<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

# Possible knight moves
KNIGHT_MOVES = [(-2, -1), (-2, 1), (-1, -2), (-1, 2), (1, -2), (1, 2), (2, -1), (2, 1)]

def bfs(start_x, start_y, target_x, target_y):
    # Perform BFS to find the shortest path from (start_x, start_y) to (target_x, target_y)
    visited = [[False] * 50 for _ in range(50)]
    queue = deque([(start_x, start_y, 0)])  # (x, y, moves)
    visited[start_x][start_y] = True
    
    while queue:
        x, y, moves = queue.popleft()
        if x == target_x and y == target_y:
            return moves
        
        for dx, dy in KNIGHT_MOVES:
            nx, ny = x + dx, y + dy
            if 0 <= nx < 50 and 0 <= ny < 50 and not visited[nx][ny]:
                visited[nx][ny] = True
                queue.append((nx, ny, moves + 1))
    
    return float('inf')  # This case shouldn't happen since all positions are reachable

def precompute_distances(kx, ky, pawns):
    # Precompute all distances between knight and pawns, and between each pair of pawns
    n = len(pawns)
    distances = [[0] * (n + 1) for _ in range(n + 1)]  # 1 extra for the knight starting position
    
    # Calculate distance from the knight to each pawn
    for i, (px, py) in enumerate(pawns):
        distances[0][i + 1] = bfs(kx, ky, px, py)
    
    # Calculate distances between every pair of pawns
    for i, (px1, py1) in enumerate(pawns):
        for j, (px2, py2) in enumerate(pawns):
            if i != j:
                distances[i + 1][j + 1] = bfs(px1, py1, px2, py2)
    
    return distances

def minmax(knight_idx, pawns_left, is_alice_turn, distances, memo):
    if not pawns_left:
        return 0
    
    # Memoization key
    state = (knight_idx, pawns_left, is_alice_turn)
    if state in memo:
        return memo[state]
    
    if is_alice_turn:
        best = -float('inf')  # Alice tries to maximize
    else:
        best = float('inf')   # Bob tries to minimize
    
    # Convert set of remaining pawns to list for easier processing
    pawns_list = list(pawns_left)
    
    # Try capturing each remaining pawn and simulate the game
    for i, pawn_idx in enumerate(pawns_list):
        remaining_pawns = pawns_left - {pawn_idx}
        moves = distances[knight_idx][pawn_idx]
        result = moves + minmax(pawn_idx, remaining_pawns, not is_alice_turn, distances, memo)
        
        if is_alice_turn:
            best = max(best, result)
        else:
            best = min(best, result)
    
    memo[state] = best
    return best



class Solution:
    def maxMoves(self, kx: int, ky: int, positions: List[List[int]]) -> int:
        # Precompute distances
        distances = precompute_distances(kx, ky, positions)
        
        # Initialize memoization
        memo = {}
        
        # Start the minimax game, starting from the knight's initial position (index 0 in distances)
        pawns_left = frozenset(range(1, len(positions) + 1))  # Pawns are indexed from 1 onward
        return minmax(0, pawns_left, True, distances, memo)
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