The Hidden Binary Search Tree:A Balanced Rotation-Free Search Tree in the AVL RAM Model

TL;DR

This paper introduces the Hidden Binary Search Tree (HBST), a balanced, rotation-free search tree that maintains logarithmic height without self-balancing, by using ideal reference values based on key intervals.

Contribution

It generalizes search trees by using reference values derived from key intervals, eliminating the need for rotations or self-balancing procedures.

Findings

HBST maintains height O(log n) without rotations.

The structure does not require input order or self-balancing.

Search operations are guaranteed logarithmic time complexity.

Abstract

In this paper we generalize the definition of "Search Trees" (ST) to enable reference values other than the key of prior inserted nodes. The idea builds on the assumption an $n$-node AVL (or Red-Black) requires to assure $O(\log_2n)$ worst-case search time, namely, a single comparison between two keys takes constant time. This means the size of each key in bits is fixed to $B=c\log_2 n$ ($c\geq1$) once $n$ is determined, otherwise the $O(1)$-time comparison assumption does not hold. Based on this we calculate \emph{ideal} reference values from the mid-point of the interval $0..2^B$. This idea follows `recursively' to assure each node along the search path is provided a reference value that guarantees an overall logarithmic time. Because the search tree property works only when keys are compared to reference values and these values are calculated only during searches, we term the data structure as the Hidden Binary Search Tree (HBST). We show elementary functions to maintain the HSBT height $O(B)=O(\log_2n)$. This result requires no special order on the input -- as does BST -- nor self-balancing procedures, as do AVL and Red-Black.