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An n-town, for a natural number n, is a group of n buildings, each occupyinga distinct position on a 2-dimensional integer grid. If we measure the distancebetween two buildings along the axis-parallel street grid, then an n-town hasoptimal shape if the sum of all pairwise Manhattan distances is minimized. Thisproblem has been studied for cities, i.e., the limiting case of very large n.For cities, it is known that the optimal shape can be described by adifferential equation, for which no closed-form is known. We show that optimaln-towns can be computed in O(n^7.5) time. This is also practically useful, asit allows us to compute optimal solutions up to n=80.

Integer Point Sets Minimizing Average Pairwise L1-Distance: What is the Optimal Shape of a Town?