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 ∈ [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 Θ(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, IRB 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).

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In 1992 Mansour proved that every size-s DNF formula is Fourier-concentrated on s O(log log s) coefficients. We improve this to s O(log log k) where k is the read number of the DNF. Since k is always at most s, our bound matches Mansour's for all DNFs and strengthens it for small-read ones. The previous best bound for read-k DNFs was s O(k 3/2 ) . For k up to Θ(log log s), we further improve our bound to the optimal poly(s); previously no such bound was known for any k = ωs(1).Our techniques involve new connections between the term structure of a DNF, viewed as a set system, and its Fourier spectrum.

We study the complexity of computing majority as a composition of local functions:where each g j : {0, 1} n → {0, 1} is an arbitrary function that queries only k n variables and h : {0, 1} m → {0, 1} is an arbitrary combining function. We prove an optimal lower bound of