Abstract
Although the maze (or gridworld) is one of the most widely used benchmark problems for real-time search algorithms, it is not sufficiently clear how the difference in the density of randomly positioned obstacles affects the structure of the state spaces and the performance of the algorithms. In particular, recent studies of the so-called phase transition phenomena that could cause dramatic change in their performance in a relatively small parameter range suggest that we should evaluate the performance in a parametric way with the parameter range wide enough to cover potential transition areas.
In this paper, we present two measures for characterizing the hardness of randomly generated mazes parameterized by obstacle ratio and relate them to the performance of real-time search algorithms. The first measure is the entropy calculated from the probability of existence of solutions. The second is a measure based on total initial heuristic error between the actual cost and its heuristic estimation. We show that the maze problems are the most complicated in both measures when the obstacle ratio is around 41\%. We then solve the parameterized maze problems with the well-known real-time search algorithms RTA*, LRTA*, and MARTA* to relate their performance to the proposed measures. Evaluating the number of steps required for a single problem solving by the three algorithms and the number of those required for the convergence of the learning process in LRTA*, we show that they all have a peak when the obstacle ratio is around 41\%. The results support the relevance of the proposed measures. We also discuss the performance of the algorithms in terms of other statistical measures to get a quantitative, deeper understanding of their behavior.
