Dynamic Programming Questions Medium
In the context of the Decode Ways 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.
In the Decode Ways problem, we are given a string of digits and we need to determine the number of ways to decode it into a sequence of letters. Each digit can be mapped to a letter (1 to A, 2 to B, and so on) and we need to find the total number of valid decodings.
To solve this problem using dynamic programming, we can break it down into smaller subproblems. Let's consider a string of length n. We can divide it into two parts: the first digit and the remaining n-1 digits, or the first two digits and the remaining n-2 digits.
Now, let's assume we have already solved the subproblems for the remaining n-1 digits and n-2 digits. We can use these solutions to construct the solution for the original problem.
If the first digit is a valid mapping to a letter (1 to 9), we can consider it as a single digit decoding and add the number of decodings for the remaining n-1 digits.
If the first two digits form a valid mapping (10 to 26), we can consider it as a double digit decoding and add the number of decodings for the remaining n-2 digits.
By considering all possible combinations of single and double digit decodings, we can find the total number of decodings for the original string.
This approach exhibits optimal substructure because the optimal solution for the original problem can be constructed by combining optimal solutions for its subproblems. By solving the subproblems and storing their solutions in a table, we can avoid redundant calculations and efficiently solve the problem using dynamic programming.