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The main idea is to use a 3D dynamic programming array dp[i][j][k] where i and j represent the current position on the grid, and k represents the number of moves remaining. The value at dp[i][j][k] represents the number of ways to move the ball out of bounds using at most k moves starting from (i, j).
Transitions are made by moving the ball in one of the four possible directions and decreasing the move count. If the ball goes out of bounds, increment the path count. The answer is the sum of paths from startRow, startColumn using maxMove moves.
Time Complexity: O(m * n * maxMove) as each grid cell and move combination is processed exactly once.
Space Complexity: O(m * n * maxMove) due to the storage of the DP array.
1using System;
2
3public class Solution {
4 public int FindPaths(int m, int n, int maxMove, int startRow, int startColumn) {
5 int MOD = 1000000007;
6 int[,,] dp = new int[m, n, maxMove + 1];
7 dp[startRow, startColumn, 0] = 1;
8 int count = 0;
9 int[][] directions = new int[][] {
10 new int[] {1, 0}, new int[] {-1, 0}, new int[] {0, 1}, new int[] {0, -1}
11 };
12 for (int move = 1; move <= maxMove; move++) {
13 for (int i = 0; i < m; i++) {
for (int j = 0; j < n; j++) {
foreach (int[] d in directions) {
int ni = i + d[0];
int nj = j + d[1];
if (ni < 0 || nj < 0 || ni >= m || nj >= n) {
count = (count + dp[i, j, move - 1]) % MOD;
} else {
dp[ni, nj, move] = (dp[ni, nj, move] + dp[i, j, move - 1]) % MOD;
}
}
}
}
}
return count;
}
public static void Main() {
var solution = new Solution();
Console.WriteLine(solution.FindPaths(2, 2, 2, 0, 0)); // Output: 6
Console.WriteLine(solution.FindPaths(1, 3, 3, 0, 1)); // Output: 12
}
}C# implementation leverages array manipulation with dynamic programming, updating possible moves and direction results. It calculates the result efficiently while maintaining the paths count variables with modulo arithmetic.
DFS with memoization is used to prevent recalculating results for already evaluated states by storing them in a map or dictionary. Each call is a recursive attempt to move in all directions from the current cell, decrementing the move count. Result of out-of-bounds attempts is stored so it can be reused, avoiding exponential time complexity.
This algorithm uses a function that considers the current position and remaining moves. For every call, explore in four possible directions and add to the path count if the ball gets out of bounds. Store already calculated counts in a dictionary for path optimization.
Time Complexity: O(m * n * maxMove) where each state is visited only once.
Space Complexity: O(m * n * maxMove) due to memoization storage.
The JavaScript implementation expands on DFS with the addition of memoization for recursive function optimization. Each progressive move in possible directions is checked, collecting results on encountering grid bounds, finally tabulating modular outcomes.