<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]:
        if aliceArrows[0] >= 11:
            bobArrows = [_+1 for _ in aliceArrows]
            bobArrows[0] -= 12
            return bobArrows
        if aliceArrows[0] + aliceArrows[1] >= 10:
            bobArrows = [_+1 for _ in aliceArrows]
            bobArrows[0] = 0
            bobArrows[1] = aliceArrows[0] + aliceArrows[1] - 10
            return bobArrows 
 
        highscore = 66 #sum(list(reversed(range(12)))[:numArrows])
        highscores = set()
        highscorecosts = [inf] * (highscore + 1)
        highscorepaths = [[] for _ in range(highscore + 1)]
        # highscorecosts[0] = 0
        # print( highscore, highscores, numArrows, aliceArrows)
        def update(s, c, i ):
            if highscorecosts[s] >  c:
                highscorecosts[s] = c
                return True
            return False
            
        for i in reversed(range(12)):
            a = aliceArrows[i]
            if i and a < numArrows:
                newhighscores = set()
                for s in sorted(highscores, reverse=True):
                    if highscorecosts[s] + a  < numArrows:
                        # print(f'new high scores: {s+i} : min({highscorecosts[s + i]}, {a+1}+{highscorecosts[s]})')
                        if update(s+i, a+1+highscorecosts[s], i ):
                            highscorepaths[s + i] = highscorepaths[s] + [i]
                            newhighscores.add(s+i)
                            # print(f'new high score path: {s+i} : {highscorepaths[s + i]}')
                # print(f'new high scores: {i} : min({highscorecosts[ i]}, {a+1})')
                if update(i, a+1, i ):
                    highscorepaths[i] = [i]
                    newhighscores.add(i)
                    # print(f'new high score path: {i} : {highscorepaths[i]}')
                highscores = highscores | newhighscores
        # print( max(highscores), highscores, highscorecosts )
        bobArrows = [0]*12
        for i in highscorepaths[max(highscores)]:
            bobArrows[i] = aliceArrows[i] + 1
        bobArrows[0] = numArrows - sum(bobArrows)
        return bobArrows
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