This paper provides a simple algorithmic template for obtaining fast distributed algorithms for a highly-dynamic setting, in which \emph{arbitrarily many} edge changes may occur in each round. Our algorithm significantly improves upon prior work in its combination of (1) having an amortized time complexity, (2) not posing any restrictions on the dynamic behavior of the environment, (3) being deterministic, (4) having strong guarantees for intermediate solutions, and (5) being applicable for a wide family of tasks which we combinatorially define. The tasks for which we deduce such an algorithm are maximal matching, -coloring, maximal independent set (which, perhaps surprisingly, seems to behave very differently from the other problems), and the seemingly unrelated problem of a 2-approximation for minimum weight vertex cover. For some of these tasks, node insertions can also be among the allowed topology changes, and for some of them also abrupt node deletions. At the core of our work is a combinatorial definition of a subclass of the celebrated family of locally-checkable labelings (LCLs) defined by Naor and Stockmeyer [SIAM J. Comp.~'95]. We call the subclass that we define \emph{locally-fixable labelings} (LFLs). Very roughly speaking, as their name suggests, these are labelings that allow a node to fix its own label and the labels of its neighbors after a topology change, based solely on their old labels.
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