2275. Largest Combination With Bitwise AND Greater Than Zero - Detailed Explanation

Problem Statement

Given an array of positive integers, your task is to choose a combination (i.e., a subset) of these integers such that the bitwise AND of all numbers in the combination is greater than 0. In other words, you want to select as many numbers as possible so that there exists at least one bit position where every selected number has a 1. The goal is to return the maximum number of integers you can choose that satisfy this property.

Example 1

• Input: candidates = [16,17,71,62,12,24,14]

• Output: 4

• Explanation:
  One possible combination of maximum size is [16,62,12,24]. Notice that each of these numbers has the 4th bit (from the right, 0-indexed) set (i.e., the bit corresponding to the value 8 is 1 in each number), ensuring that the bitwise AND of the combination is at least 8, which is greater than 0.

Example 2

• Input: candidates = [8,8]
• Output: 2
• Explanation:
  Both numbers are the same and are 8 (binary 1000). Their bitwise AND is 8, and the largest valid combination uses both numbers.

Example 3

• Input: candidates = [1,2,3,4]
• Output: 2
• Explanation:
  One optimal choice is [1,3] (both have the least significant bit set) or even [2,4] if you consider another common bit. The maximum possible valid combination here contains 2 numbers.

Constraints:

• 1 ≤ candidates.length ≤ 10⁵
• 1 ≤ candidates[i] ≤ 10⁹

Hints

1. Common Bit Observation:
   For a combination of numbers to have a bitwise AND greater than 0, they must share at least one common set bit (i.e., there exists a bit position where every number in the combination has a 1).

2. Bit-by-Bit Analysis:
   Instead of trying all possible subsets (which is infeasible for large arrays), consider each bit position (from 0 to 31, because candidates are positive integers within 32-bit range). For each bit position, count how many numbers have that bit set.

Detailed Explanation

The key insight for this problem is that the bitwise AND of a set of numbers is greater than 0 if and only if there exists at least one bit position where every number in the set has that bit set to 1. Instead of testing every possible subset, you can solve the problem by answering the following question for each bit position:

• How many numbers in the array have the i-th bit set?

If you have a count for a particular bit position ( i ), then you can choose all numbers that have that bit set and the bitwise AND of that combination will always be at least ( 2^i ) (because that bit will remain set in the AND result). Therefore, the maximum number of numbers you can choose such that their bitwise AND is non-zero is simply the maximum count among all bit positions.

Step-by-Step Walkthrough:

1. Initialize an Answer Variable:
   Begin with a variable (say, ans) that will track the maximum count of numbers sharing a common bit.

2. Iterate Over All 32 Bit Positions:
   For each bit ( i ) from 0 to 31:

   • Initialize a counter to 0.
   • For every number in the candidates array, check if the ( i )-th bit is set. This is done using the expression:

     if (number & (1 << i)) != 0:
     

     If the condition is true, increase the counter.

3. Update Maximum Count:
   After processing each bit position, update ans with the maximum value between the current ans and the count for that bit.

4. Return the Result:
   Once all bits have been processed, the variable ans contains the size of the largest combination where the bitwise AND is greater than 0.

Why This Works:

• A valid combination must have at least one bit common to all its members.
• By selecting the bit with the highest frequency in the array, you automatically ensure that every number in that subset has that bit set.
• Thus, the maximum frequency among all bits is the answer.

Complexity Analysis:

• Time Complexity: ( O(32 \times n) ) which simplifies to ( O(n) ) because there are always 32 iterations over ( n ) numbers.

• Space Complexity: ( O(1) ) additional space (only a few counter variables are used).

Code Solutions

Python Implementation

Java Implementation