2044. Count Number of Maximum Bitwise-OR Subsets - Detailed Explanation

Problem Statement

You are given an array of positive integers, nums. A subset of nums is any set that can be formed by deleting some (or none) of the elements. The bitwise OR of a subset is defined as the OR of all the numbers in that subset. Your task is to determine:

1. The maximum bitwise OR value that can be obtained from any non-empty subset of nums.
2. The number of subsets that achieve this maximum bitwise OR value.

Return the count of such subsets.

Example Inputs & Outputs

• Example 1:

  • Input: nums = [3, 1]
  • Output: 2
  • Explanation:
    The possible non-empty subsets are:
    • Subset [3] has OR = 3.
    • Subset [1] has OR = 1.
    • Subset [3, 1] has OR = 3 OR 1 = 3.
      The maximum bitwise OR value is 3, and there are 2 subsets that achieve this (i.e. [3] and [3, 1]).

• Example 2:

  • Input: nums = [2, 2, 2]
  • Output: 7
  • Explanation:
    Every non-empty subset will have OR = 2. There are (2^3 - 1 = 7) non-empty subsets, and all yield the maximum OR value 2.

• Example 3:

  • Input: nums = [3, 2, 1, 5]
  • Output: For instance, if the maximum OR is computed to be 7, count all subsets that OR to 7.
    (A detailed explanation of the subsets can be performed in a similar manner.)

Constraints

• (1 \leq \text{nums.length} \leq 16)
• (1 \leq \text{nums}[i] \leq 10^5)
• All numbers are positive integers.

Hints

1. Determine the Maximum OR Value:
   First, compute the OR of all numbers in the array. This value is the highest possible OR you can obtain from any subset because OR is a monotonically increasing operation when you add more bits.

2. Enumerate Subsets:
   Since the maximum number of elements is small (up to 16), you can afford to enumerate all non-empty subsets (using either bit manipulation or recursion) and count those whose OR equals the maximum OR.

3. Efficient Backtracking:
   Instead of iterating over every subset explicitly, consider using a backtracking (DFS) approach that accumulates the OR value as you choose to include or exclude elements.

Approach 1: Brute Force (Subset Enumeration)

Step-by-Step Walkthrough

1. Calculate Maximum OR:

   • Iterate over nums and compute the OR of all elements. This is the target value you want each subset to match.

2. Enumerate All Subsets:

   • There are (2^n - 1) non-empty subsets for an array of length (n).
   • For each subset, calculate its OR value.

3. Count Matching Subsets:

   • Compare the computed OR of each subset with the maximum OR.
   • Increment your counter if they match.

4. Return the Count.

Visual Example (Example 1: [3, 1])

• Step 1:
  Maximum OR = (3 ,|; 1 = 3).

• Step 2:
  Enumerate subsets:

  • Subset {3} → OR = 3
  • Subset {1} → OR = 1
  • Subset {3, 1} → OR = (3 ,|; 1 = 3)

• Step 3:
  Two subsets produce the maximum OR (3): {3} and {3, 1}.

Approach 2: Backtracking / DFS

Step-by-Step Walkthrough

1. Recursive Function:
   Write a DFS function that, given an index and a current OR value, makes two recursive calls: one including the current number and one excluding it.

2. Base Case:
   When you reach the end of the array, check if the current OR equals the maximum OR value computed earlier. If yes, count this subset.

3. Global Counter:
   Use a counter (or pass the count via return values) to sum up the valid subsets.

Visual Example (Using [3, 1])

• Call DFS with index = 0, current OR = 0.
  • Include 3:
    Update OR → (0 ,|; 3 = 3).
    Call DFS with index = 1.
    • Include 1:
      Update OR → (3 ,|; 1 = 3). → Valid subset.
    • Exclude 1:
      OR remains 3 → Valid subset.
  • Exclude 3:
    OR remains 0.
    Call DFS with index = 1.
    • Include 1:
      Update OR → (0 ,|; 1 = 1). → Not valid.
    • Exclude 1:
      OR remains 0 (empty subset, skip if considering only non-empty subsets).

Code Examples

Python Code