We consider the following online allocation problem: Given a unit square S, and a sequence of numbers ni∈[0, 1] with ∑j=0inj≤1; at each step i, select a region Ci of previously unassigned area ni in S. The objective is to make these regions compact in a distance-aware sense: minimize the maximum (normalized) average Manhattan distance between points from the same set Ci. Related location problems have received a considerable amount of attention; in particular, the problem of determining the "optimal shape of a city", i.e., allocating a single ni has been studied. We present an online strategy, based on an analysis of space-filling curves; for continuous shapes, we prove a factor of 1.8092, and 1.7848 for discrete point sets.
Fekete, S. P., Reinhardt, J. M., & Schweer, N. (2014). A competitive strategy for distance-aware online shape allocation. Theoretical Computer Science, 555(C), 43–54. https://doi.org/10.1016/j.tcs.2014.02.050