Minimax Algorithm in Game Theory | Set 5 (Zobrist Hashing)

Last Updated : 26 Apr, 2023

Previous posts on this topic : Minimax Algorithm in Game Theory, Evaluation Function in Game Theory, Tic-Tac-Toe AI – Finding optimal move, Alpha-Beta Pruning.
Zobrist Hashing is a hashing function that is widely used in 2 player board games. It is the most common hashing function used in transposition table. 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. We will be covering transposition tables in a later article. In this article we shall take the example of chess board and implement a hashing function for that.

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

Explanation :

The idea behind Zobrist Hashing is that for a given board state, if there is a piece on a given cell, we use the random number of that piece from the corresponding cell in the table.
If more bits are there in the random number the lesser chance of a hash collision. Therefore 64 bit numbers are commonly used as the standard and it is highly unlikely for a hash collision to occur with such large numbers. The table has to be initialized only once during the programs execution.
Also the reason why Zobrist Hashing is widely used in board games is because when a player makes a move, it is not necessary to recalculate the hash value from scratch. Due to the nature of XOR operation we can simply use few XOR operations to recalculate the hash value.

Implementation :

We shall try to find a hash value for the given board configuration.

CPP

// A program to illustrate Zobrist Hashing Algorithm
#include <bits/stdc++.h>
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<unsigned long long int>
                                 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<8; i++)
      for (int j = 0; j<8; j++)
        for (int k = 0; k<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<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;
}
 
// Main Function
int main()
{
    // Uppercase letters are white pieces
    // Lowercase letters are black pieces
    char board[8][9] =
    {
        "---K----",
        "-R----Q-",
        "--------",
        "-P----p-",
        "-----p--",
        "--------",
        "p---b--q",
        "----n--k"
    };
 
    initTable();
 
    unsigned long long int hashValue = computeHash(board);
    printf("The hash value is     : %llu\n", 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("The new hash value is : %llu\n", 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("The old hash value is : %llu\n", 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 = 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

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

Akshay L Aradhya

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Minimax Algorithm in Game Theory | Set 1 (Introduction)