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Count Number of Maximum Bitwise-OR Subsets

Problem Overview

You are given an integer array nums. The task is to find the maximum possible bitwise OR of any subset of nums and return the number of different non-empty subsets that achieve this maximum OR.

Example

Example 1

  • Input: nums = [3,1]
  • Output: 2
  • Explanation: There are two subsets with the maximum bitwise OR of 3: [3] and [3, 1].

Example 2

  • Input: nums = [2,2,2]
  • Output: 7
  • Explanation: All non-empty subsets have a bitwise OR of 2.

Step-by-Step Explanation in Different Languages

C++ Code Explanation

  1. Initialize Variables:

    • First, compute the maximum possible OR for the entire array. This is the OR of all elements combined.

  2. Backtracking Function:

    • Define a recursive backtracking function that starts with an initial OR value of 0.
    • At each step, include the current element in the OR and check if the OR matches the maximum.

  3. Count Valid Subsets:

    • If the OR equals the maximum OR, increment a global count that keeps track of how many subsets match the desired OR.

  4. Recursive Exploration:

    • The function then recursively explores all possible subsets by either including or excluding each element.

  5. Return Result:

    • Once all subsets are explored, return the total count of valid subsets.

Java Code Explanation

  1. Initialize maxOR:

    • Calculate the maximum OR by combining all elements in the array.

  2. Backtracking Helper Function:

    • Define a recursive helper method that takes the current OR value, the current index in the array, and the number of subsets found so far.

  3. Increment Count:

    • Whenever the current OR value matches the maximum OR, increment the count.

  4. Recursive Calls:

    • Use a loop to recursively explore each possible subset by including or skipping the current element at each step.

  5. Final Count:

    • After all subsets are generated, return the total count of subsets that achieved the maximum OR.

JavaScript Code Explanation

  1. Calculate Maximum OR:

    • Compute the maximum OR value by OR-ing all elements in the array.

  2. Recursive Backtracking:

    • Use a recursive function to explore each subset. For each subset, calculate its OR value and check if it matches the maximum OR.

  3. Base Case and Increment:

    • If the current subset’s OR value matches the maximum OR, increment a counter.

  4. Exploration of Subsets:

    • Recursively include each element in the subset or skip it, thus generating all possible subsets.

  5. Result Return:

    • After exploring all subsets, return the total number of subsets that achieve the maximum OR.

Python Code Explanation

  1. Determine Maximum OR:

    • Start by calculating the maximum OR for the entire array.

  2. Recursive Backtracking:

    • Define a recursive function to explore each subset, starting from an initial OR value of 0 and adding elements one by one.

  3. Check Subsets:

    • For each subset, check if its OR equals the maximum OR and increment a counter if it does.

  4. Recursion:

    • Recursively try all possible combinations of elements in the array, generating every possible subset.

  5. Return Count:

    • After exploring all subsets, return the total number that match the maximum OR.

Go Code Explanation

  1. Initialize Variables:

    • Compute the maximum OR value by OR-ing all elements in the array.

  2. Recursive Function:

    • Use a recursive backtracking function that explores all subsets. For each subset, compute its OR value and check if it matches the maximum OR.

  3. Track Count:

    • If a subset’s OR matches the maximum OR, increment a global count.

  4. Recursive Exploration:

    • Recursively include or skip elements in the subset, thus generating every possible subset.

  5. Final Count:

    • After exploring all possible subsets, return the total count of subsets that match the maximum OR.