## Integer point sets minimizing average pairwise L[subscript 1] distance: What is the optimal shape of a town?

##### Author(s)

Demaine, Erik D.; Fekete, Sandor P.; Rote, Günter; Schweer, Nils; Schymura, Daria; Zelke, Mariano; ... Show more Show less
DownloadDemaine_Integer point.pdf (294.3Kb)

OPEN_ACCESS_POLICY

# Open Access Policy

Creative Commons Attribution-Noncommercial-Share Alike

##### Alternative title

Integer point sets minimizing average pairwise L1 distance: What is the optimal shape of a town?

##### Terms of use

##### Metadata

Show full item record##### Abstract

An n-town, n[is an element of]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. For cities, it is known that the optimal shape can be described by a differential equation, for which no closed-form solution is known. We show that optimal n-towns can be computed in O(n[superscript 7.5]) time. This is also practically useful, as it allows us to compute optimal solutions up to n=80.

##### Description

Special issue of selected papers from the 21st Annual Canadian Conference on Computational Geometry

##### Date issued

2010-09##### Department

Massachusetts Institute of Technology. Computer Science and Artificial Intelligence Laboratory##### Journal

Computational Geometry: Theory and Applications

##### Publisher

Elsevier B.V.

##### Citation

Demaine, Erik D. et al. “Integer Point Sets Minimizing Average Pairwise L1 Distance: What Is the Optimal Shape of a Town?” Computational Geometry 44.2 (2011) : 82-94.

Version: Author's final manuscript

##### ISSN

0925-7721