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Optimal solver for pocket cube using iterative deepening search with additional optimizations.
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README.md

Pocket Cube Optimal Solver

This document describes version 2 of the project. For version 1 refer to branch v1.

This is an optimal solver for the Pocket Cube (2x2 Rubik's cube). For a given state of the cube, the application shows how to solve it optimally (with minimum number of moves possible), with step-by-step 3D animations.

Screenshot page 1 Screenshot page 2

Try It Yourself

https://lukapopijac.github.io/pocket-cube-optimal-solver/

How to Set Up for Development

Please, take a look at src/README.md

Motivation

This project is made as an exercise with several different goals in mind:

  • Experimenting with different search algorithms: DFS, BFS, A*, IDA*.
  • Experimenting with heuristics and pruning tables.
  • Tweaking the search to optimize the performance while minimizing the memory usage. In general, those two goals contradict each other, so the satisfying compromise was to be found.
  • Writing REST API with Node.js and Express, and deploying to AWS (only version 1).
  • Exploring Web Components for building UI.
  • Exploring 3D graphics and animation with pure CSS.
  • Preparing for the bigger challenge in the future project: Optimal solver for standard Rubik's Cube.

Pocket Cube Technicalities

The cube consists of 8 smaller cubies, each one with 3 color stickers on it. Any permutation of the cubies is possible, and 7 of them can be independently oriented in three ways. If we fix one cubie to have a chosen position and orientation, we can allow any permutation of the remaining 7 cubies and any orientation of 6 cubies (the orientation of the first cubie is fixed, 6 cubies can be independently oriented, and the orientation of the last one is determined by the other). The number of possible states is:

7! * 3^6 = 3674160

This is a fairly small amount of states, and it can be easily saved in a computer memory, in which case the search algorithm becomes trivial. As the purpose of this experiment was to try different search algorithms, the amount of memory was intentionally limited.

In case of a standard Rubik's cube, the number of possible states is vastly larger, and enumerating all the states is infeasible.

About the Search Algorithm

The idea of the implemented search algorithm is based on IDA* search. In short, IDA* works similarly like a regular iterative deepening search, but instead of exploring every node, it utilizes heuristics to prune some branches in the search three. As such it is very memory efficient and it provides an optimal solution as long as the heuristics is admissible.

Heuristic calculation is done using pattern database. For each state of the cube, the pattern can be extracted and its value can be found in the pattern database. That value represents the minimum number of moves required to solve the cube starting from the corresponding state. It is always preferable to have larger values as it would lead to better performance, but that also implies bigger database in general. The pattern database used in this project was selected to preserve good performance without using too much of memory.

IDA* search uses heuristics to prune branches which are guaranteed to lack a solution within the allowed depth. As heuristics calculation made with pattern databases is not consistent, it is possible to have a situation where IDA* will explore the subtree of what will appear to be a good node, too much. That will happen when the node has an excessively underestimated heuristics value.

To improve the IDA* search in this regard, I adjusted the search algorithm to provide more efficient pruning. I will explain the main idea with the following example.

Assume that a part of the search tree looks like in the picture below, and assume that the maximum depth is 8. Nodes A, B and C are positioned at depths 3, 4 and 5, respectively. Heuristic values of these nodes obtained from the pattern databases are h(A)=5, h(B)=4 and h(C)=8.

depth
-----          /
  3           A       h(A)=5
             /|                     maxDepth = 8
            / |...
  4        B        h(B)=4
          /|
         / |...
  5     C         h(C)=8
       /|\
       ...

During the search of this tree, when it comes to explore the node A, we can see that depth of the node plus its heuristic is acceptable: depth + heuristics <= maxDepth (3+5<=8). This leads to expansion of the node A into its children. The first child to handle is the node B. The similar situation happens with this node, as 4+4<=8, so the node B gets expanded. Again, its first child is the node C. Now, for the node C we can see that depth plus heuristic exceeds the maximum depth (5+8>8), so this part of the tree can be pruned. Furthermore, as the node C is just one step away from the node B, we know that the minimum number of steps required to get to the goal node cannot differ by more than one step. This leads to conclusion that the minimum number of steps to get to the goal from the node B is at least 7, so the heuristic value for the node B can be updated to that value. At this point, IDA* would continue with exploring the next child of the node B. My algorithm doesn't continue exploring the next child as it concludes from the updated heuristic value of the node B that the node does not satisfy requirement depth + heuristic <= maxDepth anymore: 4+7>8. The whole branch of A with the child B gets pruned at this point. The same procedure continues. As the node B is just one step away from the node A, heuristic of the node A can be updated to the value 6. Now even the node A does not satisfy the condition anymore, as 3+6>8. That means that at this point, we can prune the whole subtree which includes the node A.

It is straightforward to generalize this algorithm in the case when step costs are different than 1. I didn't bother with those details here as for the Rubik's cube, cost of one move is always 1.

There is one theoretical advantage of IDA* which is not present in my algorithm. As IDA* explores all the children of an expanded node, if the solution is not found in this subtree, it gains information about the minimum number of steps needed to find the solution in this subtree. That information can be used in the iterative deepening to set the maxDepth for the next iteration to be more than 1 step bigger than previous. However, this is just a theoretical advantage which is very unlikely to be useful in the case of Rubik's cube, as it almost always needs to increment the maximum depth of the next iteration just by one. As a result, when applied on the Rubik's cube, my algorithm basically doesn't lose any advantage of IDA*, but it improves the effectiveness of pruning mechanism.

Some Results

Note: results below refer to version 1. This version (v2) uses smaller pattern database to reduce the size of the app, as the whole app runs in the browser.

The number of moves needed to solve the Pocket Cube is 11 in the worst case. For such cases, this app written in JavaScript, finds an optimal solution in less than 10ms on my modest laptop from 2011. In other cases it works much faster, as expected. It uses pattern database which compressed has size 109KB.

The implemented search algorithm is about 30% faster than standard IDA* search. Implementation of both algorithms can be found in files search.js and search_idastar.js.

License

This software is released under the MIT license.

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