Geometric Similarity Metrics for Case-Based Reasoning

Case-based reasoning is a problem solving method that uses stored solutions to problems to aid in solving similar new problems. One of the difficulties of case-based reasoning is identifying cases that are relevant to a problem. If the problem is defined on a geometric domain — for instance, planning a route using a city map — it becomes possible to take advantage of the geometry to simplify the task of finding appropriate cases. We propose a methodology for determining a set of cases which collectively forms a good basis for a new plan, and may include partial cases, unlike most existing similarity metrics. This methodology is applicable in continuous-valued domains, where one cannot rely on the traditional method of simple role-substitution and matching. The problem of identifying relevant cases is transformed into a geometric problem with an exact solution. We construct two similar algorithms for solving the geometric problem. The first algorithm returns a correct solution, but is prohibitively slow. The second algorithm, based on the use of a Delaunay triangulation as a heuristic to model the case library, is fast, and returns an approximate solution that is within a constant factor of optimum. Both algorithms return a good set of cases for geometric planning. We have implemented the second algorithm within a real-world robotics path planning domain.