Settling the relationship between Wilber's bounds for dynamic optimality

Abstract

In FOCS 1986, Wilber proposed two combinatorial lower bounds on the operational cost of any binary search tree (BST) for a given access sequence $X \in [n]^m$. Both bounds play a central role in the ongoing pursuit of the dynamic optimality conjecture (Sleator and Tarjan, 1985), but their relationship remained unknown for more than three decades. We show that Wilber's Funnel bound dominates his Alternation bound for all $X$, and give a tight $\Theta(\lg\lg n)$ separation for some $X$, answering Wilber's conjecture and an open problem of Iacono, Demaine et. al. The main ingredient of the proof is a new "symmetric" characterization of Wilber's Funnel bound, which proves that it is invariant under rotations of $X$. We use this characterization to provide initial indication that the Funnel bound matches the Independent Rectangle bound (Demaine et al., 2009), by proving that when the Funnel bound is constant, $\mathsf{IRB}_{\diagup\hspace{-.6em}\square}$ is linear. To the best of our knowledge, our results provide the first progress on Wilber's conjecture that the Funnel bound is dynamically optimal (1986).