Settling the relationship between Wilber's bounds for dynamic optimality

TL;DR

This paper proves that Wilber's Funnel bound always exceeds his Alternation bound for binary search trees and establishes a tight separation in certain cases, advancing understanding of the bounds related to dynamic optimality.

Contribution

It demonstrates that Wilber's Funnel bound dominates the Alternation bound and provides a symmetric characterization of the Funnel bound, resolving a long-standing conjecture.

Findings

Funnel bound dominates Alternation bound for all access sequences.

A tight ll lg lg nl separation is established for some sequences.

Funnel bound is invariant under rotations of the access sequence.

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).