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

<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> <strong>inclusive</strong>.</li>
		<li>For <strong>each</strong> 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>a<sub>k</sub></code> and <code>b<sub>k</sub></code> arrows on that section respectively. If <code>a<sub>k</sub> &gt;= b<sub>k</sub></code>, then Alice takes <code>k</code> points. If <code>a<sub>k</sub> &lt; b<sub>k</sub></code>, then Bob takes <code>k</code> points.</li>
		<li>However, if <code>a<sub>k</sub> == b<sub>k</sub> == 0</code>, then <strong>nobody</strong> takes <code>k</code> points.</li>
	</ol>
	</li>
</ol>

<ul>
	<li>
	<p>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.</p>
	</li>
</ul>

<p>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 <strong>maximize</strong> the total number of points he can obtain.</p>

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

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

<p>&nbsp;</p>
<p><strong class="example">Example 1:</strong></p>
<img alt="" src="https://assets.leetcode.com/uploads/2022/02/24/ex1.jpg" style="width: 600px; height: 120px;" />
<pre>
<strong>Input:</strong> numArrows = 9, aliceArrows = [1,1,0,1,0,0,2,1,0,1,2,0]
<strong>Output:</strong> [0,0,0,0,1,1,0,0,1,2,3,1]
<strong>Explanation:</strong> 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>

<p><strong class="example">Example 2:</strong></p>
<img alt="" src="https://assets.leetcode.com/uploads/2022/02/24/ex2new.jpg" style="width: 600px; height: 117px;" />
<pre>
<strong>Input:</strong> numArrows = 3, aliceArrows = [0,0,1,0,0,0,0,0,0,0,0,2]
<strong>Output:</strong> [0,0,0,0,0,0,0,0,1,1,1,0]
<strong>Explanation:</strong> 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>

<p>&nbsp;</p>
<p><strong>Constraints:</strong></p>

<ul>
	<li><code>1 &lt;= numArrows &lt;= 10<sup>5</sup></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>

class Solution:
    def maximumBobPoints(self, numArrows: int, aliceArrows: List[int]) -> List[int]:
        # DP, table to store the max score for current section and number of arrows
        dp = [[-1 for _ in range(numArrows + 1)] for _ in range(12)]
        # minArrowsForBobToWin stores bob arrows to win in each section
        # bobMaxScore stores result bob arrows for max score
        minArrowsForBobToWin, bobMaxScore = [aliceArrows[i] + 1 for i in range(len(aliceArrows))], [0 for _ in range(12)]
        
        # Helper function to find max score for each number of arrows upto the actual number of arrows
        # for all the sections upto a given section
        def findMaxScore(section: int, numArrows: int, dp: List[List[int]], minArrowsForBobToWin: List[int]) -> int:
            # base case
            # return score 0 for arrows <= 0 or section <= 0
            if numArrows <= 0 or section <= 0:
                return 0
            # returning precalculated result
            if dp[section][numArrows] != -1:
                return dp[section][numArrows]
            # Variables to store the scores if the current section is taken or not
            taken, notTaken = 0, 0
            # if remaining arrows >= minArrowsForBobToWin[section] then we can take current section 
            # and call recursively
            if numArrows >= minArrowsForBobToWin[section]:
                taken =  section + findMaxScore(section - 1, numArrows - minArrowsForBobToWin[section], dp, minArrowsForBobToWin)
            # recursive call without taking current section
            notTaken = findMaxScore(section - 1, numArrows, dp, minArrowsForBobToWin)
            # get max score from taken and not taken case
            dp[section][numArrows] = max(taken, notTaken)
            return dp[section][numArrows]
        
        # find maximum score bob can obtain
        findMaxScore(11, numArrows, dp, minArrowsForBobToWin)
        # find bob's arrows for each section starting from max score section
        for i in range(11, 0, -1):
            if numArrows >= minArrowsForBobToWin[i] and dp[i][numArrows] > dp[i - 1][numArrows]:
                bobMaxScore[i] = minArrowsForBobToWin[i]
                numArrows -= minArrowsForBobToWin[i]
                if numArrows <= 0:
                    break
        # add remaining arrows in section 0
        if numArrows > 0:
            bobMaxScore[0] = numArrows
        return bobMaxScore
25800