A competitive strategy for distance-aware online shape allocation

We consider the following online allocation problem: Given a unit square S  , and a sequence of numbers ni∈[0,1]ni∈[0,1] with ∑j=0inj⩽1; at each step i  , select a region CiCi of previously unassigned area nini 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 CiCi. 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  nini 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.