Give a part of chessboard of 15-puzzle, how to get all of the state of the the part chessboard using BFS?

Question

the board is like this:

1 2 3 4
5 6 7 0
0 0 0 0
0 0 0 0

the '0' represents that is empty, we can move the non-zero number to the '0'. so how to get all of the state of the board using BFS?

for example, there are two state of the board:

1 2 3 4
0 0 0 0
5 6 7 0
0 0 0 0

1 2 3 0
4 0 0 0
5 0 0 0
6 7 0 0

The reason I ask this question is that I need to process all of the 15-puzzle state using Disjoint pattern database to solve the nearly most difficult state of 15-puzzle in 1 minutes.

15 14 13 12
11 10 9 8
7 6 5 4
3 1 2 0

Answer 1

I need to process all of the 15-puzzle state [..] to solve the nearly most difficult state of 15-puzzle in 1 minutes

Approach 1 - using a database and storing all states

For reasons given by Henry as well, and also supported by [1], solving this problem using a database would require generating the entire A_15 , storing all of it and then finding the shortest path, or some path between a given state and the solved state. This would require a lot of space and a lot of time. See this discussion for an outline of this approach.

Approach 2 - using a specialized depth-first search algorithm

Here is an implementation of this search strategy that uses the IDA algorithm.

Approach 3 - using computational group theory

Yet another way to handle this in a much shorter amount of time is to use GAP (which implements a variant of Schreier-Sims) in order to decompose a given word into a product of generators. There is an example in the docs that shows how to use it to solve the Rubik's cube, and it can be adapted to the 15-puzzle too [2].

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[1] Permutation Puzzles - A Mathematical Perspective by Jamie Mulholland - see page 103 and 104 for solvability criteria, and the state space being |A_15| ~ 653 billion

[2] link2 - page 37