Home * Board Representation * Bitboards * Population Count
Hamming.jpg

Population count determines the cardinality of a bitboard, also called Hamming weight or sideways sum [1]. How many one bits exists in a 64-bit computer word? In computer chess, population count is used to evaluate the mobility of pieces from their attack sets, as already applied in Chess 4.6 on the CDC 6600 and CDC Cyber.

Future or recent x86-64 processors (AMD K10 - SSE4a, Intel Nehalem - SSE4.2) provide a 64-bit popcount instruction [2], available via C++ compiler intrinsic [3][4][5][6] or inline assembly [7]. Despite different Intrinsic prototypes (_mm_popcnt_u64 vs. popcnt64), Intel and AMD popcnt instructions are binary compatible, have same encoding (F3 [REX] 0F B8 /r), and both require bit 23 set in RCX of the CPUID function 0000_0001h.
Visualization of Hamming distance [8]

Code samples in C / C++, see Defining Bitboards

Recurrence Relation

The recursive recurrence relation of population counts can be transformed to iteration as well, but lacks an arithmetical sum-formula:


However, it is helpful to initialize a lookup table, i.e. for bytes:
unsigned char popCountOfByte256[];
 
void initpopCountOfByte256()
{
   popCountOfByte256[0] = 0;
   for (int i = 1; i < 256; i++)
      popCountOfByte256[i] = popCountOfByte256[i / 2] + (i & 1);
}

Empty or Single?

Often one has to deal with sparsely populated or even empty bitboards. To determine whether a bitboard is empty or a single populated power of two value, one may use simple boolean statements rather than a complete population count.


Empty Bitboards

To test a bitboard is empty, one compares it with zero, or use the logical not operator:
if ( x == 0 ) -> bitboard is empty
if ( !x )     -> bitboard is empty
The inverse condition (not empty) is of course
if ( x != 0 ) -> bitboard is not empty
if ( x )      -> bitboard is not empty


Single Populated Bitboards

If the bitboard is not empty, we can extract the LS1B to look whether it is equal with the bitboard. Alternatively and faster, we can reset the LS1B to look whether the bitboard becomes empty.
if ( x != 0 && (x & (x-1)) == 0 ) -> population count is one, power of two value
One can skip the leading x != 0 condition to test popcount <= 1:
if ( (x & (x-1)) == 0 ) -> population count is less or equal than one
Again the inverse relation tests, whether a bitboard has more than one bit set:
if ( x & (x-1) ) -> population count is greater than one
An alternative approach to determine single populated sets, aka power of two values is based on Inclusive LS1B separation divided by two equals the ones' decrement [9]:
if ( ((x ^ (x-1)) >> 1) == (x-1) ) -> population count is one, power of two value

Loop-Approaches


Too Slow

Brute force adding all 64-bits
int popCount (U64 x) {
   int count = 0;
   for (int i = 0; i < 64; i++, x >>= 1)
      count += (int)x & 1;
   return count;
}
Of course, this is a slow algorithm, which might be improved by testing x not empty rather than i < 64. But unrolled in parallel prefix manner it already reminds on this one.


Brian Kernighan's way

Consecutively reset LS1B in a loop body and counting loop cycles until the bitset becomes empty. Brian Kernighan [10] mentioned the trick in his and Ritchie's book The C Programming_Language, 2nd Edition 1988, exercise 2-9. However, the method was first published in 1960 by Peter Wegner [11], discovered independently by Derrick Henry Lehmer, published in 1964 [12]:
int popCount (U64 x) {
   int count = 0;
   while (x) {
       count++;
       x &= x - 1; // reset LS1B
   }
   return count;
}
Despite branches - this is still one of the fastest approaches for sparsely populated bitboards. Of course the more bits that are set, the longer it takes.


