Maximizing Profits using Bitwise OR

Given two integers N, K and two integer arrays weights[] and profits[] each of size N where weights[i] is the weight associated with the i^th item and profit[i] is the profit associated with the ith item. Maximize the profit by selecting a subset of items such that the bitwise OR of their weights is less than or equal to the given threshold K.

Examples:

Input: N = 5, K = 15, weights[] = {9, 5, 12, 7, 8}, profits[] = {20, 15, 30, 12, 18}
Output: 95
Explanation: Our task is to select a subset of weights such that their bitwise OR is less than or equal to 15. The bitwise OR of 9, 5, 12, 7 and 8 is 15, satisfying the OR condition. Thus, the maximum possible sum of profits is 95, as weights 9, 5, 12, 7 and 8 are selected.

Input: N = 4, K = 6, weights[] = {3, 4, 2, 1}, profits[] = {10, 12, 8, 6}
Output: 24
Explanation: Our task is to select a subset of weights such that their bitwise OR is less than or equal to 6. The bitwise OR of 3, 2, and 1 is 3, satisfying the OR condition. Thus, the maximum possible sum of profits is 47, as weights 3, 2, and 1 are selected.

Approach: This can be solved using the following approach:

The approach is to use Recursion to evaluate all possible combinations of items to find the subset that maximizes the sum of profits while ensuring that the bitwise OR operation of the selected items does not exceed the specified threshold K

Steps to solve the problem:

• Initialize a variable max_profit to 0. This variable will be used to keep track of the maximum possible sum of profits.
• Iterate through all possible subsets of items using binary representations of numbers from 0 to 2^N - 1, where N is the number of items.
  • For each subset, calculate the bitwise OR (current_or) and sum of profit s(current_profit) of the selected items.
  • If the current_or is less than or equal to the given threshold K, update max_profit if the current_profit is greater than the max_profit.
• After processing all subsets, max_profit will hold the maximum possible sum of profits that satisfy the OR condition.
• Return max_profit as the solution to the problem.

Below is the implementation of the above approach:

95

Time Complexity: O(2^N), where N is the size of weights[] and profits[] array
Auxiliary Space: O(N)