The geometry of binary search trees

Authors:
Erik D. Demaine;Dion Harmon;John Iacono;Daniel Kane;Mihai Pătraşcu
Affiliations:
MIT Computer Science and Artificial Intelligence Laboratory, Cambridge, MA;New England Complex Systems Institute, Cambridge, MA;Polytechnic Institute of New York University, Brooklyn, NY;Harvard University, Cambridge, MA;MIT Computer Science and Artificial Intelligence Laboratory, Cambridge, MA
Venue:
SODA '09 Proceedings of the twentieth Annual ACM-SIAM Symposium on Discrete Algorithms
Year:
2009

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Abstract

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.