Algorithme Minimax dans la théorie des jeux | Ensemble 5 (hachage Zobrist)
Articles précédents sur ce sujet : Algorithme Minimax dans la théorie des jeux Fonction d'évaluation en théorie des jeux Tic-Tac-Toe AI – Trouver le mouvement optimal Taille alpha-bêta .
Zobrist Hashing est une fonction de hachage largement utilisée dans les jeux de société à 2 joueurs. Il s'agit de la fonction de hachage la plus couramment utilisée dans les tables de transposition. Les tables de transposition stockent essentiellement les valeurs évaluées des états précédents de la carte, de sorte que si elles sont rencontrées à nouveau, nous récupérons simplement la valeur stockée dans la table de transposition. Nous aborderons les tables de transposition dans un article ultérieur. Dans cet article, nous prendrons l'exemple d'un échiquier et implémenterons une fonction de hachage pour cela.
Pseudocode :
// 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
Explication :
L'idée derrière Zobrist Hashing est que pour un état de plateau donné, s'il y a une pièce sur une cellule donnée, nous utilisons le nombre aléatoire de cette pièce dans la cellule correspondante du tableau.
S'il y a plus de bits dans le nombre aléatoire, le risque de collision de hachage est moindre. Par conséquent, les nombres à 64 bits sont couramment utilisés comme norme et il est très peu probable qu'une collision de hachage se produise avec des nombres aussi grands. La table ne doit être initialisée qu'une seule fois lors de l'exécution du programme.
La raison pour laquelle Zobrist Hashing est largement utilisé dans les jeux de société est que lorsqu'un joueur effectue un mouvement, il n'est pas nécessaire de recalculer la valeur de hachage à partir de zéro. En raison de la nature de l'opération XOR, nous pouvons simplement utiliser quelques opérations XOR pour recalculer la valeur de hachage.
Mise en œuvre :
Nous allons essayer de trouver une valeur de hachage pour la configuration de carte donnée.
// 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.
Sortir :
The hash value is : 14226429382419125366 The new hash value is : 15124945578233295113 The old hash value is : 14226429382419125366