Lookup

Of course we can not use the whole bitboard as index to a lookup table - since it's size would be 18,446,744,073,709,551,616 bytes! If it is about the population count of a byte, we can simply construct a table lookup with 256 elements. For a bitboard that takes eight byte lookups we can add together:
unsigned char popCountOfByte256[];
 
void initpopCountOfByte256()
{
   popCountOfByte256[0] = 0;
   for (int i = 1; i < 256; i++)
      popCountOfByte256[i] = popCountOfByte256[i / 2] + (i & 1);
}
 
int popCount (U64 x) {
   return popCountOfByte256[ x        & 0xff] +
          popCountOfByte256[(x >>  8) & 0xff] +
          popCountOfByte256[(x >> 16) & 0xff] +
          popCountOfByte256[(x >> 24) & 0xff] +
          popCountOfByte256[(x >> 32) & 0xff] +
          popCountOfByte256[(x >> 40) & 0xff] +
          popCountOfByte256[(x >> 48) & 0xff] +
          popCountOfByte256[ x >> 56];
}
Looks quite expensive - one may use four 16-bit word-lookups with a pre-calculated 64KByte table though, but that pollutes the memory caches quite a bit. One can also treat the bitboard as array of bytes or words in memory, since endian issues don't care here - that safes all the shifts and 'ands', but has to read byte for byte from memory.
int popCount (U64 x) {
   unsigned char * p = (unsigned char *) &x;
   return popCountOfByte256[p[0]] +
          popCountOfByte256[p[1]] +
          popCountOfByte256[p[2]] +
          popCountOfByte256[p[3]] +
          popCountOfByte256[p[4]] +
          popCountOfByte256[p[5]] +
          popCountOfByte256[p[6]] +
          popCountOfByte256[p[7]];
}

SWAR-Popcount

The divide and conquer SWAR-approach deals with counting bits of duos, to aggregate the duo-counts to nibbles and bytes inside one 64-bit register in parallel, to finally sum all bytes together. According to Donald Knuth [13], a parallel population count routine was already introduced in 1957 due to Donald B. Gillies and Jeffrey C. P. Miller in the first textbook on programming, second edition: The Preparation of Programs for an Electronic Digital Computer, by Maurice Wilkes, David Wheeler and Stanley Gill, pages 191–193 [14] [15].

Counting Duo-Bits

A bit-duo (two neighboring bits) can be interpreted with bit 0 = a, and bit 1 = b as

The duo population is

which can be archived by
or

The bit-duo has up to four states with population count from zero to two as demonstrated in following table with binary digits:
x
-
x div 2

popcnt(x)
00
-
00

00
01
-
00

01
10
-
01

01
11
-
01

10
Only the lower bit is needed from x div 2 - and one don't has to worry about borrows from neighboring duos. SWAR-wise, one needs to clear all "even" bits of the div 2 subtrahend to perform a 64-bit subtraction of all 32 duos:
x = x - ((x >> 1) & 0x5555555555555555);
Note that the popcount-result of the bit-duos still takes two bits.

Counting Nibble-Bits

The next step is to add the duo-counts to populations of four neighboring bits, the 16 nibble-counts, which may range from zero to four. SWAR-wise it is done by masking odd and even duo-counts to add them together:
 x = (x & 0x3333333333333333) + ((x >> 2) & 0x3333333333333333);
Note that the popcount-result of the nibbles takes only three bits, since 100B is the maximum population (of the nibble 1111B).

Byte-Counts

You already got the idea? Now it is about to get the byte-populations from two nibble-populations. Maximum byte-population of 1000B only takes four bits, so it is safe to mask all those four bits of the sum, rather than to mask the summands:
x = (x + (x >> 4)) & 0x0f0f0f0f0f0f0f0f;

Adding the Byte-Counts

Parallel Prefix Adds

We may continue with mask-less parallel prefix SWAR-adds for byte-counts, word-counts and finally double-word-counts, to mask the least significant 8 (or 7) bits for final result in the 0..64 range:
x += (x >>  8);
x += (x >> 16);
x += (x >> 32);
return x & 255;

