&nbsp;</p>
<p><strong class="example">Example 1:</strong></p>
<pre>
<strong>Input:</strong> nums = [1,2,8], target = 7
<strong>Output:</strong> 1
<strong>Explanation:</strong> In the first operation, we choose element nums[2]. The array becomes equal to nums = [1,2,4,4].
At this stage, nums contains the subsequence [1,2,4] which sums up to 7.
It can be shown that there is no shorter sequence of operations that results in a subsequnce that sums up to 7.
</pre>
<p><strong class="example">Example 2:</strong></p>
<pre>
<strong>Input:</strong> nums = [1,32,1,2], target = 12
<strong>Output:</strong> 2
<strong>Explanation:</strong> In the first operation, we choose element nums[1]. The array becomes equal to nums = [1,1,2,16,16].
In the second operation, we choose element nums[3]. The array becomes equal to nums = [1,1,2,16,8,8]
At this stage, nums contains the subsequence [1,1,2,8] which sums up to 12.
It can be shown that there is no shorter sequence of operations that results in a subsequence that sums up to 12.</pre>
<p><strong class="example">Example 3:</strong></p>
<pre>
<strong>Input:</strong> nums = [1,32,1], target = 35
<strong>Output:</strong> -1
<strong>Explanation:</strong> It can be shown that no sequence of operations results in a subsequence that sums up to 35.
</pre>
<p>&nbsp;</p>
<p><strong>Constraints:</strong></p>
<ul>
<li><code>1 &lt;= nums.length &lt;= 1000</code></li>
<li><code>1 &lt;= nums[i] &lt;= 2<sup>30</sup></code></li>
<li><code>nums</code> consists only of non-negative powers of two.</li>
<li><code>1 &lt;= target &lt; 2<sup>31</sup></code></li>
</ul>

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leetcode-cn/problem (English)/在传球游戏中最大化函数值(English) [maximize-value-of-function-in-a-ball-passing-game].html Normal file
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<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>
<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>
<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>
<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>
<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>
<p>Return <em>an integer denoting the <strong>maximum</strong> value of the function.</em></p>
<p><strong>Note:</strong> <code>receiver</code> may contain duplicates.</p>
<p>&nbsp;</p>
<p><strong class="example">Example 1:</strong></p>
<table border="1" cellspacing="3" style="border-collapse: separate; text-align: center;">
<tbody>
<tr>
<th style="padding: 5px; border: 1px solid black;">Pass Number</th>
<th style="padding: 5px; border: 1px solid black;">Sender ID</th>
<th style="padding: 5px; border: 1px solid black;">Receiver ID</th>
<th style="padding: 5px; border: 1px solid black;">x + Receiver IDs</th>
</tr>
<tr>
<td style="padding: 5px; border: 1px solid black;">&nbsp;</td>
<td style="padding: 5px; border: 1px solid black;">&nbsp;</td>
<td style="padding: 5px; border: 1px solid black;">&nbsp;</td>
<td style="padding: 5px; border: 1px solid black;">2</td>
</tr>
<tr>
<td style="padding: 5px; border: 1px solid black;">1</td>
<td style="padding: 5px; border: 1px solid black;">2</td>
<td style="padding: 5px; border: 1px solid black;">1</td>
<td style="padding: 5px; border: 1px solid black;">3</td>
</tr>
<tr>
<td style="padding: 5px; border: 1px solid black;">2</td>
<td style="padding: 5px; border: 1px solid black;">1</td>
<td style="padding: 5px; border: 1px solid black;">0</td>
<td style="padding: 5px; border: 1px solid black;">3</td>
</tr>
<tr>
<td style="padding: 5px; border: 1px solid black;">3</td>
<td style="padding: 5px; border: 1px solid black;">0</td>
<td style="padding: 5px; border: 1px solid black;">2</td>
<td style="padding: 5px; border: 1px solid black;">5</td>
</tr>
<tr>
<td style="padding: 5px; border: 1px solid black;">4</td>
<td style="padding: 5px; border: 1px solid black;">2</td>
<td style="padding: 5px; border: 1px solid black;">1</td>
<td style="padding: 5px; border: 1px solid black;">6</td>
</tr>
</tbody>
</table>
<pre>
<strong>Input:</strong> receiver = [2,0,1], k = 4
<strong>Output:</strong> 6
<strong>Explanation:</strong> The table above shows a simulation of the game starting with the player having id x = 2.
