Large Barrier Trees for Studying Search

Barrier trees are a method for representing the landscape structure of high-dimensional discrete spaces such as those that occur in the cost function of combinatorial optimization problems. The leaves of the tree represent local optima and a vertex where subtrees join represents the lowest cost saddle-point between the local optima in the subtrees. This paper introduces an extension to existing Barrier tree methods that make them more useful for studying heuristic optimization algorithms. It is shown that every configuration in the search space can be mapped onto a vertex in the Barrier tree. This provides additional information about the landscape, such as the number of configurations in a local optimum. It also allows the computation of additional statistics such as the correlation between configurations in different parts of the Barrier tree. Furthermore, the mappings allow the dynamic behavior of a heuristic search algorithms to be visualized. This extension is illustrated using an instance of the MAX-3-SAT problem.