Minimax-algoritme i spilteori | Sæt 5 (Zobrist Hashing)
Tidligere indlæg om dette emne: Minimax-algoritme i spilteori Evalueringsfunktion i spilteori Tic-Tac-Toe AI – Find optimal bevægelse Alfa-beta beskæring .
Zobrist Hashing er en hashing-funktion, der er meget brugt i brætspil til 2 spillere. Det er den mest almindelige hashing-funktion, der bruges i transponeringstabel. Transpositionstabeller gemmer grundlæggende de evaluerede værdier af tidligere bordtilstande, så hvis de stødes på igen, henter vi simpelthen den lagrede værdi fra transpositionstabellen. Vi vil dække oversættelsestabeller i en senere artikel. I denne artikel skal vi tage eksemplet med skakbræt og implementere en hashing-funktion til det.
Pseudokode:
// A matrix with random numbers initialized once Table[#ofBoardCells][#ofPieces] // Returns Zobrist hash function for current conf- // iguration 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
Forklaring:
Ideen bag Zobrist Hashing er, at for en given brættilstand, hvis der er en brik på en given celle, bruger vi det tilfældige antal af den brik fra den tilsvarende celle i tabellen.
Hvis der er flere bits i det tilfældige tal, er der mindre chance for en hash-kollision. Derfor bruges 64 bit-tal almindeligvis som standard, og det er højst usandsynligt, at en hash-kollision opstår med så store tal. Tabellen skal kun initialiseres én gang under programmets afvikling.
Også grunden til, at Zobrist Hashing er meget brugt i brætspil, er, at når en spiller foretager et træk, er det ikke nødvendigt at genberegne hashværdien fra bunden. På grund af XOR-operationens karakter kan vi simpelthen bruge få XOR-operationer til at genberegne hashværdien.
Implementering:
Vi skal forsøge at finde en hashværdi for den givne bordkonfiguration.
// A program to illustrate Zobrist Hashing Algorithm #include using namespace std ; unsigned long long int ZobristTable [ 8 ][ 8 ][ 12 ]; mt19937 mt ( 01234567 ); // Generates a Random number from 0 to 2^64-1 unsigned long long int randomInt () { uniform_int_distribution & lt ; unsigned long long int & gt ; dist ( 0 UINT64_MAX ); return dist ( mt ); } // This function associates each piece with // a number int indexOf ( char piece ) { if ( piece == 'P' ) return 0 ; if ( piece == 'N' ) return 1 ; if ( piece == 'B' ) return 2 ; if ( piece == 'R' ) return 3 ; if ( piece == 'Q' ) return 4 ; if ( piece == 'K' ) return 5 ; if ( piece == 'p' ) return 6 ; if ( piece == 'n' ) return 7 ; if ( piece == 'b' ) return 8 ; if ( piece == 'r' ) return 9 ; if ( piece == 'q' ) return 10 ; if ( piece == 'k' ) return 11 ; else return -1 ; } // Initializes the table void initTable () { for ( int i = 0 ; i & lt ; 8 ; i ++ ) for ( int j = 0 ; j & lt ; 8 ; j ++ ) for ( int k = 0 ; k & lt ; 12 ; k ++ ) ZobristTable [ i ][ j ][ k ] = randomInt (); } // Computes the hash value of a given board unsigned long long int computeHash ( char board [ 8 ][ 9 ]) { unsigned long long int h = 0 ; for ( int i = 0 ; i & lt ; 8 ; i ++ ) { for ( int j = 0 ; j & lt ; 8 ; j ++ ) { if ( board [ i ][ j ] != '-' ) { int piece = indexOf ( board [ i ][ j ]); h ^= ZobristTable [ i ][ j ][ piece ]; } } } return h ; } // Main Function int main () { // Uppercase letters are white pieces // Lowercase letters are black pieces char board [ 8 ][ 9 ] = { & quot ; --- K ----& quot ; & quot ; - R ---- Q -& quot ; & quot ; --------& quot ; & quot ; - P ---- p -& quot ; & quot ; ----- p --& quot ; & quot ; --------& quot ; & quot ; p --- b -- q & quot ; & quot ; ---- n -- k & quot ; }; initTable (); unsigned long long int hashValue = computeHash ( board ); printf ( & quot ; The hash value is : % llu n & quot ; hashValue ); //Move the white king to the left char piece = board [ 0 ][ 3 ]; board [ 0 ][ 3 ] = '-' ; hashValue ^= ZobristTable [ 0 ][ 3 ][ indexOf ( piece )]; board [ 0 ][ 2 ] = piece ; hashValue ^= ZobristTable [ 0 ][ 2 ][ indexOf ( piece )]; printf ( & quot ; The new hash value is : % llu n & quot ; hashValue ); // Undo the white king move piece = board [ 0 ][ 2 ]; board [ 0 ][ 2 ] = '-' ; hashValue ^= ZobristTable [ 0 ][ 2 ][ indexOf ( piece )]; board [ 0 ][ 3 ] = piece ; hashValue ^= ZobristTable [ 0 ][ 3 ][ indexOf ( piece )]; printf ( & quot ; The old hash value is : % llu n & quot ; hashValue ); return 0 ; }
Java // A Java program to illustrate Zobrist Hashing Algorithm import java.util.* ; public class GFG { public static List < List < List < Integer >>> ZobristTable = new ArrayList <> (); // mt19937 mt(01234567); public static void initialise () { for ( int i = 0 ; i < 8 ; i ++ ) { ZobristTable . add ( new ArrayList <> ()); for ( int j = 0 ; j < 8 ; j ++ ) { ZobristTable . get ( i ). add ( new ArrayList <> ()); for ( int k = 0 ; k < 12 ; k ++ ) { ZobristTable . get ( i ). get ( j ). add ( 0 ); } } } } // Generates a Random number from 0 to 2^64-1 public static long randomLong () { long min = 0 L ; long max = ( long ) Math . pow ( 2 64 ); Random rnd = new Random (); return rnd . nextLong () * ( max - min ) + min ; } // This function associates each piece with a number public static int indexOf ( char piece ) { if ( piece == 'P' ) return 0 ; if ( piece == 'N' ) return 1 ; if ( piece == 'B' ) return 2 ; if ( piece == 'R' ) return 3 ; if ( piece == 'Q' ) return 4 ; if ( piece == 'K' ) return 5 ; if ( piece == 'p' ) return 6 ; if ( piece == 'n' ) return 7 ; if ( piece == 'b' ) return 8 ; if ( piece == 'r' ) return 9 ; if ( piece == 'q' ) return 10 ; if ( piece == 'k' ) return 11 ; else return - 1 ; } // Initializes the table public static void initTable () { for ( int i = 0 ; i < 8 ; i ++ ) { for ( int j = 0 ; j < 8 ; j ++ ) { for ( int k = 0 ; k < 12 ; k ++ ) { Random rnd = new Random (); ZobristTable . get ( i ). get ( j ). set ( k ( int ) randomLong ()); } } } } // Computes the hash value of a given board public static int computeHash ( List < List < Character >> board ) { int h = 0 ; for ( int i = 0 ; i < 8 ; i ++ ) { for ( int j = 0 ; j < 8 ; j ++ ) { if ( board . get ( i ). get ( j ) != '-' ) { int piece = indexOf ( board . get ( i ). get ( j )); h ^= ZobristTable . get ( i ). get ( j ). get ( piece ); } } } return h ; } public static void main ( String [] args ) { // Main Function // Uppercase letters are white pieces // Lowercase letters are black pieces List < List < Character >> board = new ArrayList <> (); board . add ( new ArrayList <> ( Arrays . asList ( '-' '-' '-' 'K' '-' '-' '-' '-' ))); board . add ( new ArrayList <> ( Arrays . asList ( '-' 'R' '-' '-' '-' '-' 'Q' '-' ))); board . add ( new ArrayList <> ( Arrays . asList ( '-' '-' '-' 'K' '-' '-' '-' '-' ))); board . add ( new ArrayList <> ( Arrays . asList ( '-' '-' '-' '-' '-' '-' '-' '-' ))); board . add ( new ArrayList <> ( Arrays . asList ( '-' 'P' '-' '-' '-' '-' 'p' '-' ))); board . add ( new ArrayList <> ( Arrays . asList ( '-' '-' '-' '-' '-' 'p' '-' '-' ))); board . add ( new ArrayList <> ( Arrays . asList ( '-' '-' '-' '-' '-' '-' '-' '-' ))); board . add ( new ArrayList <> ( Arrays . asList ( 'p' '-' '-' '-' 'b' '-' '-' 'q' ))); board . add ( new ArrayList <> ( Arrays . asList ( '-' '-' '-' '-' 'n' '-' '-' 'k' ))); initialise (); initTable (); int hashValue = computeHash ( board ); System . out . println ( 'The hash value is : ' + hashValue ); // Move the white king to the left char piece = board . get ( 0 ). get ( 3 ); board . get ( 0 ). set ( 3 '-' ); hashValue ^= ZobristTable . get ( 0 ). get ( 3 ). get ( indexOf ( piece )); board . get ( 0 ). set ( 2 piece ); hashValue ^= ZobristTable . get ( 0 ). get ( 2 ). get ( indexOf ( piece )); System . out . println ( 'The new hash value is : ' + hashValue ); // Undo the white king move piece = board . get ( 0 ). get ( 2 ); board . get ( 0 ). set ( 2 '-' ); hashValue ^= ZobristTable . get ( 0 ). get ( 2 ). get ( indexOf ( piece )); board . get ( 0 ). set ( 3 piece ); hashValue ^= ZobristTable . get ( 0 ). get ( 3 ). get ( indexOf ( piece )); System . out . println ( 'The new hash value is : ' + hashValue ); } } // This code is contributed by Vaibhav
Python3 # A program to illustrate Zobrist Hashing Algorithm # python code implementation import random # mt19937 mt(01234567); # Generates a Random number from 0 to 2^64-1 def randomInt (): min = 0 max = pow ( 2 64 ) return random . randint ( min max ) # This function associates each piece with # a number def indexOf ( piece ): if ( piece == 'P' ): return 0 elif ( piece == 'N' ): return 1 elif ( piece == 'B' ): return 2 elif ( piece == 'R' ): return 3 elif ( piece == 'Q' ): return 4 elif ( piece == 'K' ): return 5 elif ( piece == 'p' ): return 6 elif ( piece == 'n' ): return 7 elif ( piece == 'b' ): return 8 elif ( piece == 'r' ): return 9 elif ( piece == 'q' ): return 10 elif ( piece == 'k' ): return 11 else : return - 1 # Initializes the table def initTable (): # 8x8x12 array ZobristTable = [[[ randomInt () for k in range ( 12 )] for j in range ( 8 )] for i in range ( 8 )] return ZobristTable # Computes the hash value of a given board def computeHash ( board ZobristTable ): h = 0 for i in range ( 8 ): for j in range ( 8 ): if ( board [ i ][ j ] != '-' ): piece = indexOf ( board [ i ][ j ]) h ^= ZobristTable [ i ][ j ][ piece ] return h # Main Function # Uppercase letters are white pieces # Lowercase letters are black pieces board = [ '---K----' '-R----Q-' '--------' '-P----p-' '-----p--' '--------' 'p---b--q' '----n--k' ] ZobristTable = initTable () hashValue = computeHash ( board ZobristTable ) print ( 'The hash value is : ' + str ( hashValue )) #Move the white king to the left piece = board [ 0 ][ 3 ] board [ 0 ] = list ( board [ 0 ]) board [ 0 ][ 3 ] = '-' board [ 0 ] = '' . join ( board [ 0 ]) hashValue ^= ZobristTable [ 0 ][ 3 ][ indexOf ( piece )] board [ 0 ] = list ( board [ 0 ]) board [ 0 ][ 2 ] = piece board [ 0 ] = '' . join ( board [ 0 ]) hashValue ^= ZobristTable [ 0 ][ 2 ][ indexOf ( piece )] print ( 'The new hash value is : ' + str ( hashValue )) # Undo the white king move piece = board [ 0 ][ 2 ] board [ 0 ] = list ( board [ 0 ]) board [ 0 ][ 2 ] = '-' board [ 0 ] = '' . join ( board [ 0 ]) hashValue ^= ZobristTable [ 0 ][ 2 ][ indexOf ( piece )] board [ 0 ] = list ( board [ 0 ]) board [ 0 ][ 3 ] = piece board [ 0 ] = '' . join ( board [ 0 ]) hashValue ^= ZobristTable [ 0 ][ 3 ][ indexOf ( piece )] print ( 'The old hash value is : ' + str ( hashValue ))
