Postagens anteriores sobre este tema: Algoritmo Minimax na Teoria dos Jogos Função de avaliação na teoria dos jogos Tic-Tac-Toe AI – Encontrando o movimento ideal Poda Alfa-Beta .
Zobrist Hashing é uma função de hash amplamente usada em jogos de tabuleiro para 2 jogadores. É a função hash mais comum usada na tabela de transposição. As tabelas de transposição basicamente armazenam os valores avaliados dos estados anteriores do tabuleiro para que, se eles forem encontrados novamente, simplesmente recuperemos o valor armazenado da tabela de transposição. Abordaremos as tabelas de transposição em um artigo posterior. Neste artigo pegaremos o exemplo do tabuleiro de xadrez e implementaremos uma função hashing para isso.
Pseudocódigo:
// 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
Explicação:
A ideia por trás do Zobrist Hashing é que, para um determinado estado do tabuleiro, se houver uma peça em uma determinada célula, usaremos o número aleatório dessa peça da célula correspondente na tabela.
Se houver mais bits no número aleatório, menor será a chance de uma colisão de hash. Portanto, números de 64 bits são comumente usados como padrão e é altamente improvável que ocorra uma colisão de hash com números tão grandes. A tabela deve ser inicializada apenas uma vez durante a execução dos programas.
Além disso, a razão pela qual Zobrist Hashing é amplamente utilizado em jogos de tabuleiro é porque quando um jogador faz uma jogada não é necessário recalcular o valor do hash do zero. Devido à natureza da operação XOR, podemos simplesmente usar algumas operações XOR para recalcular o valor do hash.
Implementação:
Tentaremos encontrar um valor hash para a configuração da placa fornecida.
// A program to illustrate Zobrist Hashing Algorithm #include
// 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 = 0L; 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
# 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))
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.
// 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.
Saída :
The hash value is : 14226429382419125366 The new hash value is : 15124945578233295113 The old hash value is : 14226429382419125366