1,307 research outputs found
Balancing Graph Voronoi Diagrams
Abstract—Many facility location problems are concerned with minimizing operation and transportation costs by par-titioning territory into regions of similar size, each of which is served by a facility. For many optimization problems, the overall cost can be reduced by means of a partitioning into balanced subsets, especially in those cases where the cost associated with a subset is superlinear in its size. In this paper, we consider the problem of generating a Voronoi partition of a discrete graph so as to achieve balance conditions on the region sizes. Through experimentation, we first establish that the region sizes of randomly-generated graph Voronoi diagrams vary greatly in practice. We then show how to achieve a balanced partition of a graph via Voronoi site resampling. For bounded-degree graphs, where each of the n nodes has degree at most d, and for an initial randomly-chosen set of s Voronoi nodes, we prove that, by extending the set of Voronoi nodes using an algorithm by Thorup and Zwick, each Voronoi region has size at most 4dn/s+1 nodes, and that the expected size of the extended set of Voronoi nodes is at most 2s logn. Keywords-graph Voronoi diagram; balancing; facility loca-tion; territorial design I
Balancing graph Voronoi diagrams with one more vertex
Let be a graph with unit-length edges and nonnegative costs
assigned to its vertices. Being given a list of pairwise different vertices
, the {\em prioritized Voronoi diagram} of with
respect to is the partition of in subsets so
that, for every with , a vertex is in if and
only if is a closest vertex to in and there is no closest vertex
to in within the subset . For every
with , the {\em load} of vertex equals the sum of the
costs of all vertices in . The load of equals the maximum load of a
vertex in . We study the problem of adding one more vertex at the end of
in order to minimize the load. This problem occurs in the context of
optimally locating a new service facility ({\it e.g.}, a school or a hospital)
while taking into account already existing facilities, and with the goal of
minimizing the maximum congestion at a site. There is a brute-force algorithm
for solving this problem in time on -vertex -edge graphs.
We prove a matching time lower bound for the special case where
and , assuming the so called Hitting Set Conjecture of Abboud et al. On
the positive side, we present simple linear-time algorithms for this problem on
cliques, paths and cycles, and almost linear-time algorithms for trees, proper
interval graphs and (assuming to be a constant) bounded-treewidth graphs
Dynamic distributed clustering in wireless sensor networks via Voronoi tessellation control
This paper presents two dynamic and distributed clustering algorithms for Wireless Sensor Networks (WSNs). Clustering approaches are used in WSNs to improve the network lifetime and scalability by balancing the workload among the clusters. Each cluster is managed by a cluster head (CH) node. The first algorithm requires the CH nodes to be mobile: by dynamically varying the CH node positions, the algorithm is proved to converge to a specific partition of the mission area, the generalised Voronoi tessellation, in which the loads of the CH nodes are balanced. Conversely, if the CH nodes are fixed, a weighted Voronoi clustering approach is proposed with the same load-balancing objective: a reinforcement learning approach is used to dynamically vary the mission space partition by controlling the weights of the Voronoi regions. Numerical simulations are provided to validate the approaches
Aspects of Unstructured Grids and Finite-Volume Solvers for the Euler and Navier-Stokes Equations
One of the major achievements in engineering science has been the development of computer algorithms for solving nonlinear differential equations such as the Navier-Stokes equations. In the past, limited computer resources have motivated the development of efficient numerical schemes in computational fluid dynamics (CFD) utilizing structured meshes. The use of structured meshes greatly simplifies the implementation of CFD algorithms on conventional computers. Unstructured grids on the other hand offer an alternative to modeling complex geometries. Unstructured meshes have irregular connectivity and usually contain combinations of triangles, quadrilaterals, tetrahedra, and hexahedra. The generation and use of unstructured grids poses new challenges in CFD. The purpose of this note is to present recent developments in the unstructured grid generation and flow solution technology
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