Self-Adjusting Binary Search Trees: What Makes Them Tick?

TL;DR

This paper identifies key combinatorial properties that ensure self-adjusting binary search trees satisfy the access lemma, unifying and extending previous results, and providing new insights into the efficiency of various BST algorithms.

Contribution

It introduces two simple combinatorial properties that imply the access lemma for a broad class of self-adjusting BST algorithms, including splay trees and greedy algorithms.

Findings

Proves the access lemma for all minimally self-adjusting BSTs satisfying certain properties.

Addresses an open question by showing depth-halving algorithms satisfy the access lemma.

Provides a short proof for the O(log n log log n) amortized access cost of the path-balance heuristic.

Abstract

Splay trees (Sleator and Tarjan) satisfy the so-called access lemma. Many of the nice properties of splay trees follow from it. What makes self-adjusting binary search trees (BSTs) satisfy the access lemma? After each access, self-adjusting BSTs replace the search path by a tree on the same set of nodes (the after-tree). We identify two simple combinatorial properties of the search path and the after-tree that imply the access lemma. Our main result (i) implies the access lemma for all minimally self-adjusting BST algorithms for which it was known to hold: splay trees and their generalization to the class of local algorithms (Subramanian, Georgakopoulos and Mc-Clurkin), as well as Greedy BST, introduced by Demaine et al. and shown to satisfy the access lemma by Fox, (ii) implies that BST algorithms based on "strict" depth-halving satisfy the access lemma, addressing an open question that was raised several times since 1985, and (iii) yields an extremely short proof for the O(log n log log n) amortized access cost for the path-balance heuristic (proposed by Sleator), matching the best known bound (Balasubramanian and Raman) to a lower-order factor. One of our combinatorial properties is locality. We show that any BST-algorithm that satisfies the access lemma via the sum-of-log (SOL) potential is necessarily local. The other property states that the sum of the number of leaves of the after-tree plus the number of side alternations in the search path must be at least a constant fraction of the length of the search path. We show that a weak form of this property is necessary for sequential access to be linear.