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This paper studies β-approximate distance labeling schemes, which are composed of a marker algorithm for labeling the vertices of a graph with short labels, coupled with a decoder algorithm allowing one to compute a β-approximation of the distance between any two vertices directly from their labels (without using any additional information). As most applications for informative labeling schemes in general, and distance labeling schemes in particular, concern large and dynamically changing networks, it is of interest to focus on distributed dynamic labeling schemes. The paper considers the problem on dynamic weighted trees and cycles where the vertices of the tree (or the cycle) are fixed but the (positive integral) weights of the edges may change. The two models considered are the fully dynamic model, where from time to time some edge changes its weight by a fixed quanta, and the increasing dynamic model in which edge weights can only grow. The paper presents distributed β-approximate distance labeling schemes for the two models, for β 1, and establishes upper and lower bounds on the required label size and the communication complexity involved in updating the labels following a weight change.