Minimum Non-Zero Product of the Array Elements

Problem Description

You are given a positive integer p. Consider an array nums (1-indexed) that consists of the integers in the inclusive range [1, 2^p - 1] in their binary representations. You are allowed to do the following operation any number of times:

• Choose two elements x and y from nums.
• Choose a bit in x and swap it with its corresponding bit in y.

Find the minimum non-zero product of nums after performing the above operation any number of times. Return this product modulo 10^9 + 7. The answer should be the minimum product before the modulo operation is done.

Key Insights

• The elements in nums are all integers from 1 to 2^p - 1.
• The operation allows complete rearrangement of bits across all elements, meaning any bit can be swapped with any other bit of another number.
• The goal is to minimize the product of all elements, which can be achieved by strategically arranging the bits.
• The minimum non-zero product can be derived from the multiplication of the smallest non-zero elements in the array.

Space and Time Complexity

Time Complexity: O(1)
Space Complexity: O(1)

Solution

To solve the problem, we can utilize the fact that the product of all elements in the array can be minimized by rearranging the bits. The main points to consider are:

1. The largest number in the range [1, 2^p - 1] is 2^p - 1.
2. The minimum product can be calculated as:
   • The final product is 1 * 2 * ... * (2^p - 1).
   • The product can be calculated as (2^p - 1)!, but since we only need the minimum non-zero product before the modulo, we can directly derive this from the arrangement of bits.

We can utilize the properties of modulo arithmetic and factorial calculations to derive the minimum non-zero product efficiently.

Code Solutions