From the table, f(2) is equal to 6.
It can be shown that 6 is the maximum achievable value of the function.
Hence, the output is 6.
</pre>
<p><strong class="example">Example 2:</strong></p>
<table border="1" cellspacing="3" style="border-collapse: separate; text-align: center;">
<tbody>
<tr>
<th style="padding: 5px; border: 1px solid black;">Pass Number</th>
<th style="padding: 5px; border: 1px solid black;">Sender ID</th>
<th style="padding: 5px; border: 1px solid black;">Receiver ID</th>
<th style="padding: 5px; border: 1px solid black;">x + Receiver IDs</th>
</tr>
<tr>
<td style="padding: 5px; border: 1px solid black;">&nbsp;</td>
<td style="padding: 5px; border: 1px solid black;">&nbsp;</td>
<td style="padding: 5px; border: 1px solid black;">&nbsp;</td>
<td style="padding: 5px; border: 1px solid black;">4</td>
</tr>
<tr>
<td style="padding: 5px; border: 1px solid black;">1</td>
<td style="padding: 5px; border: 1px solid black;">4</td>
<td style="padding: 5px; border: 1px solid black;">3</td>
<td style="padding: 5px; border: 1px solid black;">7</td>
</tr>
<tr>
<td style="padding: 5px; border: 1px solid black;">2</td>
<td style="padding: 5px; border: 1px solid black;">3</td>
<td style="padding: 5px; border: 1px solid black;">2</td>
<td style="padding: 5px; border: 1px solid black;">9</td>
</tr>
<tr>
<td style="padding: 5px; border: 1px solid black;">3</td>
<td style="padding: 5px; border: 1px solid black;">2</td>
<td style="padding: 5px; border: 1px solid black;">1</td>
<td style="padding: 5px; border: 1px solid black;">10</td>
</tr>
</tbody>
</table>
<pre>
<strong>Input:</strong> receiver = [1,1,1,2,3], k = 3
<strong>Output:</strong> 10
<strong>Explanation:</strong> The table above shows a simulation of the game starting with the player having id x = 4.
From the table, f(4) is equal to 10.
It can be shown that 10 is the maximum achievable value of the function.
Hence, the output is 10.
</pre>
<p>&nbsp;</p>
<p><strong>Constraints:</strong></p>
<ul>
<li><code>1 &lt;= receiver.length == n &lt;= 10<sup>5</sup></code></li>
<li><code>0 &lt;= receiver[i] &lt;= n - 1</code></li>
<li><code>1 &lt;= k &lt;= 10<sup>10</sup></code></li>
</ul>

52
leetcode-cn/problem (English)/找出美丽数组的最小和(English) [find-the-minimum-possible-sum-of-a-beautiful-array].html Normal file
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<p>You are given positive integers <code>n</code> and <code>target</code>.</p>
<p>An array <code>nums</code> is <strong>beautiful</strong> if it meets the following conditions:</p>
<ul>
<li><code>nums.length == n</code>.</li>
<li><code>nums</code> consists of pairwise <strong>distinct</strong> <strong>positive</strong> integers.</li>
<li>There doesn&#39;t exist two <strong>distinct</strong> indices, <code>i</code> and <code>j</code>, in the range <code>[0, n - 1]</code>, such that <code>nums[i] + nums[j] == target</code>.</li>
</ul>
<p>Return <em>the <strong>minimum</strong> possible sum that a beautiful array could have</em>.</p>
<p>&nbsp;</p>
<p><strong class="example">Example 1:</strong></p>
<pre>
<strong>Input:</strong> n = 2, target = 3
<strong>Output:</strong> 4
<strong>Explanation:</strong> We can see that nums = [1,3] is beautiful.
