In this paper, we present a framework used to construct and analyze algorithms for online optimization problems with deadlines or with delay over a metric space. Using this framework, we present algorithms for several different problems. We present an $O(D^{2})$-competitive deterministic algorithm for online multilevel aggregation with delay on a tree of depth $D$, an exponential improvement over the $O(D^{4}2^{D})$-competitive algorithm of Bienkowski et al. (ESA 16), where the only previously-known improvement was for the special case of deadlines by Buchbinder et al. (SODA 17). We also present an $O(log^{2}n)$-competitive randomized algorithm for online service with delay over any general metric space of $n$ points, improving upon the $O(log^{4}n)$-competitive algorithm by Azar et al. (STOC 17). In addition, we present the problem of online facility location with deadlines. In this problem, requests arrive over time in a metric space, and need to be served until their deadlines by facilities that are opened momentarily for some cost. We also consider the problem of facility location with delay, in which the deadlines are replaced with arbitrary delay functions. For those problems, we present $O(log^{2}n)$-competitive algorithms, with $n$ the number of points in the metric space. The algorithmic framework we present includes techniques for the design of algorithms as well as techniques for their analysis.
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