# Competitive Online Search Trees on Trees

## TL;DR

This paper introduces an online search tree data structure for tree-structured spaces that achieves an optimal competitive ratio similar to binary search trees, using novel decomposition techniques.

## Contribution

It generalizes binary search trees to tree-structured spaces and presents an $O(\log \log n)$-competitive online search tree with new notions like Steiner-closed search trees.

## Key findings

- Achieves $O(\log \log n)$-competitiveness in the new model.
- Introduces Steiner-closed search trees for better decomposition.
- Uses a two-level decomposition approach for analysis.

## Abstract

We consider the design of adaptive data structures for searching elements of a tree-structured space. We use a natural generalization of the rotation-based online binary search tree model in which the underlying search space is the set of vertices of a tree. This model is based on a simple structure for decomposing graphs, previously known under several names including elimination trees, vertex rankings, and tubings. The model is equivalent to the classical binary search tree model exactly when the underlying tree is a path. We describe an online $O(\log \log n)$-competitive search tree data structure in this model, matching the best known competitive ratio of binary search trees. Our method is inspired by Tango trees, an online binary search tree algorithm, but critically needs several new notions including one which we call Steiner-closed search trees, which may be of independent interest. Moreover our technique is based on a novel use of two levels of decomposition, first from search space to a set of Steiner-closed trees, and secondly from these trees into paths.