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2056. Number of Valid Move Combinations On Chessboard

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 ith piece. In addition, you are given a 2D integer array positions also of length n, where positions[i] = [ri, ci] indicates that the ith piece is currently at the 1-based coordinate (ri, ci) 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), (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), (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. All pieces start traveling at the 0th second. 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​​​​​.

Notes:

  • No two pieces will start in the same square.
  • You may choose the square a piece is already on as its destination.
  • If two pieces are directly adjacent to each other, it is valid for them to move past each other and swap positions in one second.

 

Example 1:

Input: pieces = ["rook"], positions = [[1,1]]
Output: 15
Explanation: The image above shows the possible squares the piece can move to.

Example 2:

Input: pieces = ["queen"], positions = [[1,1]]
Output: 22
Explanation: The image above shows the possible squares the piece can move to.

Example 3:

Input: pieces = ["bishop"], positions = [[4,3]]
Output: 12
Explanation: The image above shows the possible squares the piece can move to.

 

Constraints:

  • n == pieces.length
  • n == positions.length
  • 1 <= n <= 4
  • pieces only contains the strings "rook", "queen", and "bishop".
  • There will be at most one queen on the chessboard.
  • 1 <= ri, ci <= 8
  • Each positions[i] is distinct.

Solutions

Solution 1: DFS

The problem has at most $4$ pieces, and each piece can move in up to $8$ directions. We can consider using DFS to search all possible move combinations.

We enumerate each piece in order. For each piece, we can choose not to move or move according to the rules. We use an array $\textit{dist}[i]$ to record the movement of the $i$-th piece, where $\textit{dist}[i][x][y]$ represents the time when the $i$-th piece passes through the coordinate $(x, y)$. We use an array $\textit{end}[i]$ to record the endpoint coordinates and time of the $i$-th piece. During the search, we need to determine whether the current piece can stop moving and whether it can continue moving in the current direction.

We define a method $\text{checkStop}(i, x, y, t)$ to determine whether the $i$-th piece can stop at coordinate $(x, y)$ at time $t$. If for all previous pieces $j$, $\textit{dist}[j][x][y] < t$, then the $i$-th piece can stop moving.

Additionally, we define a method $\text{checkPass}(i, x, y, t)$ to determine whether the $i$-th piece can pass through coordinate $(x, y)$ at time $t$. If any other piece $j$ also passes through coordinate $(x, y)$ at time $t$, or if any other piece $j$ stops at $(x, y)$ and the time does not exceed $t$, then the $i$-th piece cannot pass through coordinate $(x, y)$ at time $t$.

The time complexity is $O((n \times M)^n)$, and the space complexity is $O(n \times M)$. Here, $n$ is the number of pieces, and $M$ is the movement range of each piece.

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rook_dirs = [(1, 0), (-1, 0), (0, 1), (0, -1)]
bishop_dirs = [(1, 1), (1, -1), (-1, 1), (-1, -1)]
queue_dirs = rook_dirs + bishop_dirs


def get_dirs(piece: str) -> List[Tuple[int, int]]:
    match piece[0]:
        case "r":
            return rook_dirs
        case "b":
            return bishop_dirs
        case _:
            return queue_dirs


class Solution:
    def countCombinations(self, pieces: List[str], positions: List[List[int]]) -> int:
        def check_stop(i: int, x: int, y: int, t: int) -> bool:
            return all(dist[j][x][y] < t for j in range(i))

        def check_pass(i: int, x: int, y: int, t: int) -> bool:
            for j in range(i):
                if dist[j][x][y] == t:
                    return False
                if end[j][0] == x and end[j][1] == y and end[j][2] <= t:
                    return False
            return True

