<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>

from collections import deque
from functools import lru_cache


# Precompute distances between all points
def precompute_distances(positions, kx, ky):
    distances = {}

    # Function to compute shortest path from knight to target using BFS
    def bfs(start_pos, end_pos):
        sx, sy = start_pos
        ex, ey = end_pos
        if (sx, sy) == (ex, ey):
            return 0

        directions = [
            (-2, -1),
            (-1, -2),
            (1, -2),
            (2, -1),
            (2, 1),
            (1, 2),
            (-1, 2),
            (-2, 1),
        ]
        visited = [[False for _ in range(50)] for _ in range(50)]
        queue = deque([(sx, sy, 0)])  # (x, y, distance)
        visited[sx][sy] = True

        while queue:
            x, y, dist = queue.popleft()
            for dx, dy in directions:
                nx, ny = x + dx, y + dy
                if 0 <= nx < 50 and 0 <= ny < 50 and not visited[nx][ny]:
                    if (nx, ny) == (ex, ey):
                        return dist + 1
                    visited[nx][ny] = True
                    queue.append((nx, ny, dist + 1))
        return float("inf")  # Should never happen

    # Compute distances between knight and pawns
    knight_pos = (kx, ky)
    for pawn_pos in positions:
        distances[(knight_pos, tuple(pawn_pos))] = bfs(knight_pos, pawn_pos)

    # Compute distances between each pair of pawns
    for i in range(len(positions)):
        for j in range(len(positions)):
            if i != j:
                distances[(tuple(positions[i]), tuple(positions[j]))] = bfs(
                    positions[i], positions[j]
                )

    return distances


class Solution:
    def maxMoves(self, kx: int, ky: int, positions: list[list[int]]):
        global postions
        global distances
        postions = positions
        distances = precompute_distances(positions, kx, ky)
        num_pawns = len(positions)
        result = self.rec((1 << num_pawns) - 1, True, (kx, ky))
        return result

    @lru_cache(None)
    def rec(self, num, f, last):
        if num == 0:
            return 0
        if f:
            val = 0
            func = max
        else:
            val = float("inf")
            func = min

        for b in range(num.bit_length()):
            if num & (1 << b):
                cur = postions[b]
                dis = distances[(last, tuple(cur))]
                # print("dis-->", dis)
                val = func(val, dis + self.rec(num ^ (1 << b), not f, tuple(cur)))
        return val


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