In the incremental versions of Facility Location and k-Median, the demand points arrive one at a time and the algorithm must maintain a good solution by either adding each new demand to an existing cluster or placing it in a new singleton cluster. The algorithm can also merge some of the existing clusters at any point in time. We present the first incremental algorithm for Facility Location with uniform facility costs which achieves a constant performance ratio and the first incremental algorithm for k-Median which achieves a constant performance ratio using O(k) medians.
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