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Defining Equitable Geographic Districts in Road Networks via Stable Matching
TLDR
The problem of defining geographic districts in road networks that are equitable, in that every district has the same number of vertices and the assignment is stable in terms of geographic distance, can be solved in O(n √ n log n) time.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 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 $k$.read more
Citations
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Proceedings ArticleDOI
Balanced centroidal power diagrams for redistricting
TL;DR: The solution is, in a well-defined sense, a locally optimal solution to the problem of choosing centers in the plane and choosing an assignment of people to those 2-d centers so as to minimize the sum of squared distances subject to the assignment being balanced.
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Balanced power diagrams for redistricting.
TL;DR: A method for redistricting, decomposing a geographical area into subareas, called districts, so that the populations of the districts are as close as possible and the Districts are compact and contiguous is proposed.
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On the computational tractability of a geographic clustering problem arising in redistricting
TL;DR: If the diameter of the graph is moderately small and the number of districts is very small, the algorithm is useable, and it is shown that, under a complexity-theoretic assumption, no algorithms with running time of the form $O(c^wn^{k+1})$ exist.
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Reactive Proximity Data Structures for Graphs
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Euclidean TSP, Motorcycle Graphs, and Other New Applications of Nearest-Neighbor Chains.
Alon Efrat,David Eppstein,Daniel Frishberg,Michael T. Goodrich,Stephen G. Kobourov,Nil Mamano,Pedro Matias,Valentin Polishchuk +7 more
TL;DR: New applications of the nearest-neighbor chain algorithm are shown, a technique that originated in agglomerative hierarchical clustering and applies to a diverse class of geometric problems.
References
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Journal ArticleDOI
Voronoi diagrams—a survey of a fundamental geometric data structure
TL;DR: The Voronoi diagram as discussed by the authors divides the plane according to the nearest-neighbor points in the plane, and then divides the vertices of the plane into vertices, where vertices correspond to vertices in a plane.
Book
The design and analysis of spatial data structures
TL;DR: The design and analysis of spatial data structures and applications for predicting stock returns and remembering and imagining palestine identity and service manual are studied.
A separator theorem for planar graphs
TL;DR: In this paper, it was shown that the vertices of a planar graph can be partitioned into three sets A,B,C such that no edge joins a vertex in A with another vertex in B, neither A nor B contains more than 2n/3 vertices, and C contains no more than $2.