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As clouds move to the network edge to facilitate mobile applications, edge cloud providers are facing new challenges on resource allocation. As users may move and resource prices may vary arbitrarily, %and service delays are heterogeneous, resources in edge clouds must be allocated and adapted continuously in order to accommodate such dynamics. In this paper, we first formulate this problem with a comprehensive model that captures the key challenges, then introduce a gap-preserving transformation of the problem, and propose a novel online algorithm that optimally solves a series of subproblems with a carefully designed logarithmic objective, finally producing feasible solutions for edge cloud resource allocation over time. We further prove via rigorous analysis that our online algorithm can provide a parameterized competitive ratio, without requiring any a priori knowledge on either the resource price or …