Zone design of specific sizes using adaptive additively weighted voronoi diagrams

Territory or zone design processes entail partitioning a geographic space, organized as a set of areal units, into different regions or zones according to a specific set of criteria that are dependent on the application context. In most cases, the aim is to create zones of approximately equal sizes (zones with equal numbers of inhabitants, same average sales, etc.). However, some of the new applications that have emerged, particularly in the context of sustainable development policies, are aimed at defining zones of a predetermined, though not necessarily similar, size. In addition, the zones should be built around a given set of seeds. This type of partitioning has not been sufficiently researched; therefore, there are no known approaches for automated zone delimitation. This study proposes a new method based on a discrete version of the adaptive additively weighted Voronoi diagram that makes it possible to partition a two-dimensional space into zones of specific sizes, taking both the position and the weight of each seed into account. The method consists of repeatedly solving a traditional additively weighted Voronoi diagram, so that each seed?s weight is updated at every iteration. The zones are geographically connected using a metric based on the shortest path. Tests conducted on the extensive farming system of three municipalities in Castile-La Mancha (Spain) have established that the proposed heuristic procedure is valid for solving this type of partitioning problem. Nevertheless, these tests confirmed that the given seed position determines the spatial configuration the method must solve and this may have a great impact on the resulting partition

Weighted Voronoi Region Algorithms for Political Districting

Automated political districting shares with electronic voting the aim of preventing electoral manipulation and pursuing an impartial electoral mechanism. Political districting can be modelled as multiobjective partitioning of a graph into connected components, where population equality and compactness must hold if a majority voting rule is adopted. This leads to the formulation of combinatorial optimization problems that are extremely hard to solve exactly. We propose a class of heuristics, based on discrete weighted Voronoi regions, for obtaining compact and balanced districts, and discuss some formal properties of these algorithms. Their performance has been tested on randomly generated rectangular grids, as well as on real-life benchmarks; for the latter instances the resulting district maps are compared with the institutional ones adopted in the Italian political elections from 1994 to 2001

Weighted Voronoi Region Algorithms for Political Districting

Automated political districting shares with electronic voting the aim of preventing electoral manipulation and pursuing an impartial electoral mechanism. Political districting can be modelled as multiobjective partitioning of a graph into connected components, where population equality and compactness must hold if a majority voting rule is adopted. This leads to the formulation of combinatorial optimization problems that are extremely hard to solve exactly. We propose a class of heuristics, based on discrete weighted Voronoi regions, for obtaining compact and balanced districts, and discuss some formal properties of these algorithms. Their performance has been tested on randomly generated rectangular grids, as well as on real-life benchmarks; for the latter instances the resulting district maps are compared with the institutional ones adopted in the Italian political elections from 1994 to 2001

Defining Equitable Geographic Districts in Road Networks via Stable Matching

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 vertex-center 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 graph-based version of stable matching in which nodes are stably matched to a subset of nodes denoted as centers, prioritized by their shortest-path 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 Gale-Shapley 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 Gale-Shapley algorithm for any range of values of $k$.Comment: 9 pages, 4 figures, to appear in 25th ACM SIGSPATIAL International Conference on Advances in Geographic Information Systems (ACM SIGSPATIAL 2017) November 7-10, 2017, Redondo Beach, California, US

Estado del arte en procesos de zonificacion

Los procesos de partición espacial implican la división de un espacio geográfico en diferentes unidades o zonas según un conjunto específico de criterios. En ámbitos relacionados con las ciencias geoespaciales, la delimitación de estas zonas se realiza por agrupación de otras unidades básicas de área existentes en el espacio de trabajo. En este artículo se ofrece una revisión de los métodos de solución diseñados para este tipo de problemas, comenzando por una introducción a las técnicas heurísticas y modelos matemáticos más utilizados desde los años 60, para finalizar describiendo los recientes algoritmos aplicados a diagramas de Voronoi. También se revisan las aplicaciones en las que se han implementado algunos de estos modelos, quedando patente que son herramientas diseñadas para el tratamiento de problemas específicos, dada la dificultad de diseñar modelos genéricos y versátiles para este tipo de particiones espaciales o zonificacione

Districting Problems - New Geometrically Motivated Approaches

This thesis focuses on districting problems were the basic areas are represented by points or lines. In the context of points, it presents approaches that utilize the problem\u27s underlying geometrical information. For lines it introduces an algorithm combining features of geometric approaches, tabu search, and adaptive randomized neighborhood search that includes the routing distances explicitly. Moreover, this thesis summarizes, compares and enhances existing compactness measures

Measuring the Compactness of Political Districting Plans

The United States Supreme Court has long recognized compactness as an important principle in assessing the constitutionality of political districting plans. We propose a measure of compactness based on the distance between voters within the same district relative to the minimum distance achievable -- which we coin the relative proximity index. We prove that any compactness measure which satisfies three desirable properties (anonymity of voters, efficient clustering, and invariance to scale, population density, and number of districts) ranks districting plans identically to our index. We then calculate the relative proximity index for the 106th Congress, requiring us to solve for each state's maximal compactness; an NP-hard problem. Using two properties of maximally compact districts, we prove they are power diagrams and develop an algorithm based on these insights. The correlation between our index and the commonly-used measures of dispersion and perimeter is -.22 and -.06, respectively. We conclude by estimating seat-vote curves under maximally compact districts for several large states. The fraction of additional seats a party obtains when their average vote increases is significantly greater under maximally compact districting plans, relative to the existing plans.