- The array nums has length n = 2.
- The array nums consists of pairwise distinct positive integers.
- There doesn&#39;t exist two distinct indices, i and j, with nums[i] + nums[j] == 3.
It can be proven that 4 is the minimum possible sum that a beautiful array could have.
</pre>
<p><strong class="example">Example 2:</strong></p>
<pre>
<strong>Input:</strong> n = 3, target = 3
<strong>Output:</strong> 8
<strong>Explanation:</strong> We can see that nums = [1,3,4] is beautiful.
- The array nums has length n = 3.
- The array nums consists of pairwise distinct positive integers.
- There doesn&#39;t exist two distinct indices, i and j, with nums[i] + nums[j] == 3.
It can be proven that 8 is the minimum possible sum that a beautiful array could have.
</pre>
<p><strong class="example">Example 3:</strong></p>
<pre>
<strong>Input:</strong> n = 1, target = 1
<strong>Output:</strong> 1
<strong>Explanation:</strong> We can see, that nums = [1] is beautiful.
</pre>
<p>&nbsp;</p>
<p><strong>Constraints:</strong></p>
<ul>
<li><code>1 &lt;= n &lt;= 10<sup>5</sup></code></li>
<li><code>1 &lt;= target &lt;= 10<sup>5</sup></code></li>
</ul>

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<p>You are given a string <code>moves</code> of length <code>n</code> consisting only of characters <code>&#39;L&#39;</code>, <code>&#39;R&#39;</code>, and <code>&#39;_&#39;</code>. The string represents your movement on a number line starting from the origin <code>0</code>.</p>
<p>In the <code>i<sup>th</sup></code> move, you can choose one of the following directions:</p>
<ul>
<li>move to the left if <code>moves[i] = &#39;L&#39;</code> or <code>moves[i] = &#39;_&#39;</code></li>
<li>move to the right if <code>moves[i] = &#39;R&#39;</code> or <code>moves[i] = &#39;_&#39;</code></li>
</ul>
<p>Return <em>the <strong>distance from the origin</strong> of the <strong>furthest</strong> point you can get to after </em><code>n</code><em> moves</em>.</p>
<p>&nbsp;</p>
<p><strong class="example">Example 1:</strong></p>
<pre>
<strong>Input:</strong> moves = &quot;L_RL__R&quot;
<strong>Output:</strong> 3
<strong>Explanation:</strong> The furthest point we can reach from the origin 0 is point -3 through the following sequence of moves &quot;LLRLLLR&quot;.
</pre>
<p><strong class="example">Example 2:</strong></p>
<pre>
<strong>Input:</strong> moves = &quot;_R__LL_&quot;
<strong>Output:</strong> 5
<strong>Explanation:</strong> The furthest point we can reach from the origin 0 is point -5 through the following sequence of moves &quot;LRLLLLL&quot;.
</pre>
<p><strong class="example">Example 3:</strong></p>
<pre>
<strong>Input:</strong> moves = &quot;_______&quot;
<strong>Output:</strong> 7
<strong>Explanation:</strong> The furthest point we can reach from the origin 0 is point 7 through the following sequence of moves &quot;RRRRRRR&quot;.
</pre>
<p>&nbsp;</p>
<p><strong>Constraints:</strong></p>
<ul>
<li><code>1 &lt;= moves.length == n &lt;= 50</code></li>
<li><code>moves</code> consists only of characters <code>&#39;L&#39;</code>, <code>&#39;R&#39;</code> and <code>&#39;_&#39;</code>.</li>
</ul>

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<p>You are given positive integers <code>n</code> and <code>target</code>.</p>
<p>An array <code>nums</code> is <strong>beautiful</strong> if it meets the following conditions:</p>
<ul>
<li><code>nums.length == n</code>.</li>
<li><code>nums</code> consists of pairwise <strong>distinct</strong> <strong>positive</strong> integers.</li>
<li>There doesn&#39;t exist two <strong>distinct</strong> indices, <code>i</code> and <code>j</code>, in the range <code>[0, n - 1]</code>, such that <code>nums[i] + nums[j] == target</code>.</li>
</ul>
<p>Return <em>the <strong>minimum</strong> possible sum that a beautiful array could have</em>.</p>
<p>&nbsp;</p>
<p><strong class="example">Example 1:</strong></p>
<pre>
<strong>Input:</strong> n = 2, target = 3
<strong>Output:</strong> 4
<strong>Explanation:</strong> We can see that nums = [1,3] is beautiful.
