Dynamic Programming Questions Medium
Dynamic Programming can be used to solve the 0/1 Knapsack problem by breaking it down into smaller subproblems and solving them iteratively. The 0/1 Knapsack problem involves selecting a subset of items with maximum total value, while ensuring that the total weight of the selected items does not exceed a given capacity.
To solve this problem using Dynamic Programming, we can create a 2D array, often referred to as a memoization table, where each cell represents the maximum value that can be obtained by considering a subset of items up to a certain capacity. The rows of the table represent the items, and the columns represent the capacity.
We start by initializing the first row and column of the table with zeros, as they represent the case when there are no items or the capacity is zero. Then, for each item, we iterate through the capacities from 0 to the maximum capacity. For each cell, we have two options:
1. If the weight of the current item is less than or equal to the current capacity, we can either include or exclude the item. If we include the item, we add its value to the maximum value obtained by considering the remaining capacity (current capacity minus the weight of the item) and the previous items. If we exclude the item, we simply take the maximum value obtained by considering the previous items at the same capacity.
2. If the weight of the current item is greater than the current capacity, we cannot include the item, so we take the maximum value obtained by considering the previous items at the same capacity.
By filling the memoization table iteratively, we can find the maximum value that can be obtained by considering all the items and the given capacity. The value in the bottom-right cell of the table represents the maximum value for the entire problem.
To determine which items were selected to achieve this maximum value, we can backtrack through the memoization table. Starting from the bottom-right cell, we check if the value in the current cell is equal to the value obtained by excluding the current item at the same capacity. If it is not, it means the current item was included, and we move to the cell above and to the left. We repeat this process until we reach the top-left cell, which represents the case when there are no items and the capacity is zero.
Overall, Dynamic Programming allows us to solve the 0/1 Knapsack problem efficiently by breaking it down into smaller subproblems and solving them iteratively, using a memoization table to store and reuse the solutions to these subproblems.