Count Restricted Frog Paths on a Grid

Question

The question asks the candidate to calculate the number of restricted paths a frog can take on a grid, adhering to specific movement constraints.

Codemia · Accepted Answer

To solve the problem of counting restricted frog paths on a grid, we first need to define the grid's dimensions and the specific movement constraints imposed on the frog. For instance, if the frog can only move right or down, we can leverage combinatorial mathematics to determine the total number of unique paths.

The problem can be approached using combinatorial mathematics, specifically the concept of binomial coefficients. The number of unique paths from the top-left corner to the bottom-right corner of a grid can be represented as choosing a set number of moves down and a set number of moves right. If the grid is of size m x n, the formula to calculate the number of unique paths is given by:

$$ C(m+n-2, m-1) = \frac{(m+n-2)!}{(m-1)!(n-1)!} $$

This formula counts the ways to arrange the moves.

To implement this, we can use dynamic programming to build a solution iteratively. We create a 2D array where each cell represents the number of ways to reach that cell. The base case is that there is one way to reach any cell in the first row or the first column. The recurrence relation can be expressed as:

$$ dp[i][j] = dp[i-1][j] + dp[i][j-1] $$

This means the number of ways to reach cell (i, j) is the sum of the ways to reach the cell directly above it and the cell directly to the left.

The time complexity of this dynamic programming approach is O(m * n), where m is the number of rows and n is the number of columns in the grid. The space complexity can be reduced to O(n) by only storing the current and previous rows, as we only need the last computed values to calculate the next row.

It is important to consider edge cases such as a grid of size 1x1, where the number of paths is trivially 1. Additionally, if the frog's movement constraints are altered (e.g., allowing diagonal moves), the algorithm would need to be adjusted accordingly to account for the new valid movements.

Count Restricted Frog Paths on a Grid

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