8 puzzle Problem using Branch And Bound

We have introduced Branch and Bound and discussed the 0/1 Knapsack problem in the below posts. 

• Branch and Bound | Set 1 (Introduction with 0/1 Knapsack)
• Branch and Bound | Set 2 (Implementation of 0/1 Knapsack)

In this puzzle solution of the 8 puzzle problem is discussed. 
Given a 3×3 board with 8 tiles (every tile has one number from 1 to 8) and one empty space. The objective is to place the numbers on tiles to match the final configuration using the empty space. We can slide four adjacent (left, right, above, and below) tiles into the empty space. 

For example, 

1. DFS (Brute-Force)
We can perform a depth-first search on state-space (Set of all configurations of a given problem i.e. all states that can be reached from the initial state) tree. 

State Space Tree for 8 Puzzle

In this solution, successive moves can take us away from the goal rather than bringing us closer. The search of state-space tree follows the leftmost path from the root regardless of the initial state. An answer node may never be found in this approach.

2. BFS (Brute-Force)
We can perform a Breadth-first search on the state space tree. This always finds a goal state nearest to the root. But no matter what the initial state is, the algorithm attempts the same sequence of moves like DFS.

3. Branch and Bound
The search for an answer node can often be speeded by using an “intelligent” ranking function, also called an approximate cost function to avoid searching in sub-trees that do not contain an answer node. It is similar to the backtracking technique but uses a BFS-like search.

There are basically three types of nodes involved in Branch and Bound 
1. Live node is a node that has been generated but whose children have not yet been generated. 
2. E-node is a live node whose children are currently being explored. In other words, an E-node is a node currently being expanded. 
3. Dead node is a generated node that is not to be expanded or explored any further. All children of a dead node have already been expanded.

Cost function: 
Each node X in the search tree is associated with a cost. The cost function is useful for determining the next E-node. The next E-node is the one with the least cost. The cost function is defined as 

   C(X) = g(X) + h(X) where
   g(X) = cost of reaching the current node 
          from the root
   h(X) = cost of reaching an answer node from X.

The ideal Cost function for an 8-puzzle Algorithm : 
We assume that moving one tile in any direction will have a 1 unit cost. Keeping that in mind, we define a cost function for the 8-puzzle algorithm as below: 

   c(x) = f(x) + h(x) where
   f(x) is the length of the path from root to x 
        (the number of moves so far) and
   h(x) is the number of non-blank tiles not in 
        their goal position (the number of mis-
        -placed tiles). There are at least h(x) 
        moves to transform state x to a goal state

An algorithm is available for getting an approximation of h(x) which is an unknown value.

Complete Algorithm:

/* Algorithm LCSearch uses c(x) to find an answer node
 * LCSearch uses Least() and Add() to maintain the list 
   of live nodes
 * Least() finds a live node with least c(x), deletes
   it from the list and returns it
 * Add(x) adds x to the list of live nodes
 * Implement list of live nodes as a min-heap */

struct list_node
{
   list_node *next;

   // Helps in tracing path when answer is found
   list_node *parent; 
   float cost;
} 

algorithm LCSearch(list_node *t)
{
   // Search t for an answer node
   // Input: Root node of tree t
   // Output: Path from answer node to root
   if (*t is an answer node)
   {
       print(*t);
       return;
   }
   
   E = t; // E-node

   Initialize the list of live nodes to be empty;
   while (true)
   {
      for each child x of E
      {
          if x is an answer node
          {
             print the path from x to t;
             return;
          }
          Add (x); // Add x to list of live nodes;
          x->parent = E; // Pointer for path to root
      }

      if there are no more live nodes
      {
         print ("No answer node");
         return;
      }
       
      // Find a live node with least estimated cost
      E = Least(); 

      // The found node is deleted from the list of 
      // live nodes
   }
}

The below diagram shows the path followed by the above algorithm to reach the final configuration from the given initial configuration of the 8-Puzzle. Note that only nodes having the least value of cost function are expanded.

C++14

// Program to print path from root node to destination node
// for N*N -1 puzzle algorithm using Branch and Bound
// The solution assumes that instance of puzzle is solvable
#include <bits/stdc++.h>
using namespace std;
#define N 3

// state space tree nodes
struct Node
{
    // stores the parent node of the current node
    // helps in tracing path when the answer is found
    Node* parent;

    // stores matrix
    int mat[N][N];

    // stores blank tile coordinates
    int x, y;

    // stores the number of misplaced tiles
    int cost;

    // stores the number of moves so far
    int level;
};

// Function to print N x N matrix
int printMatrix(int mat[N][N])
{
    for (int i = 0; i < N; i++)
    {
        for (int j = 0; j < N; j++)
            printf("%d ", mat[i][j]);
        printf("\n");
    }
}

