Greedy BST on Permutation Initial Tree

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).