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Number of Valid Move Combinations On Chessboard

Difficulty: Hard

Problem Description

There is an 8 x 8 chessboard containing n pieces (rooks, queens, or bishops). You are given a string array pieces of length n, where pieces[i] describes the type (rook, queen, or bishop) of the i-th piece. In addition, you are given a 2D integer array positions also of length n, where positions[i] = [r_i, c_i] indicates that the i-th piece is currently at the 1-based coordinate (r_i, c_i) on the chessboard.

When making a move for a piece, you choose a destination square that the piece will travel toward and stop on.

  • A rook can only travel horizontally or vertically from (r, c) to the direction of (r+1, c), (r-1, c), (r, c+1), or (r, c-1).
  • A queen can only travel horizontally, vertically, or diagonally from (r, c) to the direction of (r+1, c), (r-1, c), (r, c+1), (r, c-1), (r+1, c+1), (r+1, c-1), (r-1, c+1), or (r-1, c-1).
  • A bishop can only travel diagonally from (r, c) to the direction of (r+1, c+1), (r+1, c-1), (r-1, c+1), or (r-1, c-1).

You must make a move for every piece on the board simultaneously. A move combination consists of all the moves performed on all the given pieces. Every second, each piece will instantaneously travel one square towards their destination if they are not already at it. A move combination is invalid if, at a given time, two or more pieces occupy the same square.

Return the number of valid move combinations.

Key Insights

  • Each piece has distinct movement patterns and can occupy positions based on their type.
  • The complexity arises from ensuring that no two pieces overlap during their moves.
  • There are combinatorial aspects to consider for the placement of pieces and their allowed movements.
  • Given the constraints (1 ≤ n ≤ 4), a backtracking approach could be feasible to explore all potential move combinations.

Space and Time Complexity

Time Complexity: O(4^n) - Each piece can move to multiple positions, leading to an exponential growth in combinations. Space Complexity: O(n) - To store the states of the pieces and their potential positions.

Solution

The solution can be approached using backtracking to explore all possible valid destinations for each piece. We will generate all possible moves for each piece based on their movement capabilities and check for overlaps at each time step.

  1. Generate Possible Moves: For each piece type, generate all valid destination squares on the chessboard based on their movement rules.
  2. Backtracking to Count Valid Combinations: Use a backtracking approach to simulate every possible move. For each piece, attempt to place it in every valid destination while checking for collisions with other pieces.
  3. Count Valid Configurations: Count how many configurations do not result in overlapping pieces during their movement.

Code Solutions

def count_valid_move_combinations(pieces, positions):
    def get_moves(piece_type, r, c):
        moves = set()
        if piece_type == "rook":
            for i in range(1, 9):
                moves.add((r, i))  # horizontal
                moves.add((i, c))  # vertical
        elif piece_type == "queen":
            for i in range(1, 9):
                moves.add((r, i))  # horizontal
                moves.add((i, c))  # vertical
                moves.add((r + i, c + i))  # diagonal down-right
                moves.add((r + i, c - i))  # diagonal down-left
                moves.add((r - i, c + i))  # diagonal up-right
                moves.add((r - i, c - i))  # diagonal up-left
        elif piece_type == "bishop":
            for i in range(1, 9):
                moves.add((r + i, c + i))  # diagonal down-right
                moves.add((r + i, c - i))  # diagonal down-left
                moves.add((r - i, c + i))  # diagonal up-right
                moves.add((r - i, c - i))  # diagonal up-left
        return moves

    all_moves = []
    for i in range(len(pieces)):
        all_moves.append(get_moves(pieces[i], positions[i][0], positions[i][1]))

    def backtrack(depth, occupied):
        if depth == len(pieces):
            return 1
        count = 0
        for move in all_moves[depth]:
            if move not in occupied:
                count += backtrack(depth + 1, occupied | {move})
        return count

    return backtrack(0, set())

# Example usage:
pieces = ["rook"]
positions = [[1, 1]]
print(count_valid_move_combinations(pieces, positions))  # Output: 15
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