        def dfs(i: int) -> None:
            if i >= n:
                nonlocal ans
                ans += 1
                return
            x, y = positions[i]
            dist[i][:] = [[-1] * m for _ in range(m)]
            dist[i][x][y] = 0
            end[i] = (x, y, 0)
            if check_stop(i, x, y, 0):
                dfs(i + 1)
            dirs = get_dirs(pieces[i])
            for dx, dy in dirs:
                dist[i][:] = [[-1] * m for _ in range(m)]
                dist[i][x][y] = 0
                nx, ny, nt = x + dx, y + dy, 1
                while 1 <= nx < m and 1 <= ny < m and check_pass(i, nx, ny, nt):
                    dist[i][nx][ny] = nt
                    end[i] = (nx, ny, nt)
                    if check_stop(i, nx, ny, nt):
                        dfs(i + 1)
                    nx += dx
                    ny += dy
                    nt += 1

        n = len(pieces)
        m = 9
        dist = [[[-1] * m for _ in range(m)] for _ in range(n)]
        end = [(0, 0, 0) for _ in range(n)]
        ans = 0
        dfs(0)
        return ans

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class Solution {
    int n, m = 9, ans;
    int[][][] dist;
    int[][] end;
    String[] pieces;
    int[][] positions;
    int[][] rookDirs = {{1, 0}, {-1, 0}, {0, 1}, {0, -1}};
    int[][] bishopDirs = {{1, 1}, {1, -1}, {-1, 1}, {-1, -1}};
    int[][] queenDirs = {{1, 0}, {-1, 0}, {0, 1}, {0, -1}, {1, 1}, {1, -1}, {-1, 1}, {-1, -1}};

    public int countCombinations(String[] pieces, int[][] positions) {
        n = pieces.length;
        dist = new int[n][m][m];
        end = new int[n][3];
        ans = 0;
        this.pieces = pieces;
        this.positions = positions;

        dfs(0);
        return ans;
    }

    private void dfs(int i) {
        if (i >= n) {
            ans++;
            return;
        }

        int x = positions[i][0], y = positions[i][1];
        resetDist(i);
        dist[i][x][y] = 0;
        end[i] = new int[] {x, y, 0};

        if (checkStop(i, x, y, 0)) {
            dfs(i + 1);
        }

        int[][] dirs = getDirs(pieces[i]);
        for (int[] dir : dirs) {
            resetDist(i);
            dist[i][x][y] = 0;
            int nx = x + dir[0], ny = y + dir[1], nt = 1;

            while (isValid(nx, ny) && checkPass(i, nx, ny, nt)) {
                dist[i][nx][ny] = nt;
                end[i] = new int[] {nx, ny, nt};
                if (checkStop(i, nx, ny, nt)) {
                    dfs(i + 1);
                }
                nx += dir[0];
                ny += dir[1];
                nt++;
            }
        }
    }

    private void resetDist(int i) {
        for (int j = 0; j < m; j++) {
            for (int k = 0; k < m; k++) {
                dist[i][j][k] = -1;
            }
        }
    }

    private boolean checkStop(int i, int x, int y, int t) {
        for (int j = 0; j < i; j++) {
            if (dist[j][x][y] >= t) {
                return false;
            }
        }
        return true;
    }

    private boolean checkPass(int i, int x, int y, int t) {
        for (int j = 0; j < i; j++) {
            if (dist[j][x][y] == t) {
                return false;
            }
            if (end[j][0] == x && end[j][1] == y && end[j][2] <= t) {
                return false;
            }
        }
        return true;
    }

    private boolean isValid(int x, int y) {
        return x >= 1 && x < m && y >= 1 && y < m;
    }

    private int[][] getDirs(String piece) {
        char c = piece.charAt(0);
        return switch (c) {
            case 'r' -> rookDirs;
            case 'b' -> bishopDirs;
            default -> queenDirs;
        };
    }
}