- The array nums has length n = 2.
- The array nums consists of pairwise distinct positive integers.
- There doesn&#39;t exist two distinct indices, i and j, with nums[i] + nums[j] == 3.
It can be proven that 4 is the minimum possible sum that a beautiful array could have.
</pre>
<p><strong class="example">Example 2:</strong></p>
<pre>
<strong>Input:</strong> n = 3, target = 3
<strong>Output:</strong> 8
<strong>Explanation:</strong> We can see that nums = [1,3,4] is beautiful.
- The array nums has length n = 3.
- The array nums consists of pairwise distinct positive integers.
- There doesn&#39;t exist two distinct indices, i and j, with nums[i] + nums[j] == 3.
It can be proven that 8 is the minimum possible sum that a beautiful array could have.
</pre>
<p><strong class="example">Example 3:</strong></p>
<pre>
<strong>Input:</strong> n = 1, target = 1
<strong>Output:</strong> 1
<strong>Explanation:</strong> We can see, that nums = [1] is beautiful.
</pre>
<p>&nbsp;</p>
<p><strong>Constraints:</strong></p>
<ul>
<li><code>1 &lt;= n &lt;= 10<sup>5</sup></code></li>
<li><code>1 &lt;= target &lt;= 10<sup>5</sup></code></li>
</ul>

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<p>You are given a string <code>moves</code> of length <code>n</code> consisting only of characters <code>&#39;L&#39;</code>, <code>&#39;R&#39;</code>, and <code>&#39;_&#39;</code>. The string represents your movement on a number line starting from the origin <code>0</code>.</p>
<p>In the <code>i<sup>th</sup></code> move, you can choose one of the following directions:</p>
<ul>
<li>move to the left if <code>moves[i] = &#39;L&#39;</code> or <code>moves[i] = &#39;_&#39;</code></li>
<li>move to the right if <code>moves[i] = &#39;R&#39;</code> or <code>moves[i] = &#39;_&#39;</code></li>
</ul>
<p>Return <em>the <strong>distance from the origin</strong> of the <strong>furthest</strong> point you can get to after </em><code>n</code><em> moves</em>.</p>
<p>&nbsp;</p>
<p><strong class="example">Example 1:</strong></p>
<pre>
<strong>Input:</strong> moves = &quot;L_RL__R&quot;
<strong>Output:</strong> 3
<strong>Explanation:</strong> The furthest point we can reach from the origin 0 is point -3 through the following sequence of moves &quot;LLRLLLR&quot;.
</pre>
<p><strong class="example">Example 2:</strong></p>
<pre>
<strong>Input:</strong> moves = &quot;_R__LL_&quot;
<strong>Output:</strong> 5
<strong>Explanation:</strong> The furthest point we can reach from the origin 0 is point -5 through the following sequence of moves &quot;LRLLLLL&quot;.
</pre>
<p><strong class="example">Example 3:</strong></p>
<pre>
<strong>Input:</strong> moves = &quot;_______&quot;
<strong>Output:</strong> 7
<strong>Explanation:</strong> The furthest point we can reach from the origin 0 is point 7 through the following sequence of moves &quot;RRRRRRR&quot;.
