Chess - Programming - Zobrist Hashing

Zobrist Hashing is a hashing function that is widely used in 2 player board games.

• The main purpose of Zobrist hash codes in chess programming is to get an almost unique index number for any chess position, with a very important requirement that two similar positions generate entirely different indices.
• It is the most common hashing function used in transposition tables; and sometimes opening books.
• Transposition tables basically store the evaluated values of previous board states, so that if they are encountered again we simply retrieve the stored value from the transposition table.
• These index numbers are used for faster and more space efficient Hash tables or databases.

A program to illustrate the Zobrist Hashing Algorithm

// A matrix with random numbers initialized once.
Table[#ofBoardCells][#ofPieces] 

// Returns Zobrist hash function for current configuration of board.
function findhash(board):
  hash = 0
  for each cell on the board :
    if cell is not empty :
      piece = board[cell]
      hash ^= table[cell][piece]
  return hash

At program initialization, we generate an array of pseudorandom numbers:

• One number for each piece at each square.
• One number to indicate the side to move is black.
• Four numbers to indicate the castling rights, though usually 16 (2^4) are used for speed.
• Eight numbers to indicate the file of a valid En Passant square, if any.

This results in an array with 781 (12*64 + 1 + 4 + 8) random numbers.

• Since pawns do not occur on first and eighth rank, one might be fine with 12*64 though.
• There are even proposals and implementations to use overlapping keys from unaligned access up to an array of only 12 numbers for every piece and to rotate that number by square.

Programs usually implement their own Pseudorandom number generator (PRNG), both for better quality random numbers than standard library functions, and also for reproducibility.

• This means that whatever platform the program is run on, it will use the exact same set of Zobrist keys.
• This is also useful for things like opening books, where the positions in the book can be stored by hash key and be used portably across machines, considering endianess.

To get the Zobrist hash code of a certain position, the hash key is initialized by xoring all random numbers linked to the given feature.

For example, the starting position:

[Hash for White Rook on a1] xor [White Knight on b1] xor [White Bishop on c1] xor ... ( all pieces )
... xor [White castling short] xor [White castling long] xor ... ( all castling rights )

Another example, for a White Knight that jumps from b1 to c3 capturing a Black Bishop, these operations are performed:

[Original Hash of position] xor [Hash for White Knight on b1] ... ( removing the knight from b1 )
... xor [Hash for Black Bishop on c3] ( removing the captured bishop from c3 )
... xor [Hash for White Knight on c3] ( placing the knight on the new square )
... xor [Hash for Black to move] ( change sides)

Smaller hash keys are faster and more space efficient, while larger ones reduce the risk of a hash collision.

• A collision occurs if two positions map the same key.
• Usually 64 bit is used as a standard size in modern chess programs.

Using too small a hash code, such as only 32 bits, may result in different positions being assigned the same hash code.

Programs have various strategies for deciding what to do when a clash occurs.

• Some programs attempt to store the position at the next location.
  • If that location is found to be occupied also, the program can give up or it can replace the older of the two entries just examined.
• Most programs try more than once to find an unoccupied location.

A supposedly good hashing function workaround for clashing:

• Given a 64 bit (or whatever word size you want) take the low order “n” bits where “n” is log2 (tablesize) and use this as the “first” entry (x) to probe.
• Take the next “m” bits (for Cray Blitz, m=9 and use this as a “vector stride” (y) and then load eight words, table(x), table(x+y), table(x+2y), …, table(x+7y).
• One of these will be the one chosen to be replaced by the current store, or else one of these will be the “match” for the current lookup.
• The only minor difficulty to handle is that you do not dimension the table for (n) entries [table(n)] but, rather, we have to dimension the table larger [table(n+7*512)] to handle the positions where we hash to near the end of the table, and the vector loads would then step beyond the end…
  • No, there is not a “modulus” mechanism for vector loads to make this load “wrap around” to the beginning.

Remember that a table entry contains a “key”, score, score type, best move, etc.

• This key must EXACTLY match the hash key before we consider this a “hit”.
• This means that the 64 bit hash key must match completely, even though we only used the low order “n” bits of this key to probe into the table.
• In fact, since we used the low order “n” bits to prob into the table, we do not even bother storing them since we “know” what they were already (which saves table space.)

UINT64 getZoristKey(int piece, int color, int square)
{
  int index = piece*color*square;
  const char *p = ((char *)zorbrist) + index
  return *(UNIT64 *) p;
}

This means that the array does not need:

(piece*color*square * 64) bits;

• The original and well-known implementation.

but only

(piece*color*square * 8 + 3*8) bits;

• This will reduce array of about 8th in size.

If a bit boundary is used, the array becomes:

(piece*color*square + 63) bits;

To use this: The index to the 64-bit value that is being searched for becomes an index to the first bit of the 64-bits needed.

• This gives the shortest possible array.
• The access would be 64 + 8 bits; and then shift the results according to the bit index.
• An easiest way is to use a byte boundary.

http://web.archive.org/web/20111202160335/http://www.cis.uab.edu/hyatt/collisions.html