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Can I use different class names, method names, or method signatures from those prescribed in the API? No, as usual, you will receive a serious deduction for violating the API.
Is 0 a tile? No, 0 represents the absence of a tile. Do not treat it as a tile when computing either the Hamming or Manhattan priority functions.
Can I assume that the puzzle inputs (arguments to the Board constructor and input to Solver) are valid? Yes, though it never hurts to include some basic error checking.
Do I have to implement my own stack, queue, and priority queue? No, use the versions in algs4.jar.
How do I return an Iterable<Board>? Add the items you want to a Stack<Board>, Queue<Board>, or java.util.ArrayList<Board> and return that.
How do I implement equals()? Java has some arcane rules for implementing equals(), discussed on p. 103 of Algorithms, 4th edition. Note that the argument to equals() is required to be Object. For online examples, see Date.java or Transaction.java.
Must I implement the toString() method for Board exactly as specified? Yes. Be sure to include the board dimension and use 0 for the blank square. Use String.format() to format strings—it works like StdOut.printf(), but returns the string instead of printing it to standard output. For reference, our implementation is below.
public String toString() {
StringBuilder s = new StringBuilder();
s.append(N + "\n");
for (int i = 0; i < N; i++) {
for (int j = 0; j < N; j++) {
s.append(String.format("%2d ", tileAt(i, j)));
}
s.append("\n");
}
return s.toString();
}
Should the hamming() and manhattan() methods in Board return the Hamming and Manhattan priority functions? No, you should return the sum of the Hamming and Manhattan distances from the tiles to their goal positions. Also, recall that the blank square is not considered a tile. You will compute the priority function in Solver by calling hamming() or manhattan() and adding to it the number of moves.
How do I reconstruct the solution once I've dequeued the goal search node? Since each search node records the previous search node to get there, you can chase the pointers all the way back to the initial search node (and consider them in reverse order).
Can I terminate the search as soon as a goal search node is enqueued (instead of dequeued)? No, even though it does lead to a correct solution for the slider puzzle problem using the Hamming and Manhattan priority functions, it's not technically the A* algorithm (and will not find the correct solution for other problems and other priority functions).
I noticed that the priorities of the search nodes dequeued from the priority queue never decrease. Is this a property of the A* algorithm? Yes. In the language of the A* algorithm, the Hamming and Manhattan distances (before adding in the number of moves so far) are known as heuristics. If a heuristic is both admissible (never overestimates the number of moves to the goal search node) and consistent (satisfies a certain triangle inequality), then this property is guaranteed. The Hamming and Manhattan heuristics are both admissible and consistent. You may use this property as a debugging clue: if it is ever violated, then you know you did something wrong.
Even with the critical optimization, the priority queue may contain two or more search nodes corresponding to the same board. Should I try to eliminate these? In principle, you could do so with a set data type such as SET in algs4.jar or java.util.TreeSet (provided that the Board data type were Comparable). However, almost all of the benefit from avoiding duplicate boards is already extracted from the critical optimization and the cost to identify other duplicate boards exceeds the benefit from doing so.
I run out of memory when running some of the large sample puzzles. What should I do? Be sure to ask Java for additional memory, e.g., java -Xmx1600m Solver puzzle36.txt. If your program is unable to solve certain instances, document that in your readme.txt file. You should expect to run out of memory when using the Hamming priority function. Be sure not to put the JVM option in the wrong spot or it will be treated as a command-line argument, e.g., java Solver -Xmx1600m puzzle36.txt.
My program is too slow to solve some of the large sample puzzles, even if given a huge amount of memory. Is this OK? You should not expect to solve many of the larger puzzles with the Hamming priority function. However, you should be able to solve most (but not all) of the larger puzzles with the Manhattan priority function. Also, be sure to execute from the command line (and not DrJava).
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Input files. The ftp directory contains many sample puzzles. The shortest solution to puzzle4x4-hard1.txt and puzzle4x4-hard2.txt are 38 and 47, respectively. The shortest solution to puzzle*[T].txt requires exactly T moves. Warning: puzzle36.txt, puzzle47.txt, and puzzle49.txt, and puzzle50.txt are relatively difficult.
