Basics of Bitwise Enumeration

Bitwise enumeration is a technique that explores all possible selections using bit operations (0,1). This approach is useful for:

• Enumerating all subsets of a set with N elements.
• Exploring all possible choices when there are N binary decisions (e.g., “YES” or “NO”).

These scenarios are essentially the same. For example, enumerating all subsets of a set with N elements is equivalent to determining all possible ways to either “select” or “not select” each element.

Concept of Bitwise Enumeration

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A binary decision (two choices) can be represented using binary digits (0 or 1). For example, assume three animals: dog, cat, and bird. If selecting an animal is xrepresented as 1 and not selecting as 0, then different combinations can be represented in binary:

Each pattern represents a unique subset selection, which can be efficiently generated using a simple algorithm.

When to Use Bitwise Enumeration

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Bitwise enumeration is effective when exploring all patterns involving N binary choices. Although alternative implementations exist, bitwise enumeration offers a concise and intuitive approach. However, since the computational complexity is \(N * 2^N\), this method becomes impractical for large values of N.

Implementation of Bitwise Enumeration

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The following Python implementation generates all subsets of a given list:

sample_list = ['A', 'B', 'C', 'D']

# Count elements
len_sample_list = len(sample_list)

# Explore all selection patterns using 2^N iterations
for i in range(2 ** len_sample_list):
    out_list = []
    # Check each bit position
    for j in range(len_sample_list):
        # Use bitwise AND to determine selected elements
        if (i >> j) & 1:
            out_list.append(sample_list[j])
    print(out_list)

Since the input list contains four elements, \(2^4 = 16\) different subsets are generated.