Splaying Preorders and Postorders

Abstract

Let T be a binary search tree of n nodes with root r, left subtree L = left(r), and right subtree R = right(r). The preorder and postorder of T are defined as follows: the preorder and postorder of the empty tree is the empty sequence, andwhere ⊕ denotes sequence concatenation. 1 We prove the following results about the behavior of splaying [21] preorders and postorders: 1. Inserting the nodes of preorder(T ) into an empty tree via splaying costs O(n). (Theorem 2.) 2. Inserting the nodes of postorder(T ) into an empty tree via splaying costs O(n). (Theorem 3.) 3. If T has the same keys as T and T is weight-balanced [18] then splaying either preorder(T ) or postorder(T ) starting from T costs O(n). (Theorem 4.) * Research at Princeton University partially supported by an innovation research grant from Princeton and a gift from Microsoft.

A New Path from Splay to Dynamic Optimality

“…Let Λ (X, T ) denote the crossing cost of Splay's execution of (X, T ) and let ζ(X, T ) denote the bookkeeping cost of this execution. 36 Splay's crossing cost appears to be strongly correlated with the crossing lower bound (Section 5.1). We propose two ways to exploit this (Section 5.2).…”

“…It seems likely that Theorem 2.10 could instead be proven with a slight generalization of the sequential access theorem from [55]. We believe it would likely resemble a proof that splaying (3, 2, 1)-avoiding permutations takes linear time, a topic discussed in [36,Section 5].…”

“…Splaying preorder(T ) starting from T takes linear time. (See Appendix[8,36].) OPT can substitute any other tree for preorder(T ) with cost at most the size of the initial tree and then Splay 44.…”