Count all ordered pairs (A, B) such that A and B are disjoint

Given an array arr[] of size n contains distinct elements, the task is to find the number of ordered pairs (A, B) that can be made where A and B are subsets of the given array arr[] and A ∩ B = Φ (i.e, Both A and B subset should be disjoint or no common elements between them).

Example:

Input: arr = {1, 2}
Output: 9
Explanation: The subsets of the array are {}, {1}, {2}, {3}, {1, 2}, {1, 3}, {2, 3}, and {1, 2, 3}. The ordered pairs (A, B) where A and B disjoint subsets are: ({}, {}), ({}, {1}), ({}, {2}), ({1}, {}), ({2}, {}), ({1, 2}, {}), ({}, {1, 2}, ({1}, {2}, ({2}, {1})

Input: arr = {1, 2, 3}
Output: 27

Approach:

Every element has three options, It will either be included in A, B, or not included in either A or B. Hence, there would be 3^n possible combinations for given array of size n.

Steps-by-step approach:

• Set the modulo constant MOD to 1e9 + 7.
• Define a function binpow() that calculates the power using binary exponentiation.
  • Take the modulo of the base to keep it within the specified range.
  • Initialize result to 1.
  • Iterate until the exponent is greater than 0.
  • If the current bit of the exponent is 1, multiply the result by the base.
  • Square the base and divide the exponent by 2 in each iteration.
  • Return the final result.
• Solve Function:
  • Define a function solve that takes an integer n and a vector arr.
  • Call binpow(3, n) to calculate (3^n) mod (1e9 + 7).
  • Return the result.

Below is the implementation of the above approach:

27

Time Complexity: O(log(n))
Auxiliary Space: O(log(n))