Multiplication

With todays fast 64-bit multiplication one can multiply the vector of 8-byte-counts with 0x0101010101010101 to get the final result in the most significant byte, which is then shifted right:
x = (x * 0x0101010101010101) >> 56;

Casting out

Interestingly, there is another approach to add the bytes together. As demonstrated with decimal digits (base 10) and Casting out nines [16], casting out by modulo base minus one is equivalent to taking the digit sum, which might be applied here with low range 0..8 "base 256" digits:
x = x % 255;
However, since division and modulo are usually slow instructions and modulo by constant is likely replaced by reciprocal multiplication anyway by the compiler, the multiplication by 0x0101010101010101 aka the 2-adic fraction -1/255 is the preferred method.

The PopCount routine

The Constants

Putting all together, the various SWAR-Masks and factors as defined by Donald Knuth as 2-adic fractions [17]:
const U64 k1 = C64(0x5555555555555555); /*  -1/3   */
const U64 k2 = C64(0x3333333333333333); /*  -1/5   */
const U64 k4 = C64(0x0f0f0f0f0f0f0f0f); /*  -1/17  */
const U64 kf = C64(0x0101010101010101); /*  -1/255 */
represented as bitboards:
k1  -1/3            k2  -1/5            k4  -1/17           kf  -1/255              
0x5555555555555555  0x3333333333333333  0x0f0f0f0f0f0f0f0f  0x0101010101010101
1 . 1 . 1 . 1 .     1 1 . . 1 1 . .     1 1 1 1 . . . .     1 . . . . . . .   
1 . 1 . 1 . 1 .     1 1 . . 1 1 . .     1 1 1 1 . . . .     1 . . . . . . .   
1 . 1 . 1 . 1 .     1 1 . . 1 1 . .     1 1 1 1 . . . .     1 . . . . . . .   
1 . 1 . 1 . 1 .     1 1 . . 1 1 . .     1 1 1 1 . . . .     1 . . . . . . .   
1 . 1 . 1 . 1 .     1 1 . . 1 1 . .     1 1 1 1 . . . .     1 . . . . . . .   
1 . 1 . 1 . 1 .     1 1 . . 1 1 . .     1 1 1 1 . . . .     1 . . . . . . .   
1 . 1 . 1 . 1 .     1 1 . . 1 1 . .     1 1 1 1 . . . .     1 . . . . . . .   
1 . 1 . 1 . 1 .     1 1 . . 1 1 . .     1 1 1 1 . . . .     1 . . . . . . .   

popCount

This is how the complete routine looks in C:
int popCount (U64 x) {
    x =  x       - ((x >> 1)  & k1); /* put count of each 2 bits into those 2 bits */
    x = (x & k2) + ((x >> 2)  & k2); /* put count of each 4 bits into those 4 bits */
    x = (x       +  (x >> 4)) & k4 ; /* put count of each 8 bits into those 8 bits */
    x = (x * kf) >> 56; /* returns 8 most significant bits of x + (x<<8) + (x<<16) + (x<<24) + ...  */
    return (int) x;
}

Advantage: no branches, no memory lookups, constant runtime - independent from population
Drawback: dependency chain, not much parallel speedup

slowmul_popCount

And as stated before, for computers with relatively slower multiplication, the addition can be done manually:
int slowmul_popCount (U64 x) {
    x =  x       - ((x >> 1)  & k1); /* put count of each 2 bits into those 2 bits */
    x = (x & k2) + ((x >> 2)  & k2); /* put count of each 4 bits into those 4 bits */
    x = (x       +  (x >> 4)) & k4 ; /* put count of each 8 bits into those 8 bits */
    x += x >>  8;  /* put count of each 16 bits into their lowest 8 bits */
    x += x >> 16;  /* put count of each 32 bits into their lowest 8 bits */
    x += x >> 32;  /* put count of the final 64 bits into the lowest 8 bits */
    return (int) x & 255;
}

For likely sparsely populated bitboards, the loop-wise Brian Kernighan's way might be the faster one.

