Formatted question description: https://leetcode.ca/all/1210.html

1210. Minimum Moves to Reach Target with Rotations (Hard)

In an n*n grid, there is a snake that spans 2 cells and starts moving from the top left corner at (0, 0) and (0, 1). The grid has empty cells represented by zeros and blocked cells represented by ones. The snake wants to reach the lower right corner at (n-1, n-2) and (n-1, n-1).

In one move the snake can:

  • Move one cell to the right if there are no blocked cells there. This move keeps the horizontal/vertical position of the snake as it is.
  • Move down one cell if there are no blocked cells there. This move keeps the horizontal/vertical position of the snake as it is.
  • Rotate clockwise if it's in a horizontal position and the two cells under it are both empty. In that case the snake moves from (r, c) and (r, c+1) to (r, c) and (r+1, c).
  • Rotate counterclockwise if it's in a vertical position and the two cells to its right are both empty. In that case the snake moves from (r, c) and (r+1, c) to (r, c) and (r, c+1).

Return the minimum number of moves to reach the target.

If there is no way to reach the target, return -1.

 

Example 1:

Input: grid = [[0,0,0,0,0,1],
               [1,1,0,0,1,0],
               [0,0,0,0,1,1],
               [0,0,1,0,1,0],
               [0,1,1,0,0,0],
               [0,1,1,0,0,0]]
Output: 11
Explanation:
One possible solution is [right, right, rotate clockwise, right, down, down, down, down, rotate counterclockwise, right, down].

Example 2:

Input: grid = [[0,0,1,1,1,1],
               [0,0,0,0,1,1],
               [1,1,0,0,0,1],
               [1,1,1,0,0,1],
               [1,1,1,0,0,1],
               [1,1,1,0,0,0]]
Output: 9

 

Constraints:

  • 2 <= n <= 100
  • 0 <= grid[i][j] <= 1
  • It is guaranteed that the snake starts at empty cells.

Related Topics:
Breadth-first Search

Solution 1. BFS

Since we are looking for the shortest path, BFS is our first option.

// OJ: https://leetcode.com/problems/minimum-moves-to-reach-target-with-rotations/
// Time: O(N^2)
// Space: O(N^2)
class Solution {
public:
    int minimumMoves(vector<vector<int>>& G) {
        queue<int> q;
        q.push(0);
        unordered_set<int> seen;
        seen.insert(0);
        int step = 0, N = G.size();
        auto safe = [&](int p) {
            int y = p % 1000, x = p / 1000 % 1000, dir = p / 1000000;
            if (x < 0 || y < 0 || x >= N || y >= N || G[x][y] == 1) return false;
            return (dir == 0 && y + 1 < N && G[x][y + 1] == 0)
                || (dir == 1 && x + 1 < N && G[x + 1][y] == 0);
        };
        while (q.size()) {
            int cnt = q.size();
            while (cnt--) {
                int p = q.front(), y = p % 1000, x = p / 1000 % 1000, dir = p / 1000000;
                if (x == N - 1 && y == N - 2) return step;
                q.pop();
                // move right
                int r = p + 1;
                if (seen.count(r) == 0 && safe(r)) {
                    seen.insert(r);
                    q.push(r);
                }
                // move down
                int d = p + 1000;
                if (seen.count(d) == 0 && safe(d)) {
                    seen.insert(d);
                    q.push(d);
                }
                // rotate closewise
                if (dir == 0) {
                    int p2 = 1000000 + x * 1000 + y, cx = x + 1, cy = y + 1;
                    if (seen.count(p2) || !safe(p2) || cx < 0 || cy < 0 || cx >= N || cy >= N || G[cx][cy] == 1) continue;
                    seen.insert(p2);
                    q.push(p2);
                }
                // rotate counterclosewise
                if (dir == 1) {
                    int p2 = x * 1000 + y, cx = x + 1, cy = y + 1;
                    if (seen.count(p2) || !safe(p2) || cx < 0 || cy < 0 || cx >= N || cy >= N || G[cx][cy] == 1) continue;
                    seen.insert(p2);
                    q.push(p2);
                }
            }
            ++step;
        }
        return -1;
    }
};