Testing. A good way to automatically run your program on our sample puzzles is to use the client PuzzleChecker.java.
Visualization client. You can also use SolverVisualizer.java, which takes the name of a puzzle file as a command-line argument and animates the solution.
Sample trace. The program defines two different data structures on the set of search nodes—the game tree and the priority queue. Below is a detailed trace of each data structure during the solution to puzzle04.txt.
ID Board Priority function Parent in game tree
================================================================================
A 0 1 3 Manhattan: 4 Previous: null
4 2 5 Moves: 0
7 8 6 Priority: 4 + 0 = 4
B 1 0 3 Manhattan: 3 Previous: A
4 2 5 Moves: 1
7 8 6 Priority: 3 + 1 = 4
C 4 1 3 Manhattan: 5 Previous: A
0 2 5 Moves: 1
7 8 6 Priority: 5 + 1 = 6
d 0 1 3 Manhattan: 4 Previous: B
4 2 5 Moves: 2
7 8 6 Priority: 4 + 2 = 6
E 1 2 3 Manhattan: 2 Previous: B
4 0 5 Moves: 2
7 8 6 Priority: 2 + 2 = 4
F 1 3 0 Manhattan: 4 Previous: B
4 2 5 Moves: 2
7 8 6 Priority: 4 + 2 = 6
g 1 0 3 Manhattan: 3 Previous: E
4 2 5 Moves: 3
7 8 6 Priority: 3 + 3 = 6
H 1 2 3 Manhattan: 3 Previous: E
0 4 5 Moves: 3
7 8 6 Priority: 3 + 3 = 6
I 1 2 3 Manhattan: 3 Previous: E
4 8 5 Moves: 3
7 0 6 Priority: 3 + 3 = 6
J 1 2 3 Manhattan: 1 Previous: E
4 5 0 Moves: 3
7 8 6 Priority 1 + 3 = 4
K 1 2 0 Manhattan: 2 Previous: J
4 5 3 Moves 4
7 8 6 Priority 2 + 4 = 6
l 1 2 3 Manhattan: 2 Previous: J
4 0 5 Moves: 4
7 8 6 Priority: 2 + 4 = 6
M 1 2 3 Manhattan: 0 Previous: J
4 5 6 Moves: 4
7 8 0 Priority: 0 + 4 = 4
################################################################################
Step 0
================================================================================
Game Tree
--------------------------------------------------------------------------------
A
Priority Queue
--------------------------------------------------------------------------------
pq = new MinPQ(); 0 1 2 3 4 5 6 7 8 9
- - - - - - - - - -
pq.insert(A); 0 1 2 3 4 5 6 7 8 9
- A - - - - - - - -
################################################################################
Step 1
================================================================================
Game Tree
--------------------------------------------------------------------------------
A
|
---
/ \
B C
Priority Queue
--------------------------------------------------------------------------------
pq.delMin(); 0 1 2 3 4 5 6 7 8 9
// returns A - - - - - - - - - -
pq.insert(B); 0 1 2 3 4 5 6 7 8 9
- B - - - - - - - -
pq.insert(C); 0 1 2 3 4 5 6 7 8 9
- B C - - - - - - -
################################################################################
Step 2
================================================================================
Game Tree
--------------------------------------------------------------------------------
A
|
---
/ \
B C
|
---
/ | \
d E F
Priority Queue
--------------------------------------------------------------------------------
pq.delMin(); 0 1 2 3 4 5 6 7 8 9
// returns B - C - - - - - - - -
pq.insert(E); 0 1 2 3 4 5 6 7 8 9
- E C - - - - - - -
pq.insert(F); 0 1 2 3 4 5 6 7 8 9
- E C F - - - - - -
################################################################################
Step 3
================================================================================
Game Tree
--------------------------------------------------------------------------------
A
|
---
/ \
B C
|
---
/ | \
d E F
|
-------
/ | | \
g H I J
Priority Queue
--------------------------------------------------------------------------------
pq.delMin(); 0 1 2 3 4 5 6 7 8 9
// returns E - F C - - - - - - -
pq.insert(H); 0 1 2 3 4 5 6 7 8 9
- F C H - - - - - -
pq.insert(I); 0 1 2 3 4 5 6 7 8 9
- F C H I - - - - -
pq.insert(J); 0 1 2 3 4 5 6 7 8 9
- J F H I C - - - -
################################################################################
