Optimally Tracking Labels on an Evolving Tree

TL;DR

This paper introduces an algorithm for tracking labels on an evolving tree structure, balancing update complexity and accuracy, and demonstrates near-optimal performance under various evolution scenarios.

Contribution

It provides a novel algorithmic framework for maintaining label locations on dynamic trees with provable bounds and handles both randomized and adversarial evolutions.

Findings

Maintains labels within an average distance of O(1) from actual locations.

Provides nearly matching lower bounds for the problem.

Handles both randomized and adversarial evolution scenarios.

Abstract

Motivated by the problem of maintaining data structures for a large sets of points that are evolving over the course of time, we consider the problem of maintaining a set of labels assigned to the vertices of a tree, where the locations of these labels change over time. We study the problem in the evolving data framework, where labels change over time due to the action of an agent called the evolver. The algorithm can only track these changes by explicitly probing individual nodes of the tree. This framework captures the tradeoff between the complexity of maintaining an up-to-date view of the structure and the quality of results computed with the available view. Our results allow for both randomized and adversarial evolution of the data, subject to allowing different speedup factors between the algorithm and the evolver. We show that in the limit, our algorithm maintains labels to within an average distance of $O(1)$ of their actual locations. We also present nearly matching lower bounds.