Optimal Minimax Solution for Knight and Pawns Game with BFS and Memoization
Code\npython3 []\nclass Solution:\n    def maxMoves(self, kx: int, ky: int, positions: List[List[int]]) -> int:\n        # This function calculates the shortest
# Code\n```python3 []\nclass Solution:\n    def maxMoves(self, kx: int, ky: int, positions: List[List[int]]) -> int:\n        # This function calculates the shortest distance (in knight moves) from the \'start\' position to each \'target\' pawn position.\n        def knight_moves(start, targets):\n            # List of all possible knight moves.\n            directions = [(2, 1), (2, -1), (-2, 1), (-2, -1), (1, 2), (1, -2), (-1, 2), (-1, -2)]\n            # BFS queue initialized with the knight\'s starting position and 0 steps taken.\n            queue = deque([(start[0], start[1], 0)])\n            # Dictionary to store the number of moves needed to reach each target pawn.\n            distances = {}\n            # Set to keep track of visited positions to avoid cycles.\n            visited = set([(start[0], start[1])])\n            \n            # BFS loop to explore all positions the knight can move to.\n            while queue:\n                x, y, steps = queue.popleft()\n                # If the current position contains a pawn, record the number of moves taken to reach it.\n                if (x, y) in targets:\n                    distances[(x, y)] = steps\n                # If all pawns are reached, stop searching.\n                if len(distances) == len(targets):\n                    break\n                # Explore all 8 possible knight moves.\n                for dx, dy in directions:\n                    nx, ny = x + dx, y + dy\n                    # Ensure the knight stays within the chessboard and hasn\'t visited the position already.\n                    if 0 <= nx < 50 and 0 <= ny < 50 and (nx, ny) not in visited:\n                        visited.add((nx, ny))\n                        queue.append((nx, ny, steps + 1))\n            # Return a dictionary with the number of moves required to reach each pawn.\n            return distances\n        \n        # This is the core recursive minimax function for the game simulation.\n        # pos_mask: bitmask to track which pawns have been captured.\n        # knight_pos: current position of the knight.\n        # turn: 0 for Alice\'s turn, 1 for Bob\'s turn.\n        def minimax(pos_mask, knight_pos, turn):\n            # Base case: all pawns have been captured.\n            if pos_mask == (1 << len(pawns)) - 1:\n                return 0\n            \n            # If this state has already been computed, return the stored result to avoid recomputation.\n            if (pos_mask, knight_pos, turn) in memo:\n                return memo[(pos_mask, knight_pos, turn)]\n            \n            # Alice (turn 0) tries to maximize the total moves, while Bob (turn 1) tries to minimize them.\n            best_score = float(\'-inf\') if turn == 0 else float(\'inf\')\n            \n            # Try capturing each remaining pawn.\n            for i, pawn_pos in enumerate(pawns):\n                # Skip the pawn if it has already been captured (indicated by the bitmask).\n                if pos_mask & (1 << i):\n                    continue\n                # Find the number of moves to capture this pawn.\n                moves = distances[knight_pos].get(pawn_pos, float(\'inf\'))\n                if moves == float(\'inf\'):\n                    continue\n                # Update the bitmask to indicate this pawn has been captured.\n                new_mask = pos_mask | (1 << i)\n                # Recursively calculate the total moves for the next state.\n                next_score = moves + minimax(new_mask, pawn_pos, 1 - turn)\n                \n                # Alice maximizes the moves, Bob minimizes the moves.\n                if turn == 0:\n                    best_score = max(best_score, next_score)\n                else:\n                    best_score = min(best_score, next_score)\n            \n            # Store the result of the current state in memoization table.\n            memo[(pos_mask, knight_pos, turn)] = best_score\n            return best_score\n        \n        # List of all pawns\' positions.\n        pawns = [(x, y) for x, y in positions]\n        # Combine the knight\'s initial position with the pawn positions for distance calculation.\n        all_positions = [(kx, ky)] + pawns\n        \n        # Precompute the shortest distance between all pairs of positions (knight to pawns).\n        distances = {}\n        for pos in all_positions:\n            distances[pos] = knight_moves(pos, pawns)\n        \n        # Dictionary to store the results of subproblems (memoization).\n        memo = {}\n        # Start the game with Alice\'s turn (turn 0) and return the maximum moves Alice can achieve.\n        return minimax(0, (kx, ky), 0)\n\n```
['Array', 'Math', 'Bit Manipulation', 'Breadth-First Search', 'Game Theory', 'Bitmask', 'Python3']