N Queen Problem using Branch And Bound

04 Jul, 2022
The N queens puzzle is the problem of placing N chess queens on an N×N chessboard so that no two queens threaten each other. Thus, a solution requires that no two queens share the same row, column, or diagonal.
Backtracking Algorithm for N-Queen is already discussed here. In backtracking solution we backtrack when we hit a dead end. In Branch and Bound solution, after building a partial solution, we figure out that there is no point going any deeper as we are going to hit a dead end. 
Let’s begin by describing backtracking solution. “The idea is to place queens one by one in different columns, starting from the leftmost column. When we place a queen in a column, we check for clashes with already placed queens. In the current column, if we find a row for which there is no clash, we mark this row and column as part of the solution. If we do not find such a row due to clashes, then we backtrack and return false.”
For the 1st Queen, there are total 8 possibilities as we can place 1st Queen in any row of first column. Let’s place Queen 1 on row 3.After placing 1st Queen, there are 7 possibilities left for the 2nd Queen. But wait, we don’t really have 7 possibilities. We cannot place Queen 2 on rows 2, 3 or 4 as those cells are under attack from Queen 1. So, Queen 2 has only 8 – 3 = 5 valid positions left.After picking a position for Queen 2, Queen 3 has even fewer options as most of the cells in its column are under attack from the first 2 Queens.
For the 1st Queen, there are total 8 possibilities as we can place 1st Queen in any row of first column. Let’s place Queen 1 on row 3.
After placing 1st Queen, there are 7 possibilities left for the 2nd Queen. But wait, we don’t really have 7 possibilities. We cannot place Queen 2 on rows 2, 3 or 4 as those cells are under attack from Queen 1. So, Queen 2 has only 8 – 3 = 5 valid positions left.
After picking a position for Queen 2, Queen 3 has even fewer options as most of the cells in its column are under attack from the first 2 Queens.
We need to figure out an efficient way of keeping track of which cells are under attack. In previous solution we kept an 8­-by­-8 Boolean matrix and update it each time we placed a queen, but that required linear time to update as we need to check for safe cells.Basically, we have to ensure 4 things: 1. No two queens share a column. 2. No two queens share a row. 3. No two queens share a top-right to left-bottom diagonal. 4. No two queens share a top-left to bottom-right diagonal.
Number 1 is automatic because of the way we store the solution. For number 2, 3 and 4, we can perform updates in O(1) time. The idea is to keep three Boolean arrays that tell us which rows and which diagonals are occupied.
Lets do some pre-processing first. Let’s create two N x N matrix one for / diagonal and other one for \ diagonal. Let’s call them slashCode and backslashCode respectively. The trick is to fill them in such a way that two queens sharing a same /­diagonal will have the same value in matrix slashCode, and if they share same \­diagonal, they will have the same value in backslashCode matrix.For an N x N matrix, fill slashCode and backslashCode matrix using below formula –slashCode[row][col] = row + col backslashCode[row][col] = row – col + (N-1)
Using above formula will result in below matrices  
 
The ‘N – 1’ in the backslash code is there to ensure that the codes are never negative because we will be using the codes as indices in an array.Now before we place queen i on row j, we first check whether row j is used (use an array to store row info). Then we check whether slash code ( j + i ) or backslash code ( j – i + 7 ) are used (keep two arrays that will tell us which diagonals are occupied). If yes, then we have to try a different location for queen i. If not, then we mark the row and the two diagonals as used and recurse on queen i + 1. After the recursive call returns and before we try another position for queen i, we need to reset the row, slash code and backslash code as unused again, like in the code from the previous notes.