Step 4
================================================================================
Game Tree
--------------------------------------------------------------------------------
A
|
---
/ \
B C
|
---
/ | \
d E F
|
-------
/ | | \
g H I J
|
---
/ | \
K l [M]
Priority Queue
--------------------------------------------------------------------------------
pq.delMin(); 0 1 2 3 4 5 6 7 8 9
// returns J - C F H I - - - - -
pq.insert(K); 0 1 2 3 4 5 6 7 8 9
- C F H I K - - - -
pq.insert(M); 0 1 2 3 4 5 6 7 8 9
- M F C I K H - - -
################################################################################
Step 5
================================================================================
Game Tree
--------------------------------------------------------------------------------
A
|
---
/ \
B C
|
---
/ | \
d E F
|
-------
/ | | \
g H I J
|
---
/ | \
K l [M]
Priority Queue
--------------------------------------------------------------------------------
pq.delMin(); 0 1 2 3 4 5 6 7 8 9
// returns M - H F C I K - - - -
M corresponds to a goal state, return path from root to leaf: A -> B -> E -> J -> M
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These are purely suggestions for how you might make progress. You do not have to follow these steps.
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How can I reduce the amount of memory a Board uses? For starters, recall that an N-by-N int[][] array in Java uses about 24 + 32N + 4N2 bytes; when N equals 3, this is 156 bytes. To save memory, consider using a 1D array of length N2. In principle, each board is a permutation of size N2, so you need only about lg ((N2)!) bits to represent it; when N equals 3, this is only 19 bits.
Any ways to speed up the algorithm? One useful trick is to cache the Manhattan distance of a board (or Manhattan priority of a search node). That is, maintain an extra instance variable; compute and initialize it upon construction (or first usage); afterwards, just return the cached value.
Is there an efficient way to solve the 8-puzzle and its generalizations? Finding a shortest solution to a slider puzzle is NP-hard, so it's unlikely that an efficient solution exists.
What if I'm satisfied with any solution and don't need one that uses the fewest number of moves? Yes, change the priority function to put more weight on the Manhattan distance, e.g., 100 times the Manhattan distance plus the number of moves made already. This paper describes an algorithm that guarantees to perform at most N^3 moves.
Are there better ways to solve 8- and 15-puzzle instances using the minimum number of moves? Yes, there are a number of approaches.
Can a puzzle have more than one shortest solution? Yes. See puzzle07.txt.
Solution 1
------------------------------------------------------------------------------------
1 2 3 1 2 3 1 2 3 1 2 3 1 2 3 1 2 3 1 2 3 1 2 3
7 6 7 6 7 4 6 7 4 6 4 6 4 6 4 5 6 4 5 6
5 4 8 5 4 8 5 8 5 8 7 5 8 7 5 8 7 8 7 8
Solution 2
------------------------------------------------------------------------------------
1 2 3 1 2 3 1 2 3 1 2 3 1 2 3 1 2 3 1 2 3 1 2 3
7 6 5 7 6 5 7 6 5 6 5 6 4 5 6 4 5 6 4 5 6
5 4 8 4 8 4 8 4 7 8 4 7 8 7 8 7 8 7 8
What's the maximum number of moves need to solve any 8-slider puzzle? Any 8-slider puzzle can be solved with at most 31 moves; any 15-slider puzzle can be solved with at most 80 moves.
What's the best known algorithm for determining whether a puzzle is solvable? N^2. There is a mergesort-style algorithm for counting the number of inversions of a permutation of size N in time proportional to N log N. You can also determine the parity of the number of inversions of a permutation in time proportional to N. We note that the best-known algorithm for counting the number of inversions of a permutation is N sqrt(log N)—see Counting Inversions, Offline Orthogonal Range Counting, and Related Problems.