We prove direct-sum theorems for Wilber's two lower bounds [Wilber, FOCS'86] on the cost of access sequences in the binary search tree (BST) model. These bounds are central to the question of dynamic optimality [Sleator and Tarjan, JACM'85]: the Alternation bound is the only bound to have yielded online BST algorithms beating $\log n$ competitive ratio, while the Funnel bound has repeatedly been conjectured to exactly characterize the cost of executing an access sequence using the optimal tree [Wilber, FOCS'86, Kozma'16], and has been explicitly linked to splay trees [Levy and Tarjan, SODA'19]. Previously, the direct-sum theorem for the Alternation bound was known only when approximation was allowed [Chalermsook, Chuzhoy and Saranurak, APPROX'20, ToC'24]. We use these direct-sum theorems to amplify the sequences from [Lecomte and Weinstein, ESA'20] that separate between Wilber's Alternation and Funnel bounds, increasing the Alternation and Funnel bounds while optimally maintaining the separation. As a corollary, we show that Tango trees [Demaine et al., FOCS'04] are optimal among any BST algorithms that charge their costs to the Alternation bound. This is true for any value of the Alternation bound, even values for which Tango trees achieve a competitive ratio of $o(\log \log n)$ instead of the default $O(\log \log n)$. Previously, the optimality of Tango trees was shown only for a limited range of Alternation bound [Lecomte and Weinstein, ESA'20].

翻译：我们证明了Wilber关于二叉搜索树(BST)模型中访问序列成本的两个下界[Wilber, FOCS'86]的直接和定理。这些下界对动态最优性问题[Sleator and Tarjan, JACM'85]至关重要：交替界是唯一产生在线BST算法击败$\log n$竞争比的下界，而漏斗界则被反复推测为能精确刻画使用最优树执行访问序列的成本[Wilber, FOCS'86, Kozma'16]，并已明确与伸展树相关联[Levy and Tarjan, SODA'19]。此前，交替界的直接和定理仅在允许近似的情况下已知[Chalermsook, Chuzhoy and Saranurak, APPROX'20, ToC'24]。我们利用这些直接和定理来放大[Lecomte and Weinstein, ESA'20]中区分Wilber交替界与漏斗界的序列，在最优保持分离的同时提升交替界与漏斗界的值。作为推论，我们证明探戈树[Demaine et al., FOCS'04]在所有将其成本归因于交替界的BST算法中是最优的。这对交替界的任意取值均成立，即使探戈树达到$o(\log \log n)$而非默认$O(\log \log n)$竞争比的取值亦然。此前，探戈树的最优性仅在交替界的有限范围内得到证明[Lecomte and Weinstein, ESA'20]。