Greedy BST on Permutation Initial Tree

TL;DR

This paper introduces permutation initial trees for the Greedy BST algorithm, showing that its cost on any search sequence can match that on a flat initial tree, and proves the preorder traversal conjecture for these trees.

Contribution

It defines permutation initial trees and proves the Greedy algorithm's cost equivalence to flat trees, extending the preorder traversal conjecture to this class.

Findings

Greedy cost on permutation initial trees matches flat initial trees.

Preorder traversal conjecture holds for permutation initial trees.

Linear cost bounds are maintained for preorder sequences.

Abstract

The Greedy binary search tree (BST) algorithm, like the Splay tree, is a prominent candidate for the \emph{dynamic optimality conjecture}. While Greedy satisfies many desirable properties of BST, its cost and analysis to execute a search sequence $S$ is known to depend heavily on the choice of the \emph{initial tree} configuration. Most prior analyses assume a flat (empty) initial tree, under which several tight bounds are established. In this work, we introduce the notion of a \emph{permutation initial tree}, a specific class of non-flat initial tree and prove that for any permutation search sequence $S=(s_1,s_2,\dots, s_n)$, there exists a permutation initial tree $I_p$ such that the cost of Greedy on $I_p$ is same as its cost on the flat initial tree. As an application of our result, we show that the \emph{preorder traversal conjecture} holds for Greedy when the initial tree is a permutation initial tree. While it was previously known that Greedy achieves an $O(n)$ cost on preorder sequences for flat initial tree (Chalermsook et al., FOCS 2015), our result demonstrates that the same linear bound holds when the initial tree is a permutation initial tree. This result also matches the $O(n)$ bound for Splay tree on preorder sequence when the initial tree aligns with the traversal order (Chaudhuri and H\"oft, SIGACT 1993).