Write a program to solve the 8-puzzle problem (and its natural generalizations) using the A* search algorithm.
The problem.
The 8-puzzle problem
is a puzzle popularized by Sam Loyd in the 1870s. It is
played on a 3-by-3 grid with 8 square blocks labeled 1 through 8 and a blank
square. Your goal is to rearrange the blocks so that they are in order.
You are permitted to slide blocks horizontally or vertically
into the blank square.
The following
shows a sequence of legal moves from an initial board position (left)
to the goal position (right).
1 3 1 3 1 2 3 1 2 3 1 2 3 4 2 5 => 4 2 5 => 4 5 => 4 5 => 4 5 6 7 8 6 7 8 6 7 8 6 7 8 6 7 8 initial goal
Best-first search. We now describe a solution to the problem that illustrates a general artificial intelligence methodology known as the A* search algorithm. We define a state of the game to be the board position, the number of moves made to reach there, and the previous state. First, insert the initial state (the initial board, 0 moves, and a null previous state) into a priority queue. Then, delete from the priority queue the state with the minimum priority, and insert onto the priority queue all neighboring states (those that can be reached in one move). Repeat this procedure until the state dequeued is the goal state. The success of this approach hinges on the choice of priority function.
8 1 3 1 2 3 1 2 3 4 5 6 7 8 1 2 3 4 5 6 7 8 4 2 4 5 6 ---------------------- ---------------------- 7 6 5 7 8 1 1 0 0 1 1 0 1 1 2 0 0 2 2 0 3 initial goal Hamming = 5 Manhattan = 10
We make a key oberservation: to solve the puzzle from a given state on the priority queue, the total number of moves we need to make (including those already made) is at least its priority, using either the Hamming or Manhattan priority function. (For Hamming priority, this is true because each block out of place must move at least once to reach its goal position. For Manhattan priority, this is true because each block must move its Manhattan distance from its goal position. Note that we do not count the blank tile when computing the Hamming or Manhattan priorities.)
Consequently, as soon as we dequeue a state, we have not only discovered a sequence of moves from the initial board to the board associated with the state, but one that makes the fewest number of moves. (Challenge for the mathematically inclined: prove this fact.)
A useful optimization. After implementing this procedure, you will notice one annoying feature: states corresponding to the same board position are enqueued on the priority queue many times. To prevent unnecessary exploration of useless states, when considering the neighbors of a state, don't enqueue the neighbor if its board position is the same as the previous state.
8 1 3 8 1 3 8 1 3 4 2 4 2 4 2 7 6 5 7 6 5 7 6 5 previous state disallow
Your task. Write a program Solver.java that reads the initial board from standard input and prints to standard output a sequence of board positions that solves the puzzle in the fewest number of moves. Also print out the total number of moves and the total number of states ever enqueued. The input will consist of the board dimension N followed by the N-by-N initial board position. The input format uses 0 to represent the blank square. As an example,
Note that your program should work for arbitrary N-by-N boards (for any N greater than 1), even if it is too slow to solve them.% more puzzle04.txt 3 0 1 3 4 2 5 7 8 6 % java Solver < puzzle04.txt 1 3 4 2 5 7 8 6 1 3 4 2 5 7 8 6 1 2 3 4 5 7 8 6 1 2 3 4 5 7 8 6 1 2 3 4 5 6 7 8 Number of moves = 4 Number of states enqueued = 11
Detecting infeasible puzzles. Not all initial board positions can lead to the goal state. Modify your program to detect and report such situations.
Hint: you may use the fact that board positions are divided into two equivalence classes wirth repsect to reachability: (i) those that lead to the goal position and (ii) those that lead to the goal position if we modify the initial board by swapping any pair of adjacent (non-blank) blocks. There are two ways to apply the hint:% more puzzle-impossible3x3.txt 3 1 2 3 4 5 6 8 7 0 % java Solver < puzzle3x3-impossible.txt No solution possible
Deliverables. Organize your program in an appropriate number of data types. At a minimum, you are required to implement the following APIs. Though, you are free to add additional methods or data types (such as State).
public class Board {
public Board(int[][] tiles) // construct a board from an N-by-N array of tiles
public int hamming() // return number of blocks out of place
public int manhattan() // return sum of Manhattan distances between blocks and goal
public boolean equals(Object y) // does this board equal y
public Iterable<Board> neighbors() // return an Iterable of all neighboring board positions
public String toString() // return a string representation of the board
}
public class Solver {
public Solver(Board initial) // find a solution to the initial board
public boolean isSolvable() // is the initial board solvable?
public int moves() // return min number of moves to solve the initial board
// -1 if no such solution
public void showSolution() // display the solution (as described above)
// read puzzle instance from stdin and print solution to stdout, as above
public static void main(String[] args)
}
Submit Board.java, Solver.java (with the Manhattan priority) and any other helper data types that you use besides those included in stdlib.jar and adt.jar.