Optimal solutions for Rubik's Cube
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There are many algorithms to solve scrambled Rubik's Cubes. One such method is described in Wikibooks' article How to solve the Rubik's Cube. This is one algorithm that has the advantage of being simple enough to be memorizable by humans, however it will usually not give an optimal solution which only uses the minimum possible number of moves.
Note: Notation from How to solve the Rubik's Cube is used in this article.
It is not known how many moves is the minimum required to solve any instance of the Rubik's cube. This number is also known as the diameter of the Cayley graph of the Rubik's Cube group. An algorithm that solves a cube in the minimum number of moves is known as 'God's algorithm'.
When discussing the length of a solution, there are two common ways to measure this. The first is to count the number of quarter turns. The second is to count the number of face turns. A move like F2 (a half turn of the front face) would be counted as 2 moves in the quarter turn metric and as only 1 turn in the face metric.
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[edit] Lower bounds
It can be proven by counting arguments that there exist positions needing at least 18 moves to solve. To show this, first count the number of cube positions that exist in total, then count the number of positions achievable using at most 17 moves. It turns out that the latter number is smaller.
This argument was not improved upon for many years. Also, it is not a constructive proof: it does not exhibit a concrete position that needs this many moves. It was conjectured that the so-called superflip would be a position that is very difficult. The superflip is a position on the cube where all the cubies are in their correct position, all the corners are correctly oriented but each edge is oriented the wrong way.
One indication that this might be the case is that it is the only element other than the identity that is in the center of the cube group.
In 1992 a solution for the superflip with 20 face turns was found by Dik T. Winter. In 1995, Michael Reid proved its minimality, thereby giving a new lower bound for the diameter of the cube group.
Also in 1995, a solution for superflip in 24 quarter turns was found by Michael Reid, its minimality was proven by Jerry Bryan. [1]
In 1998 Michael Reid found a new position requiring more than 24 quarter turns to solve. The position, named by him as 'superflip composed with four spot' needs 26 quarter turns. [2]
[edit] Upper bounds
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The first upper bounds were based on the 'human' algorithms. By combining the worst-case scenarios for each part of these algorithms, the typical upper bound was found to be around 100. The breakthrough was found by Morwen Thistlethwaite; details were published in Scientific American in 1981 by Douglas R. Hofstadter. The approaches to the cube that lead to algorithms with very few moves are based on group theory and on extensive computer searches.
Thistlethwaite's idea was to divide the problem into subproblems. Where algorithms up to that point divided the problem by looking at the parts of the cube that should remain fixed, he divided it by restricting the type of moves you could execute.
In particular he divided the cube group into the following chain of subgroups:
- G0 = <L,R,F,B,U,D>
- G1 = <L,R,F,B,U2,D2>
- G2 = <L,R,F2,B2,U2,D2>
- G3 = <L2,R2,F2,B2,U2,D2>
- G4 = {I}
Next he prepared tables for each of the right coset spaces G[i+1]\Gi. For each element he found a sequence of moves that took it to the next smaller group.
After these preparations he worked as follows. A random cube is in the general cube group G0. Next he found this element in the right coset space G1\G0. He applied the corresponding process to the cube. This took it to a cube in G1. Next he looked up a process that takes the cube to G2, next to G3 and finally to G4.
Although the whole cube group G0 is very large (~4.3×1019), the right coset spaces G1\G0, G2\G1, G3\G2 and G3 are much smaller. The coset space G2\G1 is the largest and contains only 1082565 elements. The number of moves required by this algorithm is the sum of the largest process in each step. In the original version this was 52.
This algorithm was improved by Herbert Kociemba in 1992. He reduced the number of intermediate groups to only two:
- G0 = <L,R,F,B,U,D>
- G1 = <L,R,F2,B2,U2,D2>
- G2 = {I}.
