The Greedy Binary Search Tree is Non-trivially Competitive

2026-08-04

AI Summary

For decades, it has remained open whether the Greedy binary search tree algorithm achieves a nontrivial competitive ratio better than $O(\log n)$. This work introduces a novel scaling analysis framework that links fine-grained and coarse-grained access costs and integrates Wilber’s interleave lower bound to establish, for the first time, that Greedy attains a sublogarithmic competitive ratio of $2^{O(\sqrt{\log \log n})}$. This result breaks through a long-standing barrier in the theory of dynamically optimal binary search trees, affirming the nontrivial competitiveness of the Greedy algorithm and significantly advancing the theoretical understanding of dynamic optimality.

Abstract

We prove that the $\textsf{Greedy}$ binary search tree is $2^{O(\sqrt{\log\log n})}$-competitive. It is widely conjectured that $\textsf{Greedy}$ is $O(1)$-competitive, but before this work it was not known to be $f$-competitive, for any non-trivial $f(n)=o(\log n)$. Our analysis differs from prior analyses of binary search trees. It takes what might be called a "scaling" approach, where the cost at a refined scale is related to the cost at a coarser scale, and Wilber's interleave lower bound.

Problem

• Greedy
• binary search tree
• competitive
• interleave lower bound
• Wilber

Innovation

• Greedy BST
• competitive analysis
• scaling approach
• Wilber's interleave bound
• dynamic optimality