Pinning Down the Strong Wilber 1 Bound for Binary Search Trees
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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(loglog 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(loglog n) approximation factor by comparing OPT with a weak variant of WB-1, that uses a fixed partitioning of the keys. This bound, however, is known to have a gap of Ω(loglog 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 all partitionings of the keys. An interesting question, mentioned by Iacono and by Kozma, is whether the O(loglog 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 Ω(loglog n/logloglog 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 (O (n^1/2^Dlog n ) ). This implies an O(1)-approximation algorithm in sub-exponential time.
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