Expected Performance of M-solution Backtracking Nl\si\ National Aeronautics and Space Administration

This paper derives upper bounds on the expected number of search tree nodes visited during an m-solution backtracking search, a search which terminates after some pre-selected number m problem solutions are found. The search behavior is assumed to have a general probabilistic structure. Our results are stated in terms of node expansion and contraction. A visited search tree node is said to be expanding if the mean number of its children visited by the search exceeds I, and is contracting otherwise. We show that if every node expands, or if every node contracts, then the number of search tree nodes visited by a search has an upper bound which is linear in the depth of the tree, in the mean number of children a node has, and in the number of solutions sought. We also derive bounds linear in the depth of the tree in some situations where an upper portion of the tree contracts (expands), while the lower portion expands (contracts). While previous analysises of I-solution backtracking have concluded that the expected performance is always linear in the tree depth, our model allows super-linear expected performance. By generalizing previous work in the expected behavior of backtracking, we are better able to identify classes of trees which can be searched in linear expected time.