Paper by Erik D. Demaine

Reference:

Erik D. Demaine, Sándor P. Fekete, Günter Rote, Nils Schweer, Daria Schymura, and Mariano Zelke, “Integer Point Sets Minimizing Average Pairwise ℓ₁ Distance: What is the Optimal Shape of a Town?”, in Proceedings of the 21st Canadian Conference on Computational Geometry (CCCG 2009), Vancouver, British Columbia, Canada, August 17–19, 2009, pages 145–148.

Abstract:

An n-town, n ∈ ℕ, is a group of n buildings, each occupying a distinct position on a 2-dimensional integer grid. If we measure the distance between two buildings along the axis-parallel street grid, then an n-town has optimal shape if the sum of all pairwise Manhattan distances is minimized. This problem has been studied for cities, i.e., the limiting case of very large n. The optimal shape can be described by a differential equation. No closed-form solution is known. We show that optimal n-towns can be computed in time polynomial in n.

Length:

The paper is 4 pages.

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