The p-median problem is a common model for optimal facility location. The
task is to place p facilities (e.g., warehouses or schools) in a
heterogeneously populated space such that the average distance from a person's
home to the nearest facility is minimized. Here we study the special case where
the population lives along a line (e.g., a road or a river). If facilities are
optimally placed, the length of the line segment served by a facility is
inversely proportional to the square root of the population density. This
scaling law is derived analytically and confirmed for concrete numerical
examples of three US Interstate highways and the Mississippi River. If facility
locations are permitted to deviate from the optimum, the number of possible
solutions increases dramatically. Using Monte Carlo simulations, we compute how
scaling is affected by an increase in the average distance to the nearest
facility. We find that the scaling exponents change and are most sensitive near
the optimum facility distribution.Comment: 7 pages, 6 figures, Physical Review E, in pres