Defining Equitable Geographic Districts in Road Networks via Stable Matching
Abstract
We introduce a novel method for defining geographic districts in road networks using stable matching. In this approach, each geographic district is defined in terms of a center, which identifies a location of interest, such as a post office or polling place, and all other network vertices must be labeled with the center to which they are associated. We focus on defining geographic districts that are equitable, in that every district has the same number of vertices and the assignment is stable in terms of geographic distance. That is, there is no unassigned vertexcenter pair such that both would prefer each other over their current assignments. We solve this problem using a version of the classic stable matching problem, called symmetric stable matching, in which the preferences of the elements in both sets obey a certain symmetry. In our case, we study a graphbased version of stable matching in which nodes are stably matched to a subset of nodes denoted as centers, prioritized by their shortestpath distances, so that each center is apportioned a certain number of nodes. We show that, for a planar graph or road network with $n$ nodes and $k$ centers, the problem can be solved in $O(n\sqrt{n}\log n)$ time, which improves upon the $O(nk)$ runtime of using the classic GaleShapley stable matching algorithm when $k$ is large. Finally, we provide experimental results on road networks for these algorithms and a heuristic algorithm that performs better than the GaleShapley algorithm for any range of values of $k$.
 Publication:

arXiv eprints
 Pub Date:
 June 2017
 arXiv:
 arXiv:1706.09593
 Bibcode:
 2017arXiv170609593E
 Keywords:

 Computer Science  Data Structures and Algorithms
 EPrint:
 9 pages, 4 figures, to appear in 25th ACM SIGSPATIAL International Conference on Advances in Geographic Information Systems (ACM SIGSPATIAL 2017) November 710, 2017, Redondo Beach, California, USA