  The evolving data framework was first proposed by Anagnostopoulos et al.,
where an evolver makes small changes to a structure behind the scenes. Instead
of taking a single input and producing a single output, an algorithm
judiciously probes the current state of the structure and attempts to
continuously maintain a sketch of the structure that is as close as possible to
its actual state. There have been a number of problems that have been studied
in the evolving framework including our own work on labeled trees. We were
motivated by the problem of maintaining a labeling in the plane, where updating
the labels require physically moving them. Applications involve tracking
evolving disease hot-spots via mobile testing units , and tracking unmanned
aerial vehicles. To be specific, we consider the problem of tracking labeled
nodes in the plane, where an evolver continuously swaps labels of any two
nearby nodes in the background unknown to us. We are tasked with maintaining a
hypothesis, an approximate sketch of the locations of these labels, which we
can only update by physically moving them over a sparse graph. We assume the
existence of an Oracle, which when suitably probed, guides us in fixing our
hypothesis.