// Function to allocate a new node
Node* newNode(int mat[N][N], int x, int y, int newX,
              int newY, int level, Node* parent)
{
    Node* node = new Node;

    // set pointer for path to root
    node->parent = parent;

    // copy data from parent node to current node
    memcpy(node->mat, mat, sizeof node->mat);

    // move tile by 1 position
    swap(node->mat[x][y], node->mat[newX][newY]);

    // set number of misplaced tiles
    node->cost = INT_MAX;

    // set number of moves so far
    node->level = level;

    // update new blank tile coordinates
    node->x = newX;
    node->y = newY;

    return node;
}

// bottom, left, top, right
int row[] = { 1, 0, -1, 0 };
int col[] = { 0, -1, 0, 1 };

// Function to calculate the number of misplaced tiles
// ie. number of non-blank tiles not in their goal position
int calculateCost(int initial[N][N], int final[N][N])
{
    int count = 0;
    for (int i = 0; i < N; i++)
      for (int j = 0; j < N; j++)
        if (initial[i][j] && initial[i][j] != final[i][j])
           count++;
    return count;
}

// Function to check if (x, y) is a valid matrix coordinate
int isSafe(int x, int y)
{
    return (x >= 0 && x < N && y >= 0 && y < N);
}

// print path from root node to destination node
void printPath(Node* root)
{
    if (root == NULL)
        return;
    printPath(root->parent);
    printMatrix(root->mat);

    printf("\n");
}

// Comparison object to be used to order the heap
struct comp
{
    bool operator()(const Node* lhs, const Node* rhs) const
    {
        return (lhs->cost + lhs->level) > (rhs->cost + rhs->level);
    }
};

// Function to solve N*N - 1 puzzle algorithm using
// Branch and Bound. x and y are blank tile coordinates
// in initial state
void solve(int initial[N][N], int x, int y,
           int final[N][N])
{
    // Create a priority queue to store live nodes of
    // search tree;
    priority_queue<Node*, std::vector<Node*>, comp> pq;

    // create a root node and calculate its cost
    Node* root = newNode(initial, x, y, x, y, 0, NULL);
    root->cost = calculateCost(initial, final);

    // Add root to list of live nodes;
    pq.push(root);

    // Finds a live node with least cost,
    // add its childrens to list of live nodes and
    // finally deletes it from the list.
    while (!pq.empty())
    {
        // Find a live node with least estimated cost
        Node* min = pq.top();

        // The found node is deleted from the list of
        // live nodes
        pq.pop();

        // if min is an answer node
        if (min->cost == 0)
        {
            // print the path from root to destination;
            printPath(min);
            return;
        }

        // do for each child of min
        // max 4 children for a node
        for (int i = 0; i < 4; i++)
        {
            if (isSafe(min->x + row[i], min->y + col[i]))
            {
                // create a child node and calculate
                // its cost
                Node* child = newNode(min->mat, min->x,
                              min->y, min->x + row[i],
                              min->y + col[i],
                              min->level + 1, min);
                child->cost = calculateCost(child->mat, final);

                // Add child to list of live nodes
                pq.push(child);
            }
        }
    }
}

// Driver code
int main()
{
    // Initial configuration
    // Value 0 is used for empty space
    int initial[N][N] =
    {
        {1, 2, 3},
        {5, 6, 0},
        {7, 8, 4}
    };

    // Solvable Final configuration
    // Value 0 is used for empty space
    int final[N][N] =
    {
        {1, 2, 3},
        {5, 8, 6},
        {0, 7, 4}
    };

    // Blank tile coordinates in initial
    // configuration
    int x = 1, y = 2;

    solve(initial, x, y, final);

    return 0;
}

Java

// Java Program to print path from root node to destination node
// for N*N -1 puzzle algorithm using Branch and Bound
// The solution assumes that instance of puzzle is solvable
import java.io.*;
import java.util.*;

class GFG
{
    public static int N = 3;
    public static class Node
    {

        // stores the parent node of the current node
        // helps in tracing path when the answer is found
        Node parent;
        int mat[][] = new int[N][N];// stores matrix
        int x, y;// stores blank tile coordinates
        int cost;// stores the number of misplaced tiles
        int level;// stores the number of moves so far
    }

    // Function to print N x N matrix
    public static void printMatrix(int mat[][]){
        for(int i = 0; i < N; i++){
            for(int j = 0; j < N; j++){
                System.out.print(mat[i][j]+" ");
            }
            System.out.println("");
        }
    }

    // Function to allocate a new node
    public static Node newNode(int mat[][], int x, int y,
                               int newX, int newY, int level,
                               Node parent){
        Node node = new Node();
        node.parent = parent;// set pointer for path to root

