Dynamic Programming Questions Medium
In the context of the Partition Equal Subset Sum 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 Partition Equal Subset Sum problem involves determining whether a given set of positive integers can be partitioned into two subsets with equal sums. To solve this problem using dynamic programming, we can break it down into smaller subproblems and build up the solution iteratively.
The optimal substructure property in this problem can be observed by considering that if we can find a subset of the given set that sums up to half of the total sum, then the remaining elements will also sum up to half of the total sum. This is because the total sum is divided equally into two subsets.
By utilizing this optimal substructure property, we can solve the problem by considering each element in the given set and determining whether it can be included in the subset with the target sum. We can build a dynamic programming table where each cell represents whether a subset with a specific sum can be formed using a subset of the given elements.
Starting with the base case of an empty set and a sum of 0, we can fill in the table by considering each element in the given set. For each element, we check if it can be included in the subset with the target sum. If it can, we mark the corresponding cell in the table as true. Otherwise, we check if the subset without the current element can achieve the target sum. If it can, we mark the corresponding cell as true.
By iteratively filling in the table, we can determine whether a subset with the target sum exists. If the cell corresponding to the target sum is true, it means that a partition with equal sums can be achieved.
In summary, the concept of optimal substructure in the context of the Partition Equal Subset Sum problem allows us to break down the problem into smaller subproblems and build up the solution iteratively. By considering the optimal solutions to these subproblems, we can determine whether a given set can be partitioned into two subsets with equal sums.