HAKMEM 169

A similar technique was proposed by Bill Gosper et al. from Massachusetts Institute of Technology, as published 1972 in Memo 239 (HAKMEM) [18] [19], to add bit-trio- rather than duo populations consecutively, and the 32 bit version relies on casting out 63. Note that the constants in the code below have octal (base-8) digits, originally written in Assembly for the PDP-6 [20]. An expanded 64-bit version, casting out 511 or 4095, is slightly less efficient than the binary SWAR version above.
int hakmem169_32(unsigned int x) {
   x = x  - ((x >> 1)  & 033333333333)
          - ((x >> 2)  & 011111111111);
   x = (x +  (x >> 3)) & 030707070707 ;
   return x % 63; /* casting out 63 */
} 

Miscellaneous


Cardinality of Multiple Sets

If we like to count arrays of sets, we can reduce 2^N-1 popcounts to N popcounts, by applying the odd-major-trick. For three sets to count we safe one, with five additional cheap instructions.
  odd   =  (x ^ y)  ^ z;
  major = ((x ^ y ) & z) | (x & y);
 
  popCount(x) + popCount(y) + popCount(z) == 2*popCount(major) + popCount(odd)
The combined popCount3 likely gains more parallel speedup, since there are two independent chains to calculate. Possible Application is to pass the union of both bishops (since usually bishops cover disjoint sets due to different square colors) plus the up to two knight move-target sets.
// return popCount(x) + popCount(y) + popCount(z)
int popCount3 (U64 x, U64 y, U64 z) {
    U64 maj = ((x ^ y ) & z) | (x & y);
    U64 odd = ((x ^ y ) ^ z);
    maj =  maj - ((maj >> 1) & k1 );
    odd =  odd - ((odd >> 1) & k1 );
    maj = (maj & k2) + ((maj >> 2) & k2);
    odd = (odd & k2) + ((odd >> 2) & k2);
    maj = (maj + (maj >> 4)) & k4;
    odd = (odd + (odd >> 4)) & k4;
    odd = ((maj + maj + odd) * kf ) >> 56;
    return (int) odd;
}

Odd and Major 7-15

Of course - reducing seven popcount to three, or even 15 popcounts to four sounds even more promising.
For N = 2^n - 1 it takes N - n odd-major pairs. Thus one add-major pair - five instructions - per saved popCount.

That is 7 - 3 = 4 pairs:
one1,two1 := oddMaj(x1,x2,x3)
one2,two2 := oddMaj(x4,x5,x6)
ones,two3 := oddMaj(x7,one1,one2)
twos,four := oddMaj(two1,two2,two3)
Or 15 - 4 = 11 pairs:
one1,two1  := oddMaj(x1,x2,x3)
one2,two2  := oddMaj(x4,x5,x6)
one3,two3  := oddMaj(x7,x8,x9)
one4,two4  := oddMaj(x10,x11,x12)
one5,two5  := oddMaj(x13,x14,x15)
one6,two6  := oddMaj(one1,one2,one3)
ones,two7  := oddMaj(one4,one5,one6)
two8,four1 := oddMaj(two1,two2,two3)
two9,four2 := oddMaj(two4,two5,two6)
twos,four3 := oddMaj(two7,two8,two9)
four,eight := oddMaj(four1,four2,four3)


