Splay trees on trees

Search trees on trees (STTs) are a far-reaching generalization of binary search trees (BSTs), allowing the efficient exploration of tree-structured domains. (BSTs are the special case in which the underlying domain is a path.) Trees on trees have been extensively studied under various guises in computer science and discrete mathematics. Recently Bose, Cardinal, Iacono, Koumoutsos, and Langerman (SODA 2020) considered adaptive STTs and observed that, apart from notable exceptions, the machinery developed for BSTs in the past decades does not readily transfer to STTs. In particular, they asked whether the optimal STT can be efficiently computed or approximated (by analogy to Knuth's algorithm for optimal BSTs), and whether natural self-adjusting BSTs such as Splay trees (Sleator, Tarjan, 1983) can be extended to this more general setting. We answer both questions affirmatively. First, we show that a $(1 + \frac{1}{t})$-approximation of the optimal STT for a given search distribution can be computed in time $O(n^{2t + 1})$ for all integers $t \geq 1$. Second, we identify a broad family of STTs with linear rotation-distance, allowing the generalization of Splay trees to the STT setting. We show that our generalized Splay satisfies a static optimality theorem, asymptotically matching the cost of the optimal STT in an online fashion, i.e. without a priori knowledge of the search distribution. Our results suggest an extension of the dynamic optimality conjecture for Splay trees to the broader setting of trees on trees.