Dynamic Programming Questions Medium
In the context of the Word Break 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 Word Break problem involves determining whether a given string can be segmented into a space-separated sequence of dictionary words. For example, given the string "applepenapple", and a dictionary containing the words "apple" and "pen", we need to determine if the string can be segmented into "apple pen apple".
To solve this problem using dynamic programming, we can break it down into smaller subproblems. We can consider each prefix of the given string and check if it can be segmented into words from the dictionary. By solving these subproblems and storing their results, we can build up the solution for the original problem.
The optimal substructure property comes into play when considering the subproblems. If we can determine that a prefix of the string can be segmented into words, we can then check if the remaining suffix can also be segmented into words. If both the prefix and suffix can be segmented, then the entire string can be segmented. This recursive relationship between the prefix and suffix subproblems allows us to build the solution for the original problem.
By utilizing the optimal substructure property, we can avoid redundant computations and improve the efficiency of the solution. We can store the results of the subproblems in a memoization table or use bottom-up dynamic programming to iteratively solve the subproblems and build the solution for the original problem.
In summary, the concept of optimal substructure in the Word Break problem allows us to break down the problem into smaller subproblems and construct an optimal solution by combining the optimal solutions to these subproblems.