Understanding the Relationship Between setbits in Bit Manipulation: A Guide to Solving Puzzles

Explore how to find pairs of numbers that satisfy the equation involving `setbits` using bit manipulation techniques in Python.
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This video is based on the question https://stackoverflow.com/q/66457128/ asked by the user 'P Shiva Rama Krishna Chowdary' ( https://stackoverflow.com/u/13698289/ ) and on the answer https://stackoverflow.com/a/66457530/ provided by the user 'harold' ( https://stackoverflow.com/u/555045/ ) at 'Stack Overflow' website. Thanks to these great users and Stackexchange community for their contributions.

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Understanding setbits and Their Relationship in Bit Manipulation

When it comes to programming challenges, especially in coding platforms like HackerEarth, you might encounter intriguing puzzles revolving around bit manipulation. One such problem that has been posed is: How can we find pairs x and y from arrays such that the equation involving the number of set bits is satisfied? The equation is represented as:

Number of setbits(x OR y) + Number of setbits(x AND y) = m

This seemingly complex problem can actually be distilled down into a more manageable form. Let's explore how this works.

Breaking Down the Problem

The equation you are trying to satisfy involves two operations: OR (|) and AND (&). To solve this problem, we can leverage an interesting property of set bits (the bits that are '1' in a binary representation).

The Setbit Property Explained

There is a key property that can help us simplify this equation:

[[See Video to Reveal this Text or Code Snippet]]

This property tells us that the total number of set bits does not change when we perform the OR and AND operations on two numbers x and y. Essentially, the set bits are merely redistributed but the total count remains the same.

Example to Illustrate the Point

Let's take a look at a few examples with just 1-bit numbers:

[[See Video to Reveal this Text or Code Snippet]]

As seen in the table above, across all combinations, the count of bits in x & y and x | y matches the total number of set bits in x and y. This means we are free to treat the number of set bits as a collective concept rather than focusing on individual operations.

Simplifying the Problem

Given this property, we can transform our original problem into a more straightforward task. Instead of focusing solely on the setbits, we transform it into finding pairs of numbers whose total set bits directly add up to m. Essentially, the problem becomes about recognizing combinations of integers rather than isolating their bit manipulations.

How to Approach It

To find pairs x and y from an array that satisfies the equation, follow these steps:

Calculate Total Set Bits: For each number in your array, calculate the total number of set bits it contains.

Pair Check: Loop through the array to find pairs of numbers. For each pair (x, y), check if the sum of their individual set bits equals the target m.

Return Pairs: Gather and return the pairs that satisfy this condition.

Conclusion

While the problem initially presents itself as complex due to the inclusion of bit manipulation operations, recognizing the underlying properties of setbits allows us to simplify and approach it effectively. By shifting our focus to the summation of set bits, we can tackle the problem with clarity and methodical precision.

The next time you encounter a similar challenge, refer back to this property, and you’ll find yourself well-equipped to handle such questions with confidence.

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