maximum-points-in-an-archery-competition

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Algorithms

Alice and Bob are opponents in an archery competition. The competition has set the following rules:

<ol>
	<li>Alice first shoots <code>numArrows</code> arrows and then Bob shoots <code>numArrows</code> arrows.</li>
	<li>The points are then calculated as follows:
	<ol>
		<li>The target has integer scoring sections ranging from <code>0</code> to <code>11</code> inclusive.</li>
		<li>For each section of the target with score <code>k</code> (in between <code>0</code> to <code>11</code>), say Alice and Bob have shot <code>ak</code> and <code>bk</code> arrows on that section respectively. If <code>ak &gt;= bk</code>, then Alice takes <code>k</code> points. If <code>ak &lt; bk</code>, then Bob takes <code>k</code> points.</li>
		<li>However, if <code>ak == bk == 0</code>, then nobody takes <code>k</code> points.</li>
	</ol>
	</li>
</ol>

<ul>
	<li>
	For example, if Alice and Bob both shot <code>2</code> arrows on the section with score <code>11</code>, then Alice takes <code>11</code> points. On the other hand, if Alice shot <code>0</code> arrows on the section with score <code>11</code> and Bob shot <code>2</code> arrows on that same section, then Bob takes <code>11</code> points.
	</li>
</ul>

You are given the integer <code>numArrows</code> and an integer array <code>aliceArrows</code> of size <code>12</code>, which represents the number of arrows Alice shot on each scoring section from <code>0</code> to <code>11</code>. Now, Bob wants to maximize the total number of points he can obtain.

Return the array <code>bobArrows</code> which represents the number of arrows Bob shot on each scoring section from <code>0</code> to <code>11</code>. The sum of the values in <code>bobArrows</code> should equal <code>numArrows</code>.

If there are multiple ways for Bob to earn the maximum total points, return any one of them.

&nbsp;
Example 1:
<img alt="" src="https://assets.leetcode.com/uploads/2022/02/24/ex1.jpg" style="width: 600px; height: 120px;" />
<pre>
Input: numArrows = 9, aliceArrows = [1,1,0,1,0,0,2,1,0,1,2,0]
Output: [0,0,0,0,1,1,0,0,1,2,3,1]
Explanation: The table above shows how the competition is scored. 
Bob earns a total point of 4 + 5 + 8 + 9 + 10 + 11 = 47.
It can be shown that Bob cannot obtain a score higher than 47 points.
</pre>

Example 2:
<img alt="" src="https://assets.leetcode.com/uploads/2022/02/24/ex2new.jpg" style="width: 600px; height: 117px;" />
<pre>
Input: numArrows = 3, aliceArrows = [0,0,1,0,0,0,0,0,0,0,0,2]
Output: [0,0,0,0,0,0,0,0,1,1,1,0]
Explanation: The table above shows how the competition is scored.
Bob earns a total point of 8 + 9 + 10 = 27.
It can be shown that Bob cannot obtain a score higher than 27 points.
</pre>

&nbsp;
Constraints:

<ul>
	<li><code>1 &lt;= numArrows &lt;= 105</code></li>
	<li><code>aliceArrows.length == bobArrows.length == 12</code></li>
	<li><code>0 &lt;= aliceArrows[i], bobArrows[i] &lt;= numArrows</code></li>
	<li><code>sum(aliceArrows[i]) == numArrows</code></li>
</ul>

maximumBobPoints

Maximum Points in an Archery Competition

Maximum Product of the Length of Two Palindromic Subsequences

maximum-product-of-the-length-of-two-palindromic-subsequences

两个回文子序列长度的最大乘积

Bit Manipulation

Array

Backtracking

Enumeration

Alice 和 Bob 是一场射箭比赛中的对手。比赛规则如下：

<ol>
	<li>Alice 先射 <code>numArrows</code> 支箭，然后 Bob 也射 <code>numArrows</code> 支箭。</li>
	<li>分数按下述规则计算：
	<ol>
		<li>箭靶有若干整数计分区域，范围从 <code>0</code> 到 <code>11</code> （含 <code>0</code> 和 <code>11</code>）。</li>
		<li>箭靶上每个区域都对应一个得分 <code>k</code>（范围是 <code>0</code> 到 <code>11</code>），Alice 和 Bob 分别在得分 <code>k</code>&nbsp;区域射中&nbsp;<code>ak</code> 和 <code>bk</code> 支箭。如果 <code>ak &gt;= bk</code> ，那么 Alice 得 <code>k</code> 分。如果 <code>ak &lt; bk</code> ，则 Bob 得 <code>k</code> 分</li>
		<li>如果 <code>ak == bk == 0</code> ，那么无人得到 <code>k</code> 分。</li>
	</ol>
	</li>
</ol>

<ul>
	<li>
	例如，Alice 和 Bob 都向计分为 <code>11</code> 的区域射 <code>2</code> 支箭，那么 Alice 得 <code>11</code> 分。如果 Alice 向计分为 <code>11</code> 的区域射 <code>0</code> 支箭，但 Bob 向同一个区域射 <code>2</code> 支箭，那么 Bob 得&nbsp;<code>11</code> 分。
	</li>
</ul>

给你整数 <code>numArrows</code> 和一个长度为 <code>12</code> 的整数数组 <code>aliceArrows</code> ，该数组表示 Alice 射中&nbsp;<code>0</code> 到 <code>11</code> 每个计分区域的箭数量。现在，Bob 想要尽可能 最大化 他所能获得的总分。

返回数组 <code>bobArrows</code> ，该数组表示 Bob 射中&nbsp;<code>0</code> 到 <code>11</code> 每个 计分区域的箭数量。且 <code>bobArrows</code> 的总和应当等于 <code>numArrows</code> 。

如果存在多种方法都可以使 Bob 获得最大总分，返回其中 任意一种 即可。

&nbsp;

示例 1：

<img alt="" src="https://pic.leetcode-cn.com/1647744752-kQKrXw-image.png" style="width: 600px; height: 120px;" />

<pre>
输入：numArrows = 9, aliceArrows = [1,1,0,1,0,0,2,1,0,1,2,0]
输出：[0,0,0,0,1,1,0,0,1,2,3,1]
解释：上表显示了比赛得分情况。
Bob 获得总分 4 + 5 + 8 + 9 + 10 + 11 = 47 。
可以证明 Bob 无法获得比 47 更高的分数。
</pre>

示例 2：

<img alt="" src="https://pic.leetcode-cn.com/1647744785-cMHzaC-image.png" style="width: 600px; height: 117px;" />

<pre>
输入：numArrows = 3, aliceArrows = [0,0,1,0,0,0,0,0,0,0,0,2]
输出：[0,0,0,0,0,0,0,0,1,1,1,0]
解释：上表显示了比赛得分情况。
Bob 获得总分 8 + 9 + 10 = 27 。
可以证明 Bob 无法获得比 27 更高的分数。
</pre>

&nbsp;

提示：

<ul>
	<li><code>1 &lt;= numArrows &lt;= 105</code></li>
	<li><code>aliceArrows.length == bobArrows.length == 12</code></li>
	<li><code>0 &lt;= aliceArrows[i], bobArrows[i] &lt;= numArrows</code></li>
	<li><code>sum(aliceArrows[i]) == numArrows</code></li>
</ul>