How Long Will It Take?

Abstract

We present a method for approximating the ex-pected number of steps required by a heuristic search algorithm to reach a goal from any ini-tial state in a problem space. The method is based on a mapping from the original state space to an abstract space in which states are charac-terized only by a syntactic “distance ” from the nearest goal. Modeling the search algorithm as a Markov process in the abstract space yields a simple system of equations for the solution time for each state. We derive some insight into the behavior of search algorithms by examining some closed form solutions for these equations; we also show that many problem spaces have a clearly de-lineated “easy zone”, inside which problems are trivial and outside which problems are impossible. The theory is borne out by experiments with both Markov and non-Markov search algorithms. Our results also bear on recent experimental data sug-gesting that heuristic repair algorithms can solve large constraint satisfaction problems easily, given a preprocessor that generates a sufficiently good initial state.