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

Optimally Tracking Labels on an Evolving Tree