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class Solution {
public:
    int countCombinations(vector<string>& pieces, vector<vector<int>>& positions) {
        int n = pieces.size();
        const int m = 9;
        int ans = 0;

        vector<vector<vector<int>>> dist(n, vector<vector<int>>(m, vector<int>(m, -1)));
        vector<vector<int>> end(n, vector<int>(3));

        const int rookDirs[4][2] = {{1, 0}, {-1, 0}, {0, 1}, {0, -1}};
        const int bishopDirs[4][2] = {{1, 1}, {1, -1}, {-1, 1}, {-1, -1}};
        const int queenDirs[8][2] = {{1, 0}, {-1, 0}, {0, 1}, {0, -1}, {1, 1}, {1, -1}, {-1, 1}, {-1, -1}};

        auto resetDist = [&](int i) {
            for (int j = 0; j < m; j++) {
                for (int k = 0; k < m; k++) {
                    dist[i][j][k] = -1;
                }
            }
        };

        auto checkStop = [&](int i, int x, int y, int t) -> bool {
            for (int j = 0; j < i; j++) {
                if (dist[j][x][y] >= t) {
                    return false;
                }
            }
            return true;
        };

        auto checkPass = [&](int i, int x, int y, int t) -> bool {
            for (int j = 0; j < i; j++) {
                if (dist[j][x][y] == t) {
                    return false;
                }
                if (end[j][0] == x && end[j][1] == y && end[j][2] <= t) {
                    return false;
                }
            }
            return true;
        };

        auto isValid = [&](int x, int y) -> bool {
            return x >= 1 && x < m && y >= 1 && y < m;
        };

        auto getDirs = [&](const string& piece) -> const int(*)[2] {
            char c = piece[0];
            if (c == 'r') {
                return rookDirs;
            }
            if (c == 'b') {
                return bishopDirs;
            }
            return queenDirs;
        };

        auto dfs = [&](this auto&& dfs, int i) -> void {
            if (i >= n) {
                ans++;
                return;
            }

            int x = positions[i][0], y = positions[i][1];
            resetDist(i);
            dist[i][x][y] = 0;
            end[i] = {x, y, 0};

            if (checkStop(i, x, y, 0)) {
                dfs(i + 1);
            }

            const int(*dirs)[2] = getDirs(pieces[i]);
            int dirsSize = (pieces[i][0] == 'q') ? 8 : 4;

            for (int d = 0; d < dirsSize; d++) {
                resetDist(i);
                dist[i][x][y] = 0;
                int nx = x + dirs[d][0], ny = y + dirs[d][1], nt = 1;

                while (isValid(nx, ny) && checkPass(i, nx, ny, nt)) {
                    dist[i][nx][ny] = nt;
                    end[i] = {nx, ny, nt};
                    if (checkStop(i, nx, ny, nt)) {
                        dfs(i + 1);
                    }
                    nx += dirs[d][0];
                    ny += dirs[d][1];
                    nt++;
                }
            }
        };

        dfs(0);
        return ans;
    }
};

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func countCombinations(pieces []string, positions [][]int) (ans int) {
    n := len(pieces)
    m := 9
    dist := make([][][]int, n)
    for i := range dist {
        dist[i] = make([][]int, m)
        for j := range dist[i] {
            dist[i][j] = make([]int, m)
        }
    }

    end := make([][3]int, n)

    rookDirs := [][2]int{{1, 0}, {-1, 0}, {0, 1}, {0, -1}}
    bishopDirs := [][2]int{{1, 1}, {1, -1}, {-1, 1}, {-1, -1}}
    queenDirs := [][2]int{{1, 0}, {-1, 0}, {0, 1}, {0, -1}, {1, 1}, {1, -1}, {-1, 1}, {-1, -1}}

    resetDist := func(i int) {
        for j := 0; j < m; j++ {
            for k := 0; k < m; k++ {
                dist[i][j][k] = -1
            }
        }
    }

    checkStop := func(i, x, y, t int) bool {
        for j := 0; j < i; j++ {
            if dist[j][x][y] >= t {
                return false
            }
        }
        return true
    }

    checkPass := func(i, x, y, t int) bool {
        for j := 0; j < i; j++ {
            if dist[j][x][y] == t {
                return false
            }
            if end[j][0] == x && end[j][1] == y && end[j][2] <= t {
                return false
            }
        }
        return true
    }