C# using System ; using System.Collections ; using System.Collections.Generic ; using System.Linq ; // C# program for the above approach // A program to illustrate Zobrist Hashing Algorithm // javascript code implementation class HelloWorld { public static List < List < List < int >>> ZobristTable = new List < List < List < int >>> (); // mt19937 mt(01234567); public static void initialise (){ for ( int i = 0 ; i < 8 ; i ++ ){ ZobristTable . Add ( new List < List < int >> ()); for ( int j = 0 ; j < 8 ; j ++ ){ ZobristTable [ i ]. Add ( new List < int > ()); for ( int k = 0 ; k < 12 ; k ++ ){ ZobristTable [ i ][ j ]. Add ( 0 ); } } } } // Generates a Random number from 0 to 2^64-1 public static int randomInt () { int min = 0 ; int max = ( int ) Math . Pow ( 2 64 ); Random rnd = new Random (); return rnd . Next () * ( max - min ) + min ; } // This function associates each piece with // a number public static int indexOf ( char piece ) { if ( piece == 'P' ) return 0 ; if ( piece == 'N' ) return 1 ; if ( piece == 'B' ) return 2 ; if ( piece == 'R' ) return 3 ; if ( piece == 'Q' ) return 4 ; if ( piece == 'K' ) return 5 ; if ( piece == 'p' ) return 6 ; if ( piece == 'n' ) return 7 ; if ( piece == 'b' ) return 8 ; if ( piece == 'r' ) return 9 ; if ( piece == 'q' ) return 10 ; if ( piece == 'k' ) return 11 ; else return - 1 ; } // Initializes the table public static void initTable () { for ( int i = 0 ; i < 8 ; i ++ ){ for ( int j = 0 ; j < 8 ; j ++ ){ for ( int k = 0 ; k < 12 ; k ++ ){ Random rnd = new Random (); ZobristTable [ i ][ j ][ k ] = rnd . Next (); } } } } // Computes the hash value of a given board public static int computeHash ( List < List < char >> board ) { int h = 0 ; for ( int i = 0 ; i < 8 ; i ++ ) { for ( int j = 0 ; j < 8 ; j ++ ) { if ( board [ i ][ j ] != '-' ) { int piece = indexOf ( board [ i ][ j ]); h ^= ZobristTable [ i ][ j ][ piece ]; } } } return h ; } static void Main () { // Main Function // Uppercase letters are white pieces // Lowercase letters are black pieces List < List < char >> board = new List < List < char >> (); board . Add ( new List < char > (){ '-' '-' '-' 'K' '-' '-' '-' '-' }); board . Add ( new List < char > (){ '-' 'R' '-' '-' '-' '-' 'Q' '-' }); board . Add ( new List < char > (){ '-' '-' '-' 'K' '-' '-' '-' '-' }); board . Add ( new List < char > (){ '-' '-' '-' '-' '-' '-' '-' '-' }); board . Add ( new List < char > (){ '-' 'P' '-' '-' '-' '-' 'p' '-' }); board . Add ( new List < char > (){ '-' '-' '-' '-' '-' 'p' '-' '-' }); board . Add ( new List < char > (){ '-' '-' '-' '-' '-' '-' '-' '-' }); board . Add ( new List < char > (){ 'p' '-' '-' '-' 'b' '-' '-' 'q' }); board . Add ( new List < char > (){ '-' '-' '-' '-' 'n' '-' '-' 'k' }); initialise (); initTable (); int hashValue = computeHash ( board ); Console . WriteLine ( 'The hash value is : ' + hashValue ); //Move the white king to the left char piece = board [ 0 ][ 3 ]; board [ 0 ][ 3 ] = '-' ; hashValue ^= ZobristTable [ 0 ][ 3 ][ indexOf ( piece )]; board [ 0 ][ 2 ] = piece ; hashValue ^= ZobristTable [ 0 ][ 2 ][ indexOf ( piece )]; Console . WriteLine ( 'The new hash value is : ' + hashValue ); // Undo the white king move piece = board [ 0 ][ 2 ]; board [ 0 ][ 2 ] = '-' ; hashValue ^= ZobristTable [ 0 ][ 2 ][ indexOf ( piece )]; board [ 0 ][ 3 ] = piece ; hashValue ^= ZobristTable [ 0 ][ 3 ][ indexOf ( piece )]; Console . WriteLine ( 'The old hash value is : ' + hashValue ); } } // The code is contributed by Nidhi goel.
