<p>There is a <code>50 x 50</code> chessboard with <strong>one</strong> knight and some pawns on it. You are given two integers <code>kx</code> and <code>ky</code> where <code>(kx, ky)</code> denotes the position of the knight, and a 2D array <code>positions</code> where <code>positions[i] = [x<sub>i</sub>, y<sub>i</sub>]</code> denotes the position of the pawns on the chessboard.</p>

<p>Alice and Bob play a <em>turn-based</em> game, where Alice goes first. In each player&#39;s turn:</p>

<ul>
	<li>The player <em>selects </em>a pawn that still exists on the board and captures it with the knight in the <strong>fewest</strong> possible <strong>moves</strong>. <strong>Note</strong> that the player can select <strong>any</strong> pawn, it <strong>might not</strong> be one that can be captured in the <strong>least</strong> number of moves.</li>
	<li><span>In the process of capturing the <em>selected</em> pawn, the knight <strong>may</strong> pass other pawns <strong>without</strong> capturing them</span>. <strong>Only</strong> the <em>selected</em> pawn can be captured in <em>this</em> turn.</li>
</ul>

<p>Alice is trying to <strong>maximize</strong> the <strong>sum</strong> of the number of moves made by <em>both</em> players until there are no more pawns on the board, whereas Bob tries to <strong>minimize</strong> them.</p>

<p>Return the <strong>maximum</strong> <em>total</em> number of moves made during the game that Alice can achieve, assuming both players play <strong>optimally</strong>.</p>

<p>Note that in one <strong>move, </strong>a chess knight has eight possible positions it can move to, as illustrated below. Each move is two cells in a cardinal direction, then one cell in an orthogonal direction.</p>

<p><img src="https://assets.leetcode.com/uploads/2024/08/01/chess_knight.jpg" style="width: 275px; height: 273px;" /></p>

<p>&nbsp;</p>
<p><strong class="example">Example 1:</strong></p>

<div class="example-block">
<p><strong>Input:</strong> <span class="example-io">kx = 1, ky = 1, positions = [[0,0]]</span></p>

<p><strong>Output:</strong> <span class="example-io">4</span></p>

<p><strong>Explanation:</strong></p>

<p><img alt="" src="https://assets.leetcode.com/uploads/2024/08/16/gif3.gif" style="width: 275px; height: 275px;" /></p>

<p>The knight takes 4 moves to reach the pawn at <code>(0, 0)</code>.</p>
</div>

<p><strong class="example">Example 2:</strong></p>

<div class="example-block">
<p><strong>Input:</strong> <span class="example-io">kx = 0, ky = 2, positions = [[1,1],[2,2],[3,3]]</span></p>

<p><strong>Output:</strong> <span class="example-io">8</span></p>

<p><strong>Explanation:</strong></p>

<p><strong><img alt="" src="https://assets.leetcode.com/uploads/2024/08/16/gif4.gif" style="width: 320px; height: 320px;" /></strong></p>

<ul>
	<li>Alice picks the pawn at <code>(2, 2)</code> and captures it in two moves: <code>(0, 2) -&gt; (1, 4) -&gt; (2, 2)</code>.</li>
	<li>Bob picks the pawn at <code>(3, 3)</code> and captures it in two moves: <code>(2, 2) -&gt; (4, 1) -&gt; (3, 3)</code>.</li>
	<li>Alice picks the pawn at <code>(1, 1)</code> and captures it in four moves: <code>(3, 3) -&gt; (4, 1) -&gt; (2, 2) -&gt; (0, 3) -&gt; (1, 1)</code>.</li>
</ul>
</div>

<p><strong class="example">Example 3:</strong></p>

<div class="example-block">
<p><strong>Input:</strong> <span class="example-io">kx = 0, ky = 0, positions = [[1,2],[2,4]]</span></p>

<p><strong>Output:</strong> <span class="example-io">3</span></p>

<p><strong>Explanation:</strong></p>

<ul>
	<li>Alice picks the pawn at <code>(2, 4)</code> and captures it in two moves: <code>(0, 0) -&gt; (1, 2) -&gt; (2, 4)</code>. Note that the pawn at <code>(1, 2)</code> is not captured.</li>
	<li>Bob picks the pawn at <code>(1, 2)</code> and captures it in one move: <code>(2, 4) -&gt; (1, 2)</code>.</li>
</ul>
</div>

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

<ul>
	<li><code>0 &lt;= kx, ky &lt;= 49</code></li>
	<li><code>1 &lt;= positions.length &lt;= 15</code></li>
	<li><code>positions[i].length == 2</code></li>
	<li><code>0 &lt;= positions[i][0], positions[i][1] &lt;= 49</code></li>
	<li>All <code>positions[i]</code> are unique.</li>
	<li>The input is generated such that <code>positions[i] != [kx, ky]</code> for all <code>0 &lt;= i &lt; positions.length</code>.</li>
</ul>

class Solution:
    def maxMoves(self, kx: int, ky: int, positions: List[List[int]]) -> int:

        cells = self.makeCells(kx, ky, positions)  # [knight, ...pawns]
        N = len(cells)

        minDis = [
            [0 for _ in range(N)] for _ in range(N)
        ]  # minDis between each two cells
        self.fill(minDis, cells)

        cache = [
            [None for _ in range(1 << 15)] for _ in range(N)
        ]  # cache[cell][availablePawns]

        return self.solve(0, (1 << N - 1) - 1, True, N, minDis, cache)

    def solve(self, idx, pawns, Alice, N, minDis, cache):
        if not pawns:
            return 0

        if cache[idx][pawns] is not None:
            return cache[idx][pawns]

        res = -1 if Alice else inf

        for i in range(1, N):
            if 1 << (i - 1) & pawns != 0:  # if pawn does not captured
                if Alice:
                    res = max(
                        res,
                        minDis[idx][i]
                        + self.solve(
                            i, pawns ^ (1 << (i - 1)), not Alice, N, minDis, cache
                        ),
                    )

                else:
                    res = min(
                        res,
                        minDis[idx][i]
                        + self.solve(
                            i, pawns ^ (1 << (i - 1)), not Alice, N, minDis, cache
                        ),
                    )

        cache[idx][pawns] = res
        return res

    def fill(self, minDis, cells):

        directions = (
            (2, 1),
            (2, -1),
            (-2, 1),
            (-2, -1),
            (1, 2),
            (1, -2),
            (-1, 2),
            (-1, -2),
        )

        def valid(x, y, board):
            return x >= 0 and x < 50 and y >= 0 and y < 50 and board[x][y] == -1

        def dfs(x, y, index):

            # board[x][y] => dis(index -> (x, y))
            # index => (x, y) => cells[i]
            board = [[-1 for _ in range(50)] for _ in range(50)]

            queue = deque([(x, y)])
            board[x][y] = 0

            seen = set(cells)  #

            while queue:
                currX, currY = queue.popleft()

                if (currX, currY) in seen:  #
                    seen.remove((currX, currY))  #

                if not seen:
                    break  #

                for direc in directions:
                    newX = currX + direc[0]
                    newY = currY + direc[1]

                    if valid(newX, newY, board):
                        board[newX][newY] = board[currX][currY] + 1
                        queue.append((newX, newY))

            for i, (p, q) in enumerate(cells):
                minDis[index][i] = board[p][q]

        for i, (x, y) in enumerate(cells):
            dfs(x, y, i)

    def makeCells(self, x, y, pos):
        res = [(x, y)]
        for p, q in pos:
            res.append((p, q))

        return res

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