Odd and Major Digit Counts

Odd-Major is probably also useful to determine digit count sets of attacks or other stuff:
U64 odd(U64 x, U64 y, U64 z) {return x^y^z;}
U64 maj(U64 x, U64 y, U64 z) {return ((x^y)&z)|(x&y);}
 
void attackCounts(U64 t[3], const U64 s[7]) {
   one1 = odd(s[0], s[1], s[2]);
   two1 = maj(s[0], s[1], s[2]);
   one2 = odd(s[3], s[4], s[5]);
   two2 = maj(s[3], s[4], s[5]);
   t[0] = odd(s[6], one1, one2);
   two3 = maj(s[6], one1, one2);
   t[1] = odd(two1, two2, two3);
   t[2] = maj(two1, two2, two3);
}
with following semantics:
exactly7attacks :=   t[2] &  t[1] &  t[0]
exactly6attacks :=   t[2] &  t[1] & ~t[0]
exactly5attacks :=   t[2] & ~t[1] &  t[0]
exactly4attacks :=   t[2] & ~t[1] & ~t[0]
exactly3attacks :=  ~t[2] &  t[1] &  t[0]
exactly2attacks :=  ~t[2] &  t[1] & ~t[0]
exactly1attack  :=  ~t[2] & ~t[1] &  t[0]
noAttack        :=  ~t[2] & ~t[1] & ~t[0]
 
atLeast4attacks :=                   t[2]
atLeast2attacks := atLeast4attacks | t[1]
atLeast1attack  := atLeast2attacks | t[0]
noAttack        := ~atLeast1attack
exactly1attack  :=  atLeast1attack  ^ atLeast2attacks

Popcount as log2 of LS1B

Assuming an architecture has a fast popcount-instruction (but no bitscan). One can isolate the LS1B, decrement it and count the remaining trailing "ones" to perform the logarithm dualis:
log2(LS1B) = popCount( LS1B - 1 );
bitIndexOfLS1B(x) = popCount( (x & -x) - 1 );

For instance, LS1B is 2^44, decrementing leaves a below LSB1 mask with exactly 44 bits set:
0x0000100000000000   0x00000FFFFFFFFFFF
. . . . . . . .      . . . . . . . .
. . . . . . . .      . . . . . . . .
. . . . 1 . . .      1 1 1 1 . . . .
. . . . . . . .      1 1 1 1 1 1 1 1
. . . . . . . .      1 1 1 1 1 1 1 1
. . . . . . . .      1 1 1 1 1 1 1 1
. . . . . . . .      1 1 1 1 1 1 1 1
. . . . . . . .      1 1 1 1 1 1 1 1

Hamming Distance

The hamming distance of two words is defined as the number of corresponding different bits.
int hammingDistance (U64 a, U64 b) {return popcnt( a ^ b);}
Hamming distance greater than one or two is an important property of codes to detect or even correct one-bit errors.

The minimum and average hamming distance over all Zobrist keys was considered as "quality"-measure of the keys. However, as long the minimum hamming distance is greater zero, linear independence (that is a small subset of all keys doesn't xor to zero), is much more important than hamming distance [21]. Maximizing the minimal hamming distance leads to very poor Zobrist keys [22].

Weighted PopCount

For a SIMD-wise kind of weighted population count, see the SSE2 dot-product.

Pre-calculated Mobility

Similar to Attacks by Occupancy Lookup to determine attack sets of sliding pieces, we may use pre-calculated population count or even center-weighted population count as a rough estimate on piece mobility [23]. It may not consider subsets of let say safe target squares.

Piece Attacks Count

As pointed out by Marco Costalba [24] [25], specialized routines to count the population (Mobility) of attack sets of king, knight and line-wise sub-sets of sliding pieces can be done more efficiently than the general SWAR-Popcount. This is similar to Flipping Mirroring and Rotating the whole bitboard versus Rank, File and Diagonal, and is based on mapping the up to eight scattered occupied bits to one byte, to perform a single byte lookup. For various mapping techniques, see:

Popcount in Hardware


See also


Publications

1949 ...

2000 ...

2010 ...


Postings

1998 ...

2000 ...

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2015 ...


