In this paper we introduce Stratified Sampling with Duplicate Detection (SSDD), an algorithm for estimating the number of state expansions performed by heuristic search algorithms seeking solutions in state spaces represented by undirected graphs. SSDD is general and can be applied to estimate other state-space properties. We test SSDD on two tasks: (i) prediction of the number of A* expansions in a given f-layer when using a consistent heuristic function, and (ii) prediction of the state-space radius. SSDD has the asymptotic guarantee of producing perfect estimates in both tasks. Our empirical results show that in task (i) SSDD produces good estimates in all four domains tested, being in most cases orders of magnitude more accurate than a competing scheme, and in task (ii) SSDD quickly produces accurate estimates of the radii of the 4x4 Sliding-Tile Puzzle and the 3x3x3 Rubik's Cube.

In this paper we advance the line of research launched by Knuth which was later improved by Chen for predicting the size of the search tree expanded by heuristic search algorithms such as IDA*. Chen's Stratified Sampling (SS) uses a partition of the nodes in the search tree called type system to guide its sampling. Recent work has shown that SS using type systems based on integer-valued heuristic functions can be quite effective. However, type systems based on real-valued heuristic functions are often too large to be practical. We use the k-means clustering algorithm for creating effective type systems for domains with real-valued heuristics. Orthogonal to the type systems, another contribution of this paper is the introduction of an algorithm called Active SS. SS allocates the same number of samples for each type. Active SS is the application of the idea of active sampling to search trees. Active SS allocates more samples to the types with higher uncertainty. Our empirical results show that (i) SS using clustering-based type systems tends to produce better predictions than competing schemes that do not use a type system, and that (ii) Active SS can produce better predictions than the regular version of SS.