</pre>
<p>&nbsp;</p>
<p><strong>Constraints:</strong></p>
<ul>
<li><code>1 &lt;= moves.length == n &lt;= 50</code></li>
<li><code>moves</code> consists only of characters <code>&#39;L&#39;</code>, <code>&#39;R&#39;</code> and <code>&#39;_&#39;</code>.</li>
</ul>

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<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>
<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>
<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>
<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>
<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>
<p>Return <em>an integer denoting the <strong>maximum</strong> value of the function.</em></p>
<p><strong>Note:</strong> <code>receiver</code> may contain duplicates.</p>
<p>&nbsp;</p>
<p><strong class="example">Example 1:</strong></p>
<table border="1" cellspacing="3" style="border-collapse: separate; text-align: center;">
<tbody>
<tr>
<th style="padding: 5px; border: 1px solid black;">Pass Number</th>
<th style="padding: 5px; border: 1px solid black;">Sender ID</th>
<th style="padding: 5px; border: 1px solid black;">Receiver ID</th>
<th style="padding: 5px; border: 1px solid black;">x + Receiver IDs</th>
</tr>
<tr>
<td style="padding: 5px; border: 1px solid black;">&nbsp;</td>
<td style="padding: 5px; border: 1px solid black;">&nbsp;</td>
<td style="padding: 5px; border: 1px solid black;">&nbsp;</td>
<td style="padding: 5px; border: 1px solid black;">2</td>
</tr>
<tr>
<td style="padding: 5px; border: 1px solid black;">1</td>
<td style="padding: 5px; border: 1px solid black;">2</td>
<td style="padding: 5px; border: 1px solid black;">1</td>
<td style="padding: 5px; border: 1px solid black;">3</td>
</tr>
<tr>
<td style="padding: 5px; border: 1px solid black;">2</td>
<td style="padding: 5px; border: 1px solid black;">1</td>
<td style="padding: 5px; border: 1px solid black;">0</td>
<td style="padding: 5px; border: 1px solid black;">3</td>
</tr>
<tr>
<td style="padding: 5px; border: 1px solid black;">3</td>
<td style="padding: 5px; border: 1px solid black;">0</td>
<td style="padding: 5px; border: 1px solid black;">2</td>
<td style="padding: 5px; border: 1px solid black;">5</td>
</tr>
<tr>
<td style="padding: 5px; border: 1px solid black;">4</td>
<td style="padding: 5px; border: 1px solid black;">2</td>
<td style="padding: 5px; border: 1px solid black;">1</td>
<td style="padding: 5px; border: 1px solid black;">6</td>
</tr>
</tbody>
</table>
<pre>
<strong>Input:</strong> receiver = [2,0,1], k = 4
<strong>Output:</strong> 6
<strong>Explanation:</strong> The table above shows a simulation of the game starting with the player having id x = 2.
From the table, f(2) is equal to 6.
It can be shown that 6 is the maximum achievable value of the function.
Hence, the output is 6.
</pre>
<p><strong class="example">Example 2:</strong></p>
<table border="1" cellspacing="3" style="border-collapse: separate; text-align: center;">
<tbody>
<tr>
<th style="padding: 5px; border: 1px solid black;">Pass Number</th>
<th style="padding: 5px; border: 1px solid black;">Sender ID</th>
<th style="padding: 5px; border: 1px solid black;">Receiver ID</th>
<th style="padding: 5px; border: 1px solid black;">x + Receiver IDs</th>
</tr>
<tr>
<td style="padding: 5px; border: 1px solid black;">&nbsp;</td>
<td style="padding: 5px; border: 1px solid black;">&nbsp;</td>
<td style="padding: 5px; border: 1px solid black;">&nbsp;</td>
<td style="padding: 5px; border: 1px solid black;">4</td>
</tr>
<tr>
<td style="padding: 5px; border: 1px solid black;">1</td>
<td style="padding: 5px; border: 1px solid black;">4</td>
<td style="padding: 5px; border: 1px solid black;">3</td>
<td style="padding: 5px; border: 1px solid black;">7</td>
</tr>
<tr>
<td style="padding: 5px; border: 1px solid black;">2</td>
<td style="padding: 5px; border: 1px solid black;">3</td>
<td style="padding: 5px; border: 1px solid black;">2</td>
<td style="padding: 5px; border: 1px solid black;">9</td>
</tr>
<tr>
<td style="padding: 5px; border: 1px solid black;">3</td>
<td style="padding: 5px; border: 1px solid black;">2</td>
<td style="padding: 5px; border: 1px solid black;">1</td>
<td style="padding: 5px; border: 1px solid black;">10</td>
</tr>
</tbody>
</table>
<pre>
<strong>Input:</strong> receiver = [1,1,1,2,3], k = 3
<strong>Output:</strong> 10
<strong>Explanation:</strong> The table above shows a simulation of the game starting with the player having id x = 4.
