The Geometry of Binary Search Trees

John Iacono, Department of Computer Science and Engineering, Polytechnic Institute of NYU

We present a novel connection between binary search trees (BSTs) and points in the plane satisfying a simple property. Using this correspondence, we achieve the following results:

1. A surprisingly clean restatement in geometric terms of many results and conjectures relating to BSTs and dynamic optimality.

2. A new lower bound for searching in the BST model, which subsumes the previous two known bounds of Wilber [FOCS’86].

3. The first proposal for dynamic optimality not based on splay trees. A natural greedy but offline algorithm was presented by Lucas [1988], and independently by Munro [2000], and was conjectured to be an (additive) approximation of the best binary search tree. We show that there exists an equal-cost online algorithm, transforming the conjecture of Lucas and Munro into the conjecture that the greedy algorithm is dynamically optimal.

Joint work with Erik Demaine (MIT) Dion Harmon (New England Complex Systems Institute) Daniel Kane (Harvard) Mihai Patrascu (IBM)