        // copy data from parent node to current node
        node.mat = new int[N][N];
        for(int i = 0; i < N; i++){
            for(int j = 0; j < N; j++){
                node.mat[i][j] = mat[i][j];
            }
        }

        // move tile by 1 position
        int temp = node.mat[x][y];
        node.mat[x][y] = node.mat[newX][newY];
        node.mat[newX][newY]=temp;

        node.cost = Integer.MAX_VALUE;// set number of misplaced tiles
        node.level = level;// set number of moves so far

        // update new blank tile coordinates
        node.x = newX;
        node.y = newY;

        return node;
    }

    // bottom, left, top, right
    public static int row[] = { 1, 0, -1, 0 };
    public static int col[] = { 0, -1, 0, 1 };

    // Function to calculate the number of misplaced tiles
    // ie. number of non-blank tiles not in their goal position
    public static int calculateCost(int initialMat[][], int finalMat[][])
    {
        int count = 0;
        for (int i = 0; i < N; i++)
          for (int j = 0; j < N; j++)
            if (initialMat[i][j]!=0 && initialMat[i][j] != finalMat[i][j])
               count++;
        return count;
    }

    // Function to check if (x, y) is a valid matrix coordinate
    public static int isSafe(int x, int y)
    {
        return (x >= 0 && x < N && y >= 0 && y < N)?1:0;
    }

    // print path from root node to destination node
    public static void printPath(Node root){
        if(root == null){
            return;
        }
        printPath(root.parent);
        printMatrix(root.mat);
        System.out.println("");
    }

    // Comparison object to be used to order the heap
    public static class comp implements Comparator<Node>{
        @Override
        public int compare(Node lhs, Node rhs){
            return (lhs.cost + lhs.level) > (rhs.cost+rhs.level)?1:-1;
        }
    }

    // Function to solve N*N - 1 puzzle algorithm using
    // Branch and Bound. x and y are blank tile coordinates
    // in initial state
    public static void solve(int initialMat[][], int x,
                             int y, int finalMat[][])
    {

        // Create a priority queue to store live nodes of search tree
        PriorityQueue<Node> pq = new PriorityQueue<>(new comp());

        // create a root node and calculate its cost
        Node root = newNode(initialMat, x, y, x, y, 0, null);
        root.cost = calculateCost(initialMat,finalMat);

        // Add root to list of live nodes;
        pq.add(root);

        // Finds a live node with least cost,
        // add its childrens to list of live nodes and
        // finally deletes it from the list.
        while(!pq.isEmpty())
        {
            Node min = pq.peek();// Find a live node with least estimated cost
            pq.poll();// The found node is deleted from the list of live nodes

            // if min is an answer node
            if(min.cost == 0){
                printPath(min);// print the path from root to destination;
                return;
            }
            // do for each child of min
            // max 4 children for a node
            for (int i = 0; i < 4; i++)
            {
                if (isSafe(min.x + row[i], min.y + col[i])>0)
                {
                    // create a child node and calculate
                    // its cost
                    Node child = newNode(min.mat, min.x, min.y, min.x + row[i],min.y + col[i], min.level + 1, min);
                    child.cost = calculateCost(child.mat, finalMat);

                    // Add child to list of live nodes
                    pq.add(child);
                }
            }
        }
    }

    //Driver Code
    public static void main (String[] args)
    {

        // Initial configuration
        // Value 0 is used for empty space
        int initialMat[][] =
        {
            {1, 2, 3},
            {5, 6, 0},
            {7, 8, 4}
        };

        // Solvable Final configuration
        // Value 0 is used for empty space
        int finalMat[][] =
        {
            {1, 2, 3},
            {5, 8, 6},
            {0, 7, 4}
        };

        // Blank tile coordinates in initial
        // configuration
        int x = 1, y = 2;

        solve(initialMat, x, y, finalMat);
    }
}

// This code is contributed by shruti456rawal

Python3

# Python3 program to print the path from root
# node to destination node for N*N-1 puzzle
# algorithm using Branch and Bound
# The solution assumes that instance of
# puzzle is solvable

# Importing copy for deepcopy function
import copy

# Importing the heap functions from python
# library for Priority Queue
from heapq import heappush, heappop

# This variable can be changed to change
# the program from 8 puzzle(n=3) to 15
# puzzle(n=4) to 24 puzzle(n=5)...
n = 3

# bottom, left, top, right
row = [ 1, 0, -1, 0 ]
col = [ 0, -1, 0, 1 ]

# A class for Priority Queue
class priorityQueue:

    # Constructor to initialize a
    # Priority Queue
    def __init__(self):
        self.heap = []

    # Inserts a new key 'k'
    def push(self, k):
        heappush(self.heap, k)

    # Method to remove minimum element
    # from Priority Queue
    def pop(self):
        return heappop(self.heap)

    # Method to know if the Queue is empty
    def empty(self):
        if not self.heap:
            return True
        else:
            return False