Below is the implementation of above idea –  
This code Takes the Dynamic Input:
C++
C
Java
Python3
Javascript
#include<bits/stdc++.h>using namespace std;int N;  // function for printing the solutionvoid printSol(vector<vector<int>>board){   for(int i  = 0;i<N;i++){       for(int j = 0;j<N;j++){           cout<<board[i][j]<<" ";       }       cout<<"\n";   }} /*  Optimized isSafe functionisSafe function to check if current row contins or current left diagonal or current right diagonal contains any queen or not if yes return false else return true*/ bool isSafe(int row ,int col ,vector<bool>rows , vector<bool>left_digonals ,vector<bool>Right_digonals){         if(rows[row] == true || left_digonals[row+col] == true || Right_digonals[col-row+N-1] == true){       return false;   }      return true;} // Recursive function to solve N-queen Problembool solve(vector<vector<int>>& board ,int col ,vector<bool>rows , vector<bool>left_digonals ,vector<bool>Right_digonals){             // base Case : If all Queens are placed       if(col>=N){           return true;       }       /* Consider this Column and move  in all rows one by one */       for(int i = 0;i<N;i++)       {           if(isSafe(i,col,rows,left_digonals,Right_digonals) == true)           {               rows[i] = true;               left_digonals[i+col] = true;               Right_digonals[col-i+N-1] = true;               board[i][col] = 1;   // placing the Queen in board[i][col]                                /* recur to place rest of the queens */                if(solve(board,col+1,rows,left_digonals,Right_digonals) == true){                   return true;               }                               // Backtracking               rows[i] = false;               left_digonals[i+col] = false;               Right_digonals[col-i+N-1] = false;               board[i][col] = 0;    // removing the Queen from board[i][col]            }       }         return false;}  int main(){   // Taking input from the user    cout<<"Enter the no of rows for the square Board : ";   cin>>N;     // board of size N*N   vector<vector<int>>board(N,vector<int>(N,0));      // array to tell which rows are occupied   vector<bool>rows(N,false);             // arrays to tell which diagonals are occupied                         vector<bool>left_digonals(2*N-1,false);   vector<bool>Right_digonals(2*N-1,false);     bool ans =  solve(board , 0, rows,left_digonals,Right_digonals);    if(ans == true){        // printing the solution Board       printSol(board);   }   else{       cout<<"Solution Does not Exist\n";   }} // This Code is Contributed by Parshant Rajput
/* C++ program to solve N Queen Problem using Branch   and Bound */#include<stdio.h>#include<string.h>#include<stdbool.h>#define N 8 /* A utility function to print solution */void printSolution(int board[N][N]){    for (int i = 0; i < N; i++)    {        for (int j = 0; j < N; j++)            printf("%2d ", board[i][j]);        printf("\n");    }} /* A Optimized function to check if a queen canbe placed on board[row][col] */bool isSafe(int row, int col, int slashCode[N][N],            int backslashCode[N][N], bool rowLookup[],      bool slashCodeLookup[], bool backslashCodeLookup[] ){    if (slashCodeLookup[slashCode[row][col]] ||        backslashCodeLookup[backslashCode[row][col]] ||        rowLookup[row])    return false;     return true;} /* A recursive utility functionto solve N Queen problem */bool solveNQueensUtil(int board[N][N], int col,    int slashCode[N][N], int backslashCode[N][N],                                  bool rowLookup[N],                            bool slashCodeLookup[],                           bool backslashCodeLookup[] ){    /* base case: If all queens are placed    then return true */    if (col >= N)        return true;     /* Consider this column and try placing       this queen in all rows one by one */    for (int i = 0; i < N; i++)    {        /* Check if queen can be placed on           board[i][col] */        if ( isSafe(i, col, slashCode,                    backslashCode, rowLookup,          slashCodeLookup, backslashCodeLookup) )        {            /* Place this queen in board[i][col] */            board[i][col] = 1;            rowLookup[i] = true;            slashCodeLookup[slashCode[i][col]] = true;            backslashCodeLookup[backslashCode[i][col]] = true;             /* recur to place rest of the queens */            if ( solveNQueensUtil(board, col + 1,                                  slashCode, backslashCode,             rowLookup, slashCodeLookup, backslashCodeLookup) )                return true;             /* If placing queen in board[i][col]            doesn't lead to a solution, then backtrack */             /* Remove queen from board[i][col] */            board[i][col] = 0;            rowLookup[i] = false;            slashCodeLookup[slashCode[i][col]] = false;            backslashCodeLookup[backslashCode[i][col]] = false;        }    }     /* If queen can not be place in any row in        this column col then return false */    return false;} /* This function solves the N Queen problem usingBranch and Bound. It mainly uses solveNQueensUtil() tosolve the problem. It returns false if queenscannot be placed, otherwise return true andprints placement of queens in the form of 1s.Please note that there may be more than onesolutions, this function prints one of thefeasible solutions.