We introduce a new technique to bound the asymptotic performance of splay trees. The basic idea is to transcribe, in an indirect fashion, the rotations performed by the splay tree as a Davenport-Schinzel sequence S, none of whose subsequences are isomorphic to fixed forbidden subsequence. We direct this technique towards Tarjan’s deque conjecture and prove that n deque operations require O(nα ∗ (n)) time, where α ∗ (n) is the minimum number of applications of the inverse-Ackermann function mapping n to a constant. We are optimistic that this approach could be directed towards other open conjectures on splay trees such as the traversal and split conjectures.

Abstract Splay trees are self-organizing binary search trees that were introduced by Sleator andTarjan [12]. In this paper we present a randomized variant of these trees. The new algorithm for reorganizing the tree is both simple and easy to implement. We prove that our randomizedsplaying scheme has the same asymptotic performance as the original deterministic scheme but improves constants in the expected running time. This is interesting in practice becausethe search time in splay trees is typically higher than the search time in skip lists and AVLtrees. We present a detailed experimental study of our algorithm. On request sequencesgenerated by fixed probability distributions, we can achieve improvements of up to 25 % over deterministic splaying. On request sequences that exhibit high locality of reference, theimprovements are minor.

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 reused in different contexts. All of our proofs begin by transcribing the individual operations of a dynamic data structure