Performance of linear-space search algorithms

Abstract

Search algorithms that use space linear in the search depth are widely employed in practice to solve difficult problems optimally, such as planning and scheduling. In this paper, we study the average-case performance of linear-space search algorithms, including depth-first branch-and-bound (DFBnB), iterative-deepening (ID), and recursive best-first search (RBFS). To facilitate our analyses, we use a random tree T(b,d) that has mean branching factor b, depth d, and node costs that are the sum of the costs of the edges from the root to the nodes. We prove that the expected number of nodes expanded by DFBnB on a random tree is no more than bd times the expected number of nodes expanded by best-first search (BFS) on the same tree, which usually requires space that is exponential in depth d. We also show that DFBnB is asymptotically optimal when BFS runs in exponential time, and ID and RBFS are asymptotically optimal when the edge costs of T( b, d) are integers. If bp 0 is the expected number of children of a node whose costs are the same as that of their parent, then the expected number of nodes expanded by these three linear-space algorithms is exponential when bp o < 1, at most O( d 4) when bp 0 = 1, and at most quadratic when bp 0 > 1. In addition, we study the heuristic branching factor of T( b, d) and the effective branching factor of BFS, DFBnB, ID, and RBFS on T( b, d). Furthermore, we use our analytic results to explain a surprising anomaly in the performance of these algorithms, and to predict the existence of a complexity transition in the Asymmetric Traveling Salesman Problem.