Top Trees: A Simpler Interface for Fully-Dynamic Trees

Title: Top Trees: A Simpler Interface for Fully-Dynamic Trees

Abstract: Top trees are a new, simpler interface for data structures that maintain information in a fully-dynamic forest. This interface is designed to be versatile and easy to use, making it suitable for a wide range of applications. For example, top trees can be used to maintain the diameter, center, and median of trees in the forest, and to support queries such as nearest common ancestor and level ancestor. The forest can be updated with insertions, deletions, and changes to vertex and edge weights, all supported in logarithmic time. Top trees can also be used to compute distances to the nearest marked vertex, which has applications in static optimization problems. This paper demonstrates how to implement top trees using Frederickson's topology trees and Sleator and Tarjan's dy-namic trees, providing quadratic improvements over previous bounds in certain cases.

Main Research Question: How can we design a simpler interface for data structures that maintains information in a fully-dynamic forest?

Methodology: The researchers developed top trees, a new interface for data structures in fully-dynamic forests. This interface is designed to be easy to use and versatile, allowing it to be applied to a wide range of applications. Top trees can maintain information such as the diameter, center, and median of trees, and support queries like nearest common ancestor and level ancestor. The forest can be updated with insertions, deletions, and changes to vertex and edge weights, all supported in logarithmic time.

Results: The researchers demonstrated how to implement top trees using Frederickson's topology trees and Sleator and Tarjan's dy-namic trees. They showed that top trees provide quadratic improvements over previous bounds in certain cases, such as maintaining the centers and medians of trees in a dynamic forest. They also provided examples of how top trees can be used to solve various problems, including finding the maximum weight of a given path and maintaining the diameters of trees in a dynamic forest.

Implications: The introduction of top trees provides a simpler interface for data structures in fully-dynamic forests, making it easier for users to access the full power of advanced techniques. This can lead to improved performance and increased flexibility in the applications that can be developed using these data structures. Additionally, the ability to maintain non-local properties in a tree, such as centers and medians, opens up new possibilities for solving problems in dynamic forests.

Link to Article: https://arxiv.org/abs/0310065v2 Authors: arXiv ID: 0310065v2