Presenting a paper by Seth Pettie, from SODA 2010
Abstract: In this paper we improve, reprove, and simplify a variety of theorems concerning the performance of data structures based on path compression and search trees. We apply a technique very familiar to computational geometers but still foreign to many researchers in (non-geometric) algorithms and data structures, namely, to bound the complexity of an object via its forbidden substructures.
To analyze an algorithm or data structure in the forbidden substructure framework one proceeds in three discrete steps. First, one transcribes the behavior of the algorithm as some combinatorial object $M$; for example, $M$ may be a graph, sequence, permutation, matrix, set system, or tree. (The size of $M$ should ideally be linear in the running time.) Second, one shows that $M$ excludes some forbidden substructure $P$, and third, one bounds the size of any object avoiding this substructure. The power of this framework derives from the fact that $M$ lies in a more pristine environment and that upper bounds on the size of a $P$-free object $M$ may be taken "off the shelf".