*/bool solveNQueens(){    int board[N][N];    memset(board, 0, sizeof board);     // helper matrices    int slashCode[N][N];    int backslashCode[N][N];     // arrays to tell us which rows are occupied    bool rowLookup[N] = {false};     //keep two arrays to tell us    // which diagonals are occupied    bool slashCodeLookup[2*N - 1] = {false};    bool backslashCodeLookup[2*N - 1] = {false};     // initialize helper matrices    for (int r = 0; r < N; r++)        for (int c = 0; c < N; c++) {          slashCode[r] = r + c,            backslashCode[r] = r - c + 7;        }     if (solveNQueensUtil(board, 0,                          slashCode, backslashCode,      rowLookup, slashCodeLookup, backslashCodeLookup) ==                                                 false )    {        printf("Solution does not exist");        return false;    }     // solution found    printSolution(board);    return true;} // Driver program to test above functionint main(){    solveNQueens();     return 0;}
// Java program to solve N Queen Problem// using Branch and Boundimport java.io.*;import java.util.Arrays; class GFG{ static int N = 8; // A utility function to print solutionstatic void printSolution(int board[][]){    int N = board.length;    for(int i = 0; i < N; i++)    {        for(int j = 0; j < N; j++)            System.out.printf("%2d ", board[i][j]);                     System.out.printf("\n");    }} // A Optimized function to check if a queen// can be placed on board[row][col]static boolean isSafe(int row, int col,                      int slashCode[][],                      int backslashCode[][],                      boolean rowLookup[],                      boolean slashCodeLookup[],                      boolean backslashCodeLookup[]){    if (slashCodeLookup[slashCode[row][col]] ||        backslashCodeLookup[backslashCode[row][col]] ||        rowLookup[row])        return false;     return true;} // A recursive utility function to// solve N Queen problemstatic boolean solveNQueensUtil(    int board[][], int col, int slashCode[][],    int backslashCode[][], boolean rowLookup[],    boolean slashCodeLookup[],    boolean backslashCodeLookup[]){         // Base case: If all queens are placed    // then return true    int N = board.length;         if (col >= N)        return true;     // Consider this column and try placing    // this queen in all rows one by one    for(int i = 0; i < N; i++)    {        // Check if queen can be placed on board[i][col]        if (isSafe(i, col, slashCode, backslashCode,                   rowLookup, slashCodeLookup,                   backslashCodeLookup))        {                         // Place this queen in board[i][col]            board[i][col] = 1;            rowLookup[i] = true;            slashCodeLookup[slashCode[i][col]] = true;            backslashCodeLookup[backslashCode[i][col]] = true;             // recur to place rest of the queens            if (solveNQueensUtil(                    board, col + 1, slashCode,                    backslashCode, rowLookup,                    slashCodeLookup,                    backslashCodeLookup))                return true;             // If placing queen in board[i][col] doesn't            // lead to a solution, then backtrack             // Remove queen from board[i][col]            board[i][col] = 0;            rowLookup[i] = false;            slashCodeLookup[slashCode[i][col]] = false;            backslashCodeLookup[backslashCode[i][col]] = false;        }    }     // If queen can not be place in any row    // in this column col then return false    return false;} /* * This function solves the N Queen problem using Branch * and Bound. It mainly uses solveNQueensUtil() to solve * the problem. It returns false if queens cannot be * placed, otherwise return true and prints placement of * queens in the form of 1s. Please note that there may * be more than one solutions, this function prints one * of the feasible solutions. */static boolean solveNQueens(){    int board[][] = new int[N][N];         // Helper matrices    int slashCode[][] = new