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 ∈ [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).

Pinning Down the Strong Wilber 1 Bound for Binary Search Trees

“…We show that, for the sequence X given by Theorem 1.1, WB (2) (X) ≥ Ω(n log log n) holds, and therefore WB (2) (X)/WB(X) ≥ Ω(log log n/ log log log n) for that sequence, implying that the gap between WB(X) and WB (2) (X) may be as large as Ω(log log n/ log log log n). We note that we only realized that our results provide this stronger separation between the two Wilber bounds after hearing the statements of the results from the independent work of Lecomte and Weinstein [LW19] mentioned above.…”

“…Independent Work. Independently from our work, Lecomte and Weinstein [LW19] showed that Wilber's Funnel bound (that we call WB-2 bound and discuss below) dominates the WB-1 bound, and moreover, they show an access sequence X for which the two bounds have a gap of Ω(log log n). In particular, their result implies that the gap between WB(X) and OPT(X) is Ω(log log n) for that access sequence.…”

“…In particular, their result implies that the gap between WB(X) and OPT(X) is Ω(log log n) for that access sequence. We note that the access sequence X that is used in our negative results also provides a gap of Ω(log log n/ log log log n) between the WB-2 and the WB-1 bounds, although we only realized this after hearing the statement of the results of [LW19]. Additionally, Lecomte and Weinstein show that the WB-2 bound is invariant under rotations, and use this to show that, when the WB-2 bound is constant, then the Independent Rectangle bound of [DHI + 09] is linear, thus making progress on establishing the relationship between the two bounds.…”