Novel O(H(N)+N/H(N)) Algorithmic Techniques for Several Types of Queries and Updates on Rooted Trees and Lists

TL;DR

This paper introduces new algorithmic techniques with a time complexity of O(H(N)+N/H(N)) for efficient queries and updates on rooted trees, binary search trees, and lists, including a novel compressed super-node tree and H-list data structure.

Contribution

It presents a new compressed super-node tree representation, a global rebuilding approach for balanced binary search trees, and the H-list data structure, all achieving improved efficiency.

Findings

Achieves O(sqrt(N)) time complexity with H(N)=sqrt(N).

Introduces a compressed super-node tree for rooted trees.

Develops the H-list supporting concatenation, split, and queries.

Abstract

In this paper we present novel algorithmic techniques with a O(H(N)+N/H(N)) time complexity for performing several types of queries and updates on general rooted trees, binary search trees and lists of size N. For rooted trees we introduce a new compressed super-node tree representation which can be used for efficiently addressing a wide range of applications. For binary search trees we discuss the idea of globally rebuilding the entire tree in a fully balanced manner whenever the height of the tree exceeds the value of a conveniently chosen function of the number of tree nodes. In the end of the paper we introduce the H-list data structure which supports concatenation, split and several types of queries. Note that when choosing H(N)=sqrt(N) we obtain O(H(N)+N/H(N))=O(sqrt(N)).