<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

# Minimax function to simulate the game
def minimax(knight_pos, pawns, is_alice_turn, memo, distances):
    if not pawns:
        return 0  # No more pawns to capture
    key = (tuple(knight_pos), tuple(sorted([tuple(i) for i in pawns])))

    # key = (knight_pos, tuple(pawns), is_alice_turn)
    if key in memo:
        return memo[key]
    
    if is_alice_turn:
        max_moves = float('-inf')
        for i, pawn in enumerate(pawns):
            next_pawns = pawns[:i] + pawns[i+1:]
            moves = distances[(knight_pos, tuple(pawn))]
            max_moves = max(max_moves, moves + minimax(pawn, next_pawns, False, memo, distances))
        memo[key] = max_moves
    else:
        min_moves = float('inf')
        for i, pawn in enumerate(pawns):
            next_pawns = pawns[:i] + pawns[i+1:]
            moves = distances[(knight_pos, tuple(pawn))]
            min_moves = min(min_moves, moves + minimax(pawn, next_pawns, True, memo, distances))
        memo[key] = min_moves
    
    return memo[key]

class Solution:
    def maxMoves(self, kx: int, ky: int, positions: list[list[int]]) -> int:
        memo = {}
        
        # Precompute distances between knight and pawns, and between pawns
        distances = precompute_distances(positions, kx, ky)
        
        # Use minimax to simulate the game with memoization
        return minimax((kx, ky), [tuple(pos) for pos in positions], True, memo, distances)


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