Near Optimal Solving of the (N2-1)-puzzle Using Heuristics Based on Artificial Neural Networks

We address the design of heuristics for near-optimal solving of the (N2–1)-puzzle using the A* search algorithm in this paper. The A* search algorithm explores configurations of the puzzle in the order determined by a heuristic that tries to estimate the minimum number of moves needed to reach the goal from the given configuration. To guarantee finding an optimal solution, the A* algorithm requires heuristics that estimate the number of moves from below. Common heuristics for the (N2–1)-puzzle often underestimate the true number of moves greatly in order to meet the admissibility requirement. The worse the estimation is the more configurations the search algorithm needs to explore. We therefore relax from the admissibility requirement and design a novel heuristic that tries estimating the minimum number of moves remaining as precisely as possible while overestimation of the true distance is permitted. Our heuristic called ANN-distance is based on a deep artificial neural network (ANN). We experimentally show that with a well trained ANN-distance heuristic, whose inputs are just the positions of the tiles, we are able to achieve better accuracy of estimation than with conventional heuristics such as those derived from the Manhattan distance or pattern database heuristics. Though we cannot guarantee admissibility of ANN-distance due to possible overestimation of the true number of moves, an experimental evaluation on random 15-puzzles shows that in most cases the ANN-distance calculates the true minimum distance from the goal or an estimation that is very close to the true distance. Consequently, A* search with the ANN-distance heuristic usually finds an optimal solution or a solution that is very close to the optimum. Moreover, the underlying neural network in ANN-distance consumes much less memory than a comparable pattern database. We also show that a deep artificial neural network can be more powerful than a shallow artificial neural network, and also trained our heuristic to prefer underestimating the optimal solution cost, which pushed the solutions towards better optimality.