JavaScript // A program to illustrate Zobrist Hashing Algorithm // javascript code implementation let ZobristTable = new Array ( 8 ); for ( let i = 0 ; i < 8 ; i ++ ){ ZobristTable [ i ] = new Array ( 8 ); } for ( let i = 0 ; i < 8 ; i ++ ){ for ( let j = 0 ; j < 8 ; j ++ ){ ZobristTable [ i ][ j ] = new Array ( 12 ); } } // mt19937 mt(01234567); // Generates a Random number from 0 to 2^64-1 function randomInt () { let min = 0 ; let max = Math . pow ( 2 64 ); return Math . floor ( Math . random () * ( max - min ) + min ); } // This function associates each piece with // a number function indexOf ( piece ) { if ( piece == 'P' ) return 0 ; if ( piece == 'N' ) return 1 ; if ( piece == 'B' ) return 2 ; if ( piece == 'R' ) return 3 ; if ( piece == 'Q' ) return 4 ; if ( piece == 'K' ) return 5 ; if ( piece == 'p' ) return 6 ; if ( piece == 'n' ) return 7 ; if ( piece == 'b' ) return 8 ; if ( piece == 'r' ) return 9 ; if ( piece == 'q' ) return 10 ; if ( piece == 'k' ) return 11 ; else return - 1 ; } // Initializes the table function initTable () { for ( let i = 0 ; i < 8 ; i ++ ) for ( let j = 0 ; j < 8 ; j ++ ) for ( let k = 0 ; k < 12 ; k ++ ){ ZobristTable [ i ][ j ][ k ] = randomInt (); } } // Computes the hash value of a given board function computeHash ( board ) { let h = 0 ; for ( let i = 0 ; i < 8 ; i ++ ) { for ( let j = 0 ; j < 8 ; j ++ ) { if ( board [ i ][ j ] != '-' ) { let piece = indexOf ( board [ i ][ j ]); h ^= ZobristTable [ i ][ j ][ piece ]; } } } return h ; } // Main Function // Uppercase letters are white pieces // Lowercase letters are black pieces let board = [ '---K----' '-R----Q-' '--------' '-P----p-' '-----p--' '--------' 'p---b--q' '----n--k' ]; initTable (); let hashValue = computeHash ( board ); console . log ( 'The hash value is : ' + hashValue ); //Move the white king to the left let piece = board [ 0 ][ 3 ]; board [ 0 ] = board [ 0 ]. split ( '' ); board [ 0 ][ 3 ] = '-' ; board [ 0 ] = board [ 0 ]. join ( '' ); hashValue ^= ZobristTable [ 0 ][ 3 ][ indexOf ( piece )]; board [ 0 ] = board [ 0 ]. split ( '' ); board [ 0 ][ 2 ] = piece ; board [ 0 ] = board [ 0 ]. join ( '' ); hashValue ^= ZobristTable [ 0 ][ 2 ][ indexOf ( piece )]; console . log ( 'The new hash value is : ' + hashValue ); // Undo the white king move piece = board [ 0 ][ 2 ]; board [ 0 ] = board [ 0 ]. split ( '' ); board [ 0 ][ 2 ] = '-' ; board [ 0 ] = board [ 0 ]. join ( '' ); hashValue ^= ZobristTable [ 0 ][ 2 ][ indexOf ( piece )]; board [ 0 ][ 3 ] = piece ; hashValue ^= ZobristTable [ 0 ][ 3 ][ indexOf ( piece )]; console . log ( 'The old hash value is : ' + hashValue ); // The code is contributed by Nidhi goel.
Output:
The hash value is : 14226429382419125366 The new hash value is : 15124945578233295113 The old hash value is : 14226429382419125366
