 Iterative lengthening search

Iterative lengthening search repeatedly runs cost-bounded depth-first search, starting with threshold zero and replacing each unsuccessful threshold by the smallest rejected path cost. It uses a depth-first search stack rather than the potentially large search frontier of uniform-cost search.

Suppose an optimal plan has cost $C^{\ast }$ and each pass terminates. At every unsuccessful threshold $L<C^{\ast }$, some first rejected prefix of an optimal plan has path cost at most $C^{\ast }$. Therefore the next threshold is no larger than $C^{\ast }$. Any first returned goal state has cost at most $C^{\ast }$ and hence exactly $C^{\ast }$. Finite branching factor and a positive lower bound on step costs ensure that only finitely many search nodes precede that cost, giving completeness as well as optimality. With arbitrary real step costs, the threshold increments can be much smaller than the smallest step cost: the difference between two path costs need not itself be a step cost.