Multi-Dimensional Heuristic Searching

A heuristic improvement technique referred to as multi-dimensional heuristics is presented. Instead of only applying the heuristic between two states X 1 and X 2 , when a distance esti­ mate of X 1 X 2 is needed, this technique uses a reference state R and applies the heuristic func­ tion to {X 1 ,R) and (X' 2 ,R) and compares the resulting values. If two states are close to each other, then they should also be approximately equidistant to a third reference state. It is pos­ sible to use many such reference states to im­ prove some heuristics. The reference states are used to map the search into an N-dimensional search space. The process of choosing reference states can be automated and is in fact a learn­ ing procedure. Test results using the 15-puzzle are presented in support of the effectiveness of multi-dimensional heuristics. This method has been shown to improve both a weak 15-puzzle heuristic, the tile reversal heuristic, as well as the stronger Manhattan distance heuristic. 1 Introduction Traditional heuristic search involves ordering state ex­ pansions relative to their estimated costs of participat­ ing in a solution. This cost is computed by a function f = g + h where g is the known cost (depth) arid h is an estimate of the remaining cost or distance to the goal [Hart et a/., 1968]. This paper proposes a general method of improving h. Instead of estimating h by eval­ uating a given state X with respect to the goal G. several new reference states are used to gain perspective [Nelson, 1988]. The relative position of X and G among the refer­ ence states will be used to estimate the distance between X and G. Assuming h is of a general nature and can be used to estimate the distance between any two states in the search space, then h can be used to estimate the distances from the reference states to both G and X. if the goal is used as a reference state r K , then ARk is just the traditional heuristic estimate h. These differ­ ence values AR i , where 1 < i < n and n is the number of reference states, will be used to give a better estimate of the actual distance from the X to G. The new esti­ mate will be referred to as H n and will …