Splaying Preorders and Postorders

Abstract

Let T be a binary search tree of n nodes with root r, left subtree $L=\text {left}(r)$, and right subtree $R=\text {right}(r)$. The preorder and postorder of T are defined as follows: the preorder and postorder of the empty tree is the empty sequence, and

$$\begin{aligned} \text {preorder}(T)&= (r)\oplus \text {preorder}(L)\oplus \text {preorder}(R)\\ \text {postorder}(T)&= \text {postorder}(L)\oplus \text {postorder}(R)\oplus (r), \end{aligned}$$

where $\oplus$ denotes sequence concatenation. (We will refer to any such sequence as a preorder or a postorder). 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.)

For 1 and 2, we use the fact that preorders and postorders are pattern-avoiding: i.e. they contain no subsequences that are order-isomorphic to (2, 3, 1) and (3, 1, 2), respectively. Pattern-avoidance implies certain constraints on the manner in which items are inserted. We exploit this structure with a simple potential function that counts inserted nodes lying on access paths to uninserted nodes. Our methods can likely be extended to permutations that avoid more general patterns. The proof of 3 uses the fact that preorders and postorders of balanced search trees do not contain many large “jumps” in symmetric order, and exploits this fact using the dynamic finger theorem [5, 6]. Items 2 and 3 are both novel. Item 1 was originally proved by Chaudhuri and Höft [4]; our proof simplifies theirs. These results provide further evidence in favor of the elusive dynamic optimality conjecture [21].

Research at Princeton University partially supported by an innovation research grant from Princeton and a gift from Microsoft.