Dynamic Programming Questions Medium
In the context of the Unique Paths problem, the concept of optimal substructure refers to the property that an optimal solution to the problem can be constructed from optimal solutions to its subproblems.
The Unique Paths problem involves finding the number of unique paths from the top-left corner to the bottom-right corner of a grid, where you can only move down or right. To solve this problem using dynamic programming, we can break it down into smaller subproblems.
Let's consider a grid with dimensions m x n. To reach the bottom-right corner from the top-left corner, we can either move down or move right. Therefore, the number of unique paths to reach a specific cell (i, j) in the grid can be calculated by adding the number of unique paths to reach the cell above it (i-1, j) and the number of unique paths to reach the cell to its left (i, j-1).
By breaking down the problem into smaller subproblems, we can calculate the number of unique paths for each cell in the grid, starting from the top-left corner and moving towards the bottom-right corner. This way, we can build up the solution to the original problem by utilizing the solutions to its subproblems.
The optimal substructure property comes into play when we consider the overall number of unique paths. Since the number of unique paths to reach a specific cell depends on the number of unique paths to reach its neighboring cells, we can use this information to calculate the total number of unique paths from the top-left corner to the bottom-right corner.
In summary, the concept of optimal substructure in the context of the Unique Paths problem means that the optimal solution to the problem can be obtained by combining the optimal solutions to its subproblems, which in this case involves calculating the number of unique paths for each cell in the grid.