Java

  • class Solution {
        public int minimumMoves(int[][] grid) {
            final int BLOCK = -1;
            final int WHITE = 0;
            final int GRAY = 1;
            final int BLACK = 2;
            int rows = grid.length, columns = grid[0].length;
            if (grid[rows - 1][columns - 1] == 1 || grid[rows - 1][columns - 2] == -1)
                return -1;
            int[][][] colors = new int[rows][columns][2];
            int[][][] distances = new int[rows][columns][2];
            for (int i = 0; i < rows; i++)
                colors[i][columns - 1][0] = BLOCK;
            for (int i = 0; i < columns; i++)
                colors[rows - 1][i][1] = BLOCK;
            for (int i = 0; i < rows; i++) {
                for (int j = 0; j < columns; j++) {
                    if (grid[i][j] == 1) {
                        colors[i][j][0] = BLOCK;
                        colors[i][j][1] = BLOCK;
                        if (i > 0)
                            colors[i - 1][j][1] = BLOCK;
                        if (j > 0)
                            colors[i][j - 1][0] = BLOCK;
                    }
                    distances[i][j][0] = Integer.MAX_VALUE;
                    distances[i][j][1] = Integer.MAX_VALUE;
                }
            }
            colors[0][0][0] = GRAY;
            distances[0][0][0] = 0;
            Queue<int[]> queue = new LinkedList<int[]>();
            queue.offer(new int[]{0, 0, 0});
            while (!queue.isEmpty()) {
                int[] status = queue.poll();
                int row = status[0], column = status[1], direction = status[2];
                int distance = distances[row][column][direction];
                int newRow = row + 1, newColumn = column + 1, newDirection = 1 - direction;
                int newDistance = distance + 1;
                if (newRow < rows && colors[newRow][column][direction] == WHITE) {
                    colors[newRow][column][direction] = GRAY;
                    distances[newRow][column][direction] = newDistance;
                    queue.offer(new int[]{newRow, column, direction});
                }
                if (newColumn < columns && colors[row][newColumn][direction] == WHITE) {
                    colors[row][newColumn][direction] = GRAY;
                    distances[row][newColumn][direction] = newDistance;
                    queue.offer(new int[]{row, newColumn, direction});
                }
                if (colors[row][column][newDirection] == WHITE) {
                    if (row < rows - 1 && column < columns - 1 && grid[row + 1][column + 1] == 0) {
                        colors[row][column][newDirection] = GRAY;
                        distances[row][column][newDirection] = newDistance;
                        queue.offer(new int[]{row, column, newDirection});
                    }
                }
                colors[row][column][direction] = BLACK;
            }
            int totalMoves = distances[rows - 1][columns - 2][0];
            return totalMoves == Integer.MAX_VALUE ? -1 : totalMoves;
        }
    }
    
  • // OJ: https://leetcode.com/problems/minimum-moves-to-reach-target-with-rotations/
    // Time: O(N^2)
    // Space: O(N^2)
    class Solution {
    public:
        int minimumMoves(vector<vector<int>>& G) {
            queue<int> q;
            q.push(0);
            unordered_set<int> seen;
            seen.insert(0);
            int step = 0, N = G.size();
            auto safe = [&](int p) {
                int y = p % 1000, x = p / 1000 % 1000, dir = p / 1000000;
                if (x < 0 || y < 0 || x >= N || y >= N || G[x][y] == 1) return false;
                return (dir == 0 && y + 1 < N && G[x][y + 1] == 0)
                    || (dir == 1 && x + 1 < N && G[x + 1][y] == 0);
            };
            while (q.size()) {
                int cnt = q.size();
                while (cnt--) {
                    int p = q.front(), y = p % 1000, x = p / 1000 % 1000, dir = p / 1000000;
                    if (x == N - 1 && y == N - 2) return step;
                    q.pop();
                    // move right
                    int r = p + 1;
                    if (seen.count(r) == 0 && safe(r)) {
                        seen.insert(r);
                        q.push(r);
                    }
                    // move down
                    int d = p + 1000;
                    if (seen.count(d) == 0 && safe(d)) {
                        seen.insert(d);
                        q.push(d);
                    }
                    // rotate closewise
                    if (dir == 0) {
                        int p2 = 1000000 + x * 1000 + y, cx = x + 1, cy = y + 1;
                        if (seen.count(p2) || !safe(p2) || cx < 0 || cy < 0 || cx >= N || cy >= N || G[cx][cy] == 1) continue;
                        seen.insert(p2);
                        q.push(p2);
                    }
                    // rotate counterclosewise
                    if (dir == 1) {
                        int p2 = x * 1000 + y, cx = x + 1, cy = y + 1;
                        if (seen.count(p2) || !safe(p2) || cx < 0 || cy < 0 || cx >= N || cy >= N || G[cx][cy] == 1) continue;
                        seen.insert(p2);
                        q.push(p2);
                    }
                }
                ++step;
            }
            return -1;
        }
    };
    
  • # 1210. Minimum Moves to Reach Target with Rotations
    # https://leetcode.com/problems/minimum-moves-to-reach-target-with-rotations
    
    class Solution:
        def minimumMoves(self, grid: List[List[int]]) -> int:
            n = len(grid)
            start = (0, 0, 0, 1)
            end = (n - 1, n - 2, n - 1, n - 1)
            curr_level = {start}
            moves = 0
            visited = set()
            
            while curr_level:
                next_level = set()
                for pos in curr_level:
                    visited.add(pos)
                    r1, c1, r2, c2 = pos
                    
                    # down
                    if c1 + 1 < n and grid[r1][c1+1] == 0 and c2 + 1 < n and grid[r2][c2+1] == 0:
                        if (r1, c1 + 1, r2, c2 + 1) not in visited:
                            next_level.add((r1, c1 + 1, r2, c2 + 1))
                    
                    # right
                    if r1 + 1 < n and grid[r1+1][c1] == 0 and r2 + 1 < n and grid[r2+1][c2] == 0:
                        if (r1 + 1, c1, r2 + 1, c1) not in visited:
                            next_level.add((r1 + 1, c1, r2 + 1, c2))
                    
                    # clockwise
                    if r1 == r2 and c2 == c1 + 1 and r1 + 1 < n and grid[r1+1][c1] + grid[r1+1][c1+1] == 0 :
                        if (r1, c1, r1 + 1, c1) not in visited:
                            next_level.add((r1, c1, r1 + 1, c1))
                    
                    # counter-clockwise
                    if c1 == c2 and r2 == r1 + 1 and c1 + 1 < n and grid[r1][c1+1] + grid[r1+1][c1+1] == 0:
                        if (r1, c1, r1, c1 + 1) not in visited:
                            next_level.add((r1, c1, r1, c1 + 1))
                            
                if end in next_level: return moves + 1
                
                curr_level = next_level
                moves += 1
                
            return -1
    
    

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