Pinning Down the Strong Wilber 1 Bound for Binary Search Trees

The famous dynamic optimality conjecture of Sleator and Tarjan from 1985 conjectures the existence of an $O(1)$-competitive algorithm for binary search trees (BST's). Even the simpler problem of (offline) approximation of the optimal cost of a BST for a given input, that we denote by $OPT$, is still widely open, with the best current algorithm achieving an $O(\log\log n)$-approximation. A major challenge in designing such algorithms is to obtain a tight lower bound on $OPT$ that is algorithm friendly. Although several candidate lower bounds were suggested in the past, such as WB-1 and WB-2 bounds by Wilber, and Independent Rectangles bound by Demaine et al., the only currently known non-trivial approximation algorithm achieves an $O(\log\log n)$ approximation factor by comparing $OPT$ with a weak variant of WB-1, that uses a \emph{fixed} partitioning of the keys. This bound, however, is known to have a gap of $\Omega(\log\log n)$, and therefore it cannot yield better approximation algorithms. To overcome this obstacle, it is natural to consider a stronger variant of WB-1, that maximizes the bound over \emph{all} partitionings of the keys. An interesting question, mentioned by Iacono and by Kozma, is whether the $O(\log\log n)$-approximation can be improved by using this stronger bound. In this paper, we show that the gap between the stronger WB-1 bound and $OPT$ may be as large as $\Omega(\log\log n/\log\log\log n)$. This rules out the hope of obtaining better approximation algorithms via the only known algorithmic approach, combined with the stronger WB-1 bound. We also provide algorithmic results: for any parameter $D$, we present a simple $O(D)$-approximation algorithm with running time $\exp\left(O\left (n^{1/2^D}\log n\right )\right )$. This implies an $O(1)$-approximation algorithm in sub-exponential time.