From the table, f(4) is equal to 10.
It can be shown that 10 is the maximum achievable value of the function.
Hence, the output is 10.
</pre>
<p>&nbsp;</p>
<p><strong>Constraints:</strong></p>
<ul>
<li><code>1 &lt;= receiver.length == n &lt;= 10<sup>5</sup></code></li>
<li><code>0 &lt;= receiver[i] &lt;= n - 1</code></li>
<li><code>1 &lt;= k &lt;= 10<sup>10</sup></code></li>
</ul>

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<p>You are given a <strong>0-indexed</strong> array <code>nums</code> consisting of <strong>non-negative</strong> powers of <code>2</code>, and an integer <code>target</code>.</p>
<p>In one operation, you must apply the following changes to the array:</p>
<ul>
<li>Choose any element of the array <code>nums[i]</code> such that <code>nums[i] &gt; 1</code>.</li>
<li>Remove <code>nums[i]</code> from the array.</li>
<li>Add <strong>two</strong> occurrences of <code>nums[i] / 2</code> to the <strong>end</strong> of <code>nums</code>.</li>
</ul>
<p>Return the <em><strong>minimum number of operations</strong> you need to perform so that </em><code>nums</code><em> contains a <strong>subsequence</strong> whose elements sum to</em> <code>target</code>. If it is impossible to obtain such a subsequence, return <code>-1</code>.</p>
<p>A <strong>subsequence</strong> is an array that can be derived from another array by deleting some or no elements without changing the order of the remaining elements.</p>
<p>&nbsp;</p>
<p><strong class="example">Example 1:</strong></p>
<pre>
<strong>Input:</strong> nums = [1,2,8], target = 7
<strong>Output:</strong> 1
<strong>Explanation:</strong> In the first operation, we choose element nums[2]. The array becomes equal to nums = [1,2,4,4].
At this stage, nums contains the subsequence [1,2,4] which sums up to 7.
It can be shown that there is no shorter sequence of operations that results in a subsequnce that sums up to 7.
</pre>
<p><strong class="example">Example 2:</strong></p>
<pre>
<strong>Input:</strong> nums = [1,32,1,2], target = 12
<strong>Output:</strong> 2
<strong>Explanation:</strong> In the first operation, we choose element nums[1]. The array becomes equal to nums = [1,1,2,16,16].
In the second operation, we choose element nums[3]. The array becomes equal to nums = [1,1,2,16,8,8]
At this stage, nums contains the subsequence [1,1,2,8] which sums up to 12.
It can be shown that there is no shorter sequence of operations that results in a subsequence that sums up to 12.</pre>
<p><strong class="example">Example 3:</strong></p>
<pre>
<strong>Input:</strong> nums = [1,32,1], target = 35
<strong>Output:</strong> -1
<strong>Explanation:</strong> It can be shown that no sequence of operations results in a subsequence that sums up to 35.
</pre>
<p>&nbsp;</p>
<p><strong>Constraints:</strong></p>
<ul>
<li><code>1 &lt;= nums.length &lt;= 1000</code></li>
<li><code>1 &lt;= nums[i] &lt;= 2<sup>30</sup></code></li>
<li><code>nums</code> consists only of non-negative powers of two.</li>
<li><code>1 &lt;= target &lt; 2<sup>31</sup></code></li>
</ul>