External Links


References

  1. ^ Cryptography is also a significant application of the /R function symbol, which counts the number of one bits in a word; Turing refers to this as the "sideways adder" in his quick-reference summary. from Alan Turing (1949). Alan Turing's Manual for the Ferranti Mk. I. transcribed in 2000 by Robert Thau, pdf from The Computer History Museum, 9.4 The position of the most significant digit » Ferranti Mark 1
  2. ^ Extending the World’s Most Popular Processor Architecture by R.M. Ramanathan, Intel, covers SSE4 and popcnt
  3. ^ _mm_popcnt_u64
  4. ^ _popcnt16, _popcnt, _popcnt64 C-Intrinsic MSDN Library
  5. ^ Miscellaneous Intrinsic
  6. ^ __builtin_popcountll GCC Intrinsic
  7. ^ Stockfish POPCNT support with gcc by Marco Costalba, CCC, January 31, 2010
  8. ^ Color of each pixel is Hamming distance between the binary representations of its x and y coordinates, modulo 16, in the 16-color system, by Josiedraus, June 8, 2007, Richard Hamming from Wikipedia
  9. ^ Matters Computational - ideas, algorithms, source code (pdf) Ideas and Source Code by Jörg Arndt, 1.7 Functions related to the base-2 logarithm, function one_bit_q(), pp 18
  10. ^ Counting bits set, Brian Kernighan's way from Bit Twiddling Hacks by Sean Eron Anderson
  11. ^ Peter Wegner (1960). A technique for counting ones in a binary computer. Communications of the ACM, Volume 3, 1960
  12. ^ Edwin Ford Beckenbach (editor) (1964). Applied combinatorial mathematics, John Wiley
  13. ^ Donald Knuth (2009). The Art of Computer Programming, Volume 4, Fascicle 1: Bitwise tricks & techniques, as Pre-Fascicle 1a postscript, Sideways addition, p 11
  14. ^ Maurice Wilkes, David Wheeler, Stanley Gill (1951). The Preparation of Programs for an Electronic Digital Computer. Addison-Wesley Press; 1st edition, amazon.com; 2nd edition 1957, amazon.com
  15. ^ Electronic Delay Storage Automatic Calculator from Wikipedia
  16. ^ Casting Out Nines by Bill the Lizard, June 13, 2009
  17. ^ Donald Knuth (2009). The Art of Computer Programming, Volume 4, Fascicle 1: Bitwise tricks & techniques, as Pre-Fascicle 1a postscript, p 9
  18. ^ HAKMEM - ITEM 169 To count the ones in a PDP-6/10 word (in order of one-ups-manship: Gosper, Mann, Lenard, [Root and Mann])
  19. ^ HAKMEMC -- HAKMEM Programming hacks in C by Alan Mycroft
  20. ^ HAKMEM 169 for PDP-6/PDP-10 36-bit words
  21. ^ Re: About random numbers and hashing by Sven Reichard, CCC, December 05, 2001
  22. ^ Zobrist key random numbers by Robert Hyatt from CCC, January 21, 2009
  23. ^ Magic and precomputation by Onno Garms from Winboard Programming Forum, September 23, 2007
  24. ^ fast mobility count through hashing by Marco Costalba from CCC, May 09, 2009
  25. ^ Piece attacks count by Marco Costalba from CCC, May 18, 2009
  26. ^ Sideways Add / Population Count by Jitze Couperus and Steve Bellovin et al., cryptography@c2.net, January 28, 1999
  27. ^ sse-popcount/popcnt-avx512-harley-seal.cpp at master · WojciechMula/sse-popcount · GitHub
  28. ^ David A. Wagner, Steven M. Bellovin (1994). A Programmable Plaintext Recognizer.
  29. ^ AMD Athlon Processor x86 Code Optimization Guide (pdf) Efficient 64-Bit Population Count Using MMX™ Instructions Page 184
  30. ^ National Security Agency from Wikipedia
  31. ^ Michael Beeler, Bill Gosper and Rich Schroeppel (1972). HAKMEM, Memo 239, Artificial Intelligence Laboratory, Massachusetts Institute of Technology, made Web-available by Henry Baker
  32. ^ Ali Haurand from Wikipedia.de (German)

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