# Node structure
class node:

    def __init__(self, parent, mat, empty_tile_pos,
                 cost, level):

        # Stores the parent node of the
        # current node helps in tracing
        # path when the answer is found
        self.parent = parent

        # Stores the matrix
        self.mat = mat

        # Stores the position at which the
        # empty space tile exists in the matrix
        self.empty_tile_pos = empty_tile_pos

        # Stores the number of misplaced tiles
        self.cost = cost

        # Stores the number of moves so far
        self.level = level

    # This method is defined so that the
    # priority queue is formed based on
    # the cost variable of the objects
    def __lt__(self, nxt):
        return self.cost < nxt.cost

# Function to calculate the number of
# misplaced tiles ie. number of non-blank
# tiles not in their goal position
def calculateCost(mat, final) -> int:

    count = 0
    for i in range(n):
        for j in range(n):
            if ((mat[i][j]) and
                (mat[i][j] != final[i][j])):
                count += 1

    return count

def newNode(mat, empty_tile_pos, new_empty_tile_pos,
            level, parent, final) -> node:

    # Copy data from parent matrix to current matrix
    new_mat = copy.deepcopy(mat)

    # Move tile by 1 position
    x1 = empty_tile_pos[0]
    y1 = empty_tile_pos[1]
    x2 = new_empty_tile_pos[0]
    y2 = new_empty_tile_pos[1]
    new_mat[x1][y1], new_mat[x2][y2] = new_mat[x2][y2], new_mat[x1][y1]

    # Set number of misplaced tiles
    cost = calculateCost(new_mat, final)

    new_node = node(parent, new_mat, new_empty_tile_pos,
                    cost, level)
    return new_node

# Function to print the N x N matrix
def printMatrix(mat):

    for i in range(n):
        for j in range(n):
            print("%d " % (mat[i][j]), end = " ")

        print()

# Function to check if (x, y) is a valid
# matrix coordinate
def isSafe(x, y):

    return x >= 0 and x < n and y >= 0 and y < n

# Print path from root node to destination node
def printPath(root):

    if root == None:
        return

    printPath(root.parent)
    printMatrix(root.mat)
    print()

# Function to solve N*N - 1 puzzle algorithm
# using Branch and Bound. empty_tile_pos is
# the blank tile position in the initial state.
def solve(initial, empty_tile_pos, final):

    # Create a priority queue to store live
    # nodes of search tree
    pq = priorityQueue()

    # Create the root node
    cost = calculateCost(initial, final)
    root = node(None, initial,
                empty_tile_pos, cost, 0)

    # Add root to list of live nodes
    pq.push(root)

    # Finds a live node with least cost,
    # add its children to list of live
    # nodes and finally deletes it from
    # the list.
    while not pq.empty():

        # Find a live node with least estimated
        # cost and delete it form the list of
        # live nodes
        minimum = pq.pop()

        # If minimum is the answer node
        if minimum.cost == 0:

            # Print the path from root to
            # destination;
            printPath(minimum)
            return

        # Generate all possible children
        for i in range(4):
            new_tile_pos = [
                minimum.empty_tile_pos[0] + row[i],
                minimum.empty_tile_pos[1] + col[i], ]

            if isSafe(new_tile_pos[0], new_tile_pos[1]):

                # Create a child node
                child = newNode(minimum.mat,
                                minimum.empty_tile_pos,
                                new_tile_pos,
                                minimum.level + 1,
                                minimum, final,)

                # Add child to list of live nodes
                pq.push(child)

# Driver Code

# Initial configuration
# Value 0 is used for empty space
initial = [ [ 1, 2, 3 ],
            [ 5, 6, 0 ],
            [ 7, 8, 4 ] ]

# Solvable Final configuration
# Value 0 is used for empty space
final = [ [ 1, 2, 3 ],
          [ 5, 8, 6 ],
          [ 0, 7, 4 ] ]

# Blank tile coordinates in
# initial configuration
empty_tile_pos = [ 1, 2 ]

# Function call to solve the puzzle
solve(initial, empty_tile_pos, final)

# This code is contributed by Kevin Joshi

Output : 

1 2 3 
5 6 0 
7 8 4 

1 2 3 
5 0 6 
7 8 4 

1 2 3 
5 8 6 
7 0 4 

1 2 3 
5 8 6 
0 7 4

Sources:
www.cs.umsl.edu/~sanjiv/classes/cs5130/lectures/bb.pdf
https://www.seas.gwu.edu/~bell/csci212/Branch_and_Bound.pdf

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