int[N][N];    int backslashCode[][] = new int[N][N];     // Arrays to tell us which rows are occupied    boolean[] rowLookup = new boolean[N];     // Keep two arrays to tell us    // which diagonals are occupied    boolean slashCodeLookup[] = new boolean[2 * N - 1];    boolean backslashCodeLookup[] = new boolean[2 * N - 1];     // Initialize helper matrices    for(int r = 0; r < N; r++)        for(int c = 0; c < N; c++)        {            slashCode[r] = r + c;            backslashCode[r] = r - c + 7;        }     if (solveNQueensUtil(board, 0, slashCode,                         backslashCode, rowLookup,                         slashCodeLookup,                         backslashCodeLookup) == false)    {        System.out.printf("Solution does not exist");        return false;    }         // Solution found    printSolution(board);    return true;} // Driver codepublic static void main(String[] args){    solveNQueens();}} // This code is contributed by sujitmeshram
""" Python3 program to solve N Queen Problemusing Branch or Bound """ N = 8 """ A utility function to print solution """def printSolution(board):    for i in range(N):        for j in range(N):            print(board[i][j], end = " ")        print() """ A Optimized function to check ifa queen can be placed on board[row][col] """def isSafe(row, col, slashCode, backslashCode,           rowLookup, slashCodeLookup,                       backslashCodeLookup):    if (slashCodeLookup[slashCode[row][col]] or        backslashCodeLookup[backslashCode[row][col]] or        rowLookup[row]):        return False    return True """ A recursive utility function   to solve N Queen problem """def solveNQueensUtil(board, col, slashCode, backslashCode,                     rowLookup, slashCodeLookup,                     backslashCodeLookup):                             """ base case: If all queens are       placed then return True """    if(col >= N):        return True    for i in range(N):        if(isSafe(i, col, slashCode, backslashCode,                  rowLookup, slashCodeLookup,                  backslashCodeLookup)):                                 """ Place this queen in board[i][col] """            board[i][col] = 1            rowLookup[i] = True            slashCodeLookup[slashCode[i][col]] = True            backslashCodeLookup[backslashCode[i][col]] = True                         """ recur to place rest of the queens """            if(solveNQueensUtil(board, col + 1,                                slashCode, backslashCode,                                rowLookup, slashCodeLookup,                                backslashCodeLookup)):                return True                         """ If placing queen in board[i][col]            doesn't lead to a solution,then backtrack """                         """ Remove queen from board[i][col] """            board[i][col] = 0            rowLookup[i] = False            slashCodeLookup[slashCode[i][col]] = False            backslashCodeLookup[backslashCode[i][col]] = False                 """ If queen can not be place in any row in    this column col then return False """    return False """ This function solves the N Queen problem usingBranch or Bound. It mainly uses solveNQueensUtil()tosolve the problem. It returns False if queenscannot be placed,otherwise return True orprints placement of queens in the form of 1s.Please note that there may be more than onesolutions,this function prints one of thefeasible solutions."""def solveNQueens():    board = [[0 for i in range(N)]                for j in range(N)]         # helper matrices    slashCode = [[0 for i in range(N)]                    for j in range(N)]    backslashCode = [[0 for i in range(N)]                        for j in range(N)]         # arrays to tell us which rows are occupied    rowLookup = [False] * N         # keep two arrays to tell us    # which diagonals are occupied    x = 2 * N - 1    slashCodeLookup = [False] * x    backslashCodeLookup = [False] * x         # initialize helper matrices    for rr in range(N):        for cc in range(N):            slashCode[rr][cc] = rr + cc            backslashCode[rr][cc] = rr - cc + 7         if(solveNQueensUtil(board, 0, slashCode, backslashCode,                        rowLookup, slashCodeLookup,                        backslashCodeLookup) == False):        print("Solution does not exist")        return False             # solution found    printSolution(board)    return True # Driver CdesolveNQueens() # This code is contributed by SHUBHAMSINGH10
