Out of Boundary Paths

Problem Statement

In a given problem, we are working with an m x n grid where a ball is placed initially at a specified location [startRow, startColumn]. The core task is to compute how many distinct paths are available for this ball to exit the grid, given that it can only make a limited number of moves, precisely maxMove moves. A move is defined as shifting the ball to any of its four adjacent grid cells (Up, Down, Left, Right). It should be noted that moving the ball out of the grid's boundary is regarded as exiting the grid. The complexity of computing the total number of such paths is simplified by introducing the modulus operation with 10^9 + 7 to handle potentially very large numbers.

Examples

Example 1

Input:

m = 2, n = 2, maxMove = 2, startRow = 0, startColumn = 0

Output:

6

Example 2

Input:

m = 1, n = 3, maxMove = 3, startRow = 0, startColumn = 1

Output:

12

Constraints

• 1 <= m, n <= 50
• 0 <= maxMove <= 50
• 0 <= startRow < m
• 0 <= startColumn < n

Approach and Intuition

To solve this problem efficiently, we can use a dynamic programming (DP) approach that leverages the ability to recursively calculate the number of exit paths based on smaller, previously computed subproblems.

1. Initialize a DP table where dp[i][j][k] represents the number of ways to reach the cell (i, j) using exactly k moves.
2. Start with the initial state set at the starting position with dp[startRow][startColumn][0] = 1 since there is one way to be at the start without any moves.
3. For each move from 1 to maxMove, update the DP table based on the potential moves from each cell:
   • Move up (i-1, j)
   • Move down (i+1, j)
   • Move left (i, j-1)
   • Move right (i, j+1)
4. For every move when updating the state, consider if moving to or from a boundary cell. If a move takes the ball out of bounds, accumulate those outcomes since they represent valid exit paths.
5. Since the result can grow large, use modular arithmetic to keep the counts manageable, as specified by the problem with 10^9 + 7.

Through this DP approach alongside boundary checks and accumulation of exit valid paths, we can compute the total number of ways the ball can exit the grid within the specified move limit. This provides a clear and perceptive method to solve the problem by dynamically building upon simpler subproblem solutions.

Solutions

class Solution {
  public int calculatePaths(int rows, int cols, int maxMove, int startRow, int startCol) {
    int modVal = (int)(1e9 + 7);
    int[][] counts = new int[rows][cols];
    counts[startRow][startCol] = 1;
    int result = 0;
    for (int step = 1; step <= maxMove; step++) {
      int[][] nextCounts = new int[rows][cols];
      for (int i = 0; i < rows; i++) {
        for (int j = 0; j < cols; j++) {
          if (i == rows - 1) result = (result + counts[i][j]) % modVal;
          if (j == cols - 1) result = (result + counts[i][j]) % modVal;
          if (i == 0) result = (result + counts[i][j]) % modVal;
          if (j == 0) result = (result + counts[i][j]) % modVal;
          nextCounts[i][j] = (
              ((i > 0 ? counts[i - 1][j] : 0) + (i < rows - 1 ? counts[i + 1][j] : 0)) % modVal +
              ((j > 0 ? counts[i][j - 1] : 0) + (j < cols - 1 ? counts[i][j + 1] : 0)) % modVal
          ) % modVal;
        }
      }
      counts = nextCounts;
    }
    return result;
  }
}