<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 getDist(self, pos1, pos2):
            if pos2 < pos1:
                pos1, pos2 = pos2, pos1

            if (pos1, pos2) in self.dstmap:
                return self.dstmap[(pos1, pos2)]

            dp_distance = [[-1 for i in range(50)] for j in range(50)]
            dp_distance[pos1[0]][pos1[1]] = 0

            queue = deque()
            queue.append(pos1)

            visited = set()
            visited.add(pos1)
            
            dirs = [(-2, -1), (-2, 1), (-1, -2), (-1, 2), (1, -2), (1, 2), (2, -1), (2, 1)]
            while queue:
                cur = queue.popleft()
                for d in dirs:
                    new_x, new_y = cur[0] + d[0], cur[1] + d[1]
                    if (new_x, new_y) in visited:
                        continue
                    if new_x >= 0 and new_y >= 0 and new_x < 50 and new_y < 50:
                        dp_distance[new_x][new_y] = dp_distance[cur[0]][cur[1]] + 1
                        if new_x == pos2[0] and new_y == pos2[1]:
                            self.dstmap[(pos1, pos2)] = dp_distance[new_x][new_y]
                            return self.dstmap[(pos1, pos2)]

                        queue.append((new_x, new_y))
                        visited.add((new_x, new_y))
    @cache
    def AliceMove(self, pawn_bm, knight_pos):
        if pawn_bm == 0:
            return 0
        # print("AliceMove: pawn_bm =", pawn_bm, "knight_pos =", knight_pos)
        ans = 0
        for i in range(self.n):
            if pawn_bm & (1 << i):
                remain_mask = pawn_bm ^ (1 << i)
                ans = max(ans, self.BobMove(remain_mask, self.positions[i]) + self.getDist(knight_pos, self.positions[i]))
                # print("\ti =", i, ", ans =", ans)
        
        return ans
    @cache
    def BobMove(self, pawn_bm, knight_pos):
        if pawn_bm == 0:
            return 0
        # print("BobMove: pawn_bm =", pawn_bm, "knight_pos =", knight_pos)

        ans = float('inf')
        for i in range(self.n):
            if pawn_bm & (1 << i):
                remain_mask = pawn_bm ^ (1 << i)
                ans = min(ans, self.AliceMove(remain_mask, self.positions[i]) + self.getDist(knight_pos, self.positions[i]))
                # print("\ti =", i, ", ans =", ans)
        return ans


    def maxMoves(self, kx: int, ky: int, positions: List[List[int]]) -> int:
        self.dstmap = dict()

        positions = [(p[0], p[1]) for p in positions]
        self.positions = positions

        for i in range(len(positions)-1):
            self.getDist(positions[i], positions[i+1])

        self.n = len(positions)

        pawn_bm = (1 << self.n) - 1
        return self.AliceMove(pawn_bm, (kx, ky))
        

        # retval = 0
        # for i in range(self.n):
        #     remain_pawn_bm = pawn_bm ^ (1 << i)



        # print(self.dstmap)
        # return 0
7766