    isValid := func(x, y int) bool {
        return x >= 1 && x < m && y >= 1 && y < m
    }

    getDirs := func(piece string) [][2]int {
        switch piece[0] {
        case 'r':
            return rookDirs
        case 'b':
            return bishopDirs
        default:
            return queenDirs
        }
    }

    var dfs func(i int)
    dfs = func(i int) {
        if i >= n {
            ans++
            return
        }

        x, y := positions[i][0], positions[i][1]
        resetDist(i)
        dist[i][x][y] = 0
        end[i] = [3]int{x, y, 0}

        if checkStop(i, x, y, 0) {
            dfs(i + 1)
        }

        dirs := getDirs(pieces[i])
        for _, dir := range dirs {
            resetDist(i)
            dist[i][x][y] = 0
            nx, ny, nt := x+dir[0], y+dir[1], 1

            for isValid(nx, ny) && checkPass(i, nx, ny, nt) {
                dist[i][nx][ny] = nt
                end[i] = [3]int{nx, ny, nt}
                if checkStop(i, nx, ny, nt) {
                    dfs(i + 1)
                }
                nx += dir[0]
                ny += dir[1]
                nt++
            }
        }
    }

    dfs(0)
    return
}

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const rookDirs: [number, number][] = [
    [1, 0],
    [-1, 0],
    [0, 1],
    [0, -1],
];
const bishopDirs: [number, number][] = [
    [1, 1],
    [1, -1],
    [-1, 1],
    [-1, -1],
];
const queenDirs = [...rookDirs, ...bishopDirs];

function countCombinations(pieces: string[], positions: number[][]): number {
    const n = pieces.length;
    const m = 9;
    let ans = 0;

    const dist = Array.from({ length: n }, () =>
        Array.from({ length: m }, () => Array(m).fill(-1)),
    );

    const end: [number, number, number][] = Array(n).fill([0, 0, 0]);

    const resetDist = (i: number) => {
        for (let j = 0; j < m; j++) {
            for (let k = 0; k < m; k++) {
                dist[i][j][k] = -1;
            }
        }
    };

    const checkStop = (i: number, x: number, y: number, t: number): boolean => {
        for (let j = 0; j < i; j++) {
            if (dist[j][x][y] >= t) {
                return false;
            }
        }
        return true;
    };

    const checkPass = (i: number, x: number, y: number, t: number): boolean => {
        for (let j = 0; j < i; j++) {
            if (dist[j][x][y] === t) {
                return false;
            }
            if (end[j][0] === x && end[j][1] === y && end[j][2] <= t) {
                return false;
            }
        }
        return true;
    };

    const isValid = (x: number, y: number): boolean => {
        return x >= 1 && x < m && y >= 1 && y < m;
    };

    const getDirs = (piece: string): [number, number][] => {
        switch (piece[0]) {
            case 'r':
                return rookDirs;
            case 'b':
                return bishopDirs;
            default:
                return queenDirs;
        }
    };

    const dfs = (i: number) => {
        if (i >= n) {
            ans++;
            return;
        }

        const [x, y] = positions[i];
        resetDist(i);
        dist[i][x][y] = 0;
        end[i] = [x, y, 0];

        if (checkStop(i, x, y, 0)) {
            dfs(i + 1);
        }

        const dirs = getDirs(pieces[i]);
        for (const [dx, dy] of dirs) {
            resetDist(i);
            dist[i][x][y] = 0;
            let nx = x + dx,
                ny = y + dy,
                nt = 1;

            while (isValid(nx, ny) && checkPass(i, nx, ny, nt)) {
                dist[i][nx][ny] = nt;
                end[i] = [nx, ny, nt];
                if (checkStop(i, nx, ny, nt)) {
                    dfs(i + 1);
                }
                nx += dx;
                ny += dy;
                nt++;
            }
        }
    };

    dfs(0);
    return ans;
}

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