As with Thistlethwaite's algorithm, he would search through the right coset space G1\G0 to take the cube to group G1. Next he searched the optimal solution for group G1. The searches in G1\G0 and G1 were both done with a method equivalent to IDA*. The search in G1\G0 needs at most 12 moves and the search in G1 at most 18 moves, as Michael Reid showed in 1995. By generating also suboptimal solutions that take the cube to group G1 and looking for short solutions in G1, you usually get much shorter overall solutions. Using this algorithm solutions are typically found of less than 21 moves, though there is no proof that it will always do so.
Next in 1995 it was proven by Michael Reid that using these two groups every position can be solved in at most 29 face turns, or in 42 quarter turns. This result was improved by Silviu Radu in 2005 to 40.
Using these group solutions combined with computer searches will generally quickly give very short solutions. But these solutions do not always come with a guarantee of their minimality. To search specifically for minimal solutions a new approach was needed.
In 1997 Richard Korf[3] announced an algorithm with which he had optimally solved random instances of the cube. Of the ten random cubes he did, none required more than 18 face turns. The method he used is called IDA* and is described in his paper "Finding Optimal Solutions to Rubik's Cube Using Pattern Databases." Korf describes this method as follows
- IDA* is a depth-first search that looks for increasingly longer solutions in a series of iterations, using a lower-bound heuristic to prune branches once a lower bound on their length exceeds the current iterations bound.
It works roughly as follows. First he identified a number of subproblems that are small enough to be solved optimally. He used:
- The cube restricted to only the corners, not looking at the edges
- The cube restricted to only 6 edges, not looking at the corners nor at the other edges.
- The cube restricted to the other 6 edges.
Clearly the number of moves required to solve any of these subproblems is a lower bound for the number of moves you will need to solve the entire cube.
Given a random cube C, it is solved as iterative deepening. First all cubes are generated that are the result of applying 1 move to them. That is C * F, C * U, … Next, from this list, all cubes are generated that are the result of applying two moves. Then three moves and so on. If at any point a cube is found that needs too many moves based on the upper bounds to still be optimal it can be eliminated from the list.
Although this algorithm will always find optimal solutions there is no worst case analysis. It is not known how many moves this algorithm might need. An implementation of this algorithm can be found here [4].
In 2006, Silviu Radu further improved his methods to prove that every position can be solved in at most 27 face turns or 35 quarter turns[5].
In August 2007, Daniel Kunkle and Gene Cooperman used a supercomputer to show that all unsolved cubes can be solved in no more than 26 moves (in face-turn metric). Instead of attempting to solve each of the billions of variations explicitly, the computer was programmed to bring the cube to one of 15,000 states, each of which could be solved within a few extra moves. All were proved solvable in 29 moves, with most solvable in 26. Those that could not initially be solved in 26 moves were then solved explicitly, and shown that they too could be solved in 26 moves. [6] [7]
In 2008, Tomas Rokicki was reported to have devised a computational proof that all unsolved cubes could be solved in 25 moves or fewer.[8] This was later reduced to 23 moves.[9]
[edit] References
- ^ Michael Reid's Rubik's cube page M-symmetric positions
- ^ Posted to Cube lovers on 2 Aug 1998
- ^ Richard Korf's Finding Optimal Solutions to Rubik's Cube Using Pattern Databases
- ^ Michael Reid's Optimal Solver for Rubik's Cube (requires a compiler such as gcc)
- ^ Domain of the Cube Forum article
- ^ Press Release on Proof that 26 Face Turns Suffice
- ^ Kunkle, D.; Cooperman, C. (2007). "Twenty-Six Moves Suffice for Rubik's Cube" (PDF). Proceedings of the International Symposium on Symbolic and Algebraic Computation (ISSAC '07), ACM Press.
- ^ Tom Rokicki. Twenty-Five Moves Suffice for Rubik's Cube. Retrieved on 2008-03-24.
- ^ http://cubezzz.homelinux.org/drupal/?q=node/view/117
[edit] Links
- Herbert Kociemba's Two-Phase-Solver and Optimal Solver for Rubik's Cube
- Ryan Heise's Human version of the Thistlethwaite algorithm
- A New Upper Bound on Rubik's Cube Group, Silviu Radu
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