algorithm - How to count the number of set bits in a 32-bit integer?

8 bits representing the number 7 look like this:

00000111

Three bits are set.

What are algorithms to determine the number of set bits in a 32-bit integer?

Answer

This is known as the 'Hamming Weight', 'popcount' or 'sideways addition'.

The 'best' algorithm really depends on which CPU you are on and what your usage pattern is.

Some CPUs have a single built-in instruction to do it and others have parallel instructions which act on bit vectors. The parallel instructions (like x86's popcnt, on CPUs where it's supported) will almost certainly be fastest. Some other architectures may have a slow instruction implemented with a microcoded loop that tests a bit per cycle (citation needed).

A pre-populated table lookup method can be very fast if your CPU has a large cache and/or you are doing lots of these instructions in a tight loop. However it can suffer because of the expense of a 'cache miss', where the CPU has to fetch some of the table from main memory.

If you know that your bytes will be mostly 0's or mostly 1's then there are very efficient algorithms for these scenarios.

I believe a very good general purpose algorithm is the following, known as 'parallel' or 'variable-precision SWAR algorithm'. I have expressed this in a C-like pseudo language, you may need to adjust it to work for a particular language (e.g. using uint32_t for C++ and >>> in Java):

int numberOfSetBits(int i)
{
     // Java: use >>> instead of >>
     // C or C++: use uint32_t
     i = i - ((i >> 1) & 0x55555555);
     i = (i & 0x33333333) + ((i >> 2) & 0x33333333);
     return (((i + (i >> 4)) & 0x0F0F0F0F) * 0x01010101) >> 24;
}

This has the best worst-case behaviour of any of the algorithms discussed, so will efficiently deal with any usage pattern or values you throw at it.

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This bitwise-SWAR algorithm could parallelize to be done in multiple vector elements at once, instead of in a single integer register, for a speedup on CPUs with SIMD but no usable popcount instruction. (e.g. x86-64 code that has to run on any CPU, not just Nehalem or later.)

However, the best way to use vector instructions for popcount is usually by using a variable-shuffle to do a table-lookup for 4 bits at a time of each byte in parallel. (The 4 bits index a 16 entry table held in a vector register).

On Intel CPUs, the hardware 64bit popcnt instruction can outperform an SSSE3 PSHUFB bit-parallel implementation by about a factor of 2, but only if your compiler gets it just right. Otherwise SSE can come out significantly ahead. Newer compiler versions are aware of the popcnt false dependency problem on Intel.

References:

https://graphics.stanford.edu/~seander/bithacks.html

https://en.wikipedia.org/wiki/Hamming_weight

http://gurmeet.net/puzzles/fast-bit-counting-routines/

http://aggregate.ee.engr.uky.edu/MAGIC/#Population%20Count%20(Ones%20Count)