Solving a Rubik s Cube with IDA* Search and Neural Networks

Transcription

1 Solving a Rubik s Cube with IDA* Search and Neural Networks Justin Schneider CS 539 Yu Hen Hu Fall 2017

2 1 Introduction: A Rubik s Cube is a style of tactile puzzle, wherein 26 external cubes referred to as cubies surround a single core (see Figure 1). Each of the six faces of the cube can be rotated in multiples of 90, both clockwise and counterclockwise. On the outward-facing edges of each cubie, there are colored stickers that uniquely define each cubie from the rest. In the solved position, all six faces of the cube are composed of solid colors, but the colors can be in many different patterns approximately 43 quintillion (Ref. 3) when the cube is scrambled. The goal of this project is to find an optimal solution for a scrambled Rubik s Cube using a time- and space-efficient search algorithm such as Iterative Figure 1 Deepening A-Star Search (IDA*). Additionally, since IDA* search uses a heuristic function in addition to a distance parameter to choose the next best choice (Ref. 2), a neural network will be trained using self-generated data to serve as the aforementioned heuristic. Work Performed: Firstly, the data required to train and test the neural network was generated through a virtual state space in Java (see Appendix for code). The program maintained the state of a Rubik s Cube through a 20-element array, which represented the 12 edge cubies and 8 corner cubies present on the Cube. Since the 6 other cubies are only center pieces, and are simply used to determine which side is which color, knowing the location and orientation of those 20 movable cubies completely determines the state, relative to a specific rotation. The array contained characters, with uppercase for the corners and lowercase for the edges, complying with the

3 2 standard in Rubik s Cube notation. These names were given to the cubies via a very specific pattern of rotating around the Cube while traversing the alphabet (see Figure 2 and 3). When transitioning from a given state array to a training vector, the letters were all converted into their corresponding numerical values with respect to their position in the alphabet. To create each state array, a random set of moves was applied to a solved cube, and the number of moves used was saved as the target vector for the training vector generated from the Figure 2 state. Data was generated with increasing depths, up to a depth of 9 moves away from a solved state, which was the maximum depth possible before the program ran out of usable memory since the set of possible states expanded so rapidly as moves were added. Once the data was created, the neural network had to be trained, which presented the problem of imbalanced classes (Ref. 1). To combat this issue, the minority Figure 3 classes data sets from lower depths were oversampled to match the size of the majority classes, and then training data was taken via a random sampling of each set. Results and discussions: The neural networks used in this project were trained using the data generated as described above, and then tested using a IDA* search program designed within the same Java Rubik s Cube virtual state space mentioned earlier. Furthermore, the performance of the networks outputs as heuristics were compared against the Manhattan Distance Heuristic, which is a standard heuristic used when dealing with the Rubik s Cube (Ref. 3). The first neural network

4 3 trained contained a single hidden layer of ten nodes, and yielded a correlation coefficient of approximately 0.78, but classification accuracy of only 2.47 percent (see Figure 4). Since the network was so inaccurate, a scaling constant could not be found that would transform the output into a useable and admissible (never overestimates the true cost) heuristic function. If the output was scaled low enough such that it became admissible, its value became of little to no worth in the depth cutoff when compared to the distance parameter, so the program would get stuck and fail to converge to a solution. In an effort to remedy the errors in the network, it was reformatted to contain fifteen hidden nodes, and retrained. After the second round of training, the network yielded a correlation of 0.81 and classification accuracy of about percent, which was an increase of over 10-fold (see Figure 5). Due to the increased accuracy, in some low depth cases the output of the neural network could be both usable and admissible after scaling. However, in the majority of states tested, the output still remained unusable as a functioning IDA* search heuristic. Figure 4 Figure 5

5 4 Conclusion: Although the second neural network yielded much better results than the first, they both paled in comparison to the Manhattan Distance Heuristic, due to their extremely large rates of error. Both of the networks were trained on data sets of over 1.3 million vectors, but due to memory constraints the data only covered instances up to a maximum depth of nine. The improvement between the nets indicates the potential for creating an accurate network that can serve as a useable and admissible heuristic, but the resources required to train such a network and store the related data set are far beyond those available to most. In absence of these resources, however, the Manhattan Distance Heuristic can be used with no data generation or training, quick state evaluations, and relatively good accuracy when solving a Rubik s Cube with IDA* Search. Appendix The code utilized in this project is contained in the attached zip file.

6 5 References 1. Fawcett, Tom. Learning from Imbalanced Classes. Silicon Valley Data Science. Published August 25, IDA-Star (IDA*) Algorithm In General. Algorithms Insight (Wordpress). Updated April 9, Kaur, Harpreet. Algorithms For Solving The Rubik s Cube. DiVA-Portal. Published May 27, 2015.