<script> // JavaScript program to solve N Queen Problem// using Branch and Bound let N = 8; // A utility function to print solutionfunction printSolution(board){    let N = board.length;    for(let i = 0; i < N; i++)    {        for(let j = 0; j < N; j++)            document.write(board[i][j]+" ");                      document.write("<br>");    }} // A Optimized function to check if a queen// can be placed on board[row][col]function isSafe(row,col,slashCode,backslashCode,rowLookup,slashCodeLookup,backslashCodeLookup){    if (slashCodeLookup[slashCode[row][col]] ||        backslashCodeLookup[backslashCode[row][col]] ||        rowLookup[row])        return false;      return true;} // A recursive utility function to// solve N Queen problemfunction solveNQueensUtil(board,col,slashCode,backslashCode,rowLookup,slashCodeLookup,backslashCodeLookup){    // Base case: If all queens are placed    // then return true    //let N = board.length;          if (col >= N)        return true;      // Consider this column and try placing    // this queen in all rows one by one    for(let i = 0; i < N; i++)    {        // Check if queen can be placed on board[i][col]        if (isSafe(i, col, slashCode, backslashCode,                   rowLookup, slashCodeLookup,                   backslashCodeLookup))        {                          // Place this queen in board[i][col]            board[i][col] = 1;            rowLookup[i] = true;            slashCodeLookup[slashCode[i][col]] = true;            backslashCodeLookup[backslashCode[i][col]] = true;              // recur to place rest of the queens            if (solveNQueensUtil(                    board, col + 1, slashCode,                    backslashCode, rowLookup,                    slashCodeLookup,                    backslashCodeLookup))                return true;              // If placing queen in board[i][col] doesn't            // lead to a solution, then backtrack              // Remove queen from board[i][col]            board[i][col] = 0;            rowLookup[i] = false;            slashCodeLookup[slashCode[i][col]] = false;            backslashCodeLookup[backslashCode[i][col]] = false;        }    }      // If queen can not be place in any row    // in this column col then return false    return false;} /* * This function solves the N Queen problem using Branch * and Bound. It mainly uses solveNQueensUtil() to solve * the problem. It returns false if queens cannot be * placed, otherwise return true and prints placement of * queens in the form of 1s. Please note that there may * be more than one solutions, this function prints one * of the feasible solutions. */function solveNQueens(){    let board = new Array(N);          // Helper matrices    let slashCode = new Array(N);    let backslashCode = new Array(N);          for(let i=0;i<N;i++)    {        board[i]=new Array(N);        slashCode[i]=new Array(N);        backslashCode[i]=new Array(N);        for(let j=0;j<N;j++)        {            board[i][j]=0;            slashCode[i][j]=0;            backslashCode[i][j]=0;        }    }              // Arrays to tell us which rows are occupied    let rowLookup = new Array(N);     for(let i=0;i<N;i++)        rowLookup[i]=false;         // Keep two arrays to tell us    // which diagonals are occupied    let slashCodeLookup = new Array(2 * N - 1);    let backslashCodeLookup = new Array(2 * N - 1);     for(let i=0;i<2*N-1;i++)    {        slashCodeLookup[i]=false;        backslashCodeLookup[i]=false;    }      // Initialize helper matrices    for(let r = 0; r < N; r++)        for(let c = 0; c < N; c++)        {            slashCode[r] = r + c;            backslashCode[r] = r - c + 7;        }      if (solveNQueensUtil(board, 0, slashCode,                         backslashCode, rowLookup,                         slashCodeLookup,                         backslashCodeLookup) == false)    {        document.write("Solution does not exist");        return false;    }          // Solution found    printSolution(board);    return true;} // Driver codesolveNQueens();  // This code is contributed by avanitrachhadiya2155 </script>
Input:
Enter the no of rows for the square Board : 8
Output : 
 1  0  0  0  0  0  0  0 
 0  0  0  0  0  0  1  0 
 0  0  0  0  1  0  0  0 
 0  0  0  0  0  0  0  1 
 0  1  0  0  0  0  0  0 
 0  0  0  1  0  0  0  0 
 0  0  0  0  0  1  0  0 
 0  0  1  0  0  0  0  0
References : https://en.wikipedia.org/wiki/Eight_queens_puzzle www.cs.cornell.edu/~wdtseng/icpc/notes/bt2.pdfThis article is contributed by Aditya Goel. If you like GeeksforGeeks and would like to contribute, you can also write an article and mail your article to review-team@geeksforgeeks.org. See your article appearing on the GeeksforGeeks main page and help other Geeks.Please write comments if you find anything incorrect, or you want to share more information about the topic discussed above
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