75,121 research outputs found
Approximation Schemes for Partitioning: Convex Decomposition and Surface Approximation
We revisit two NP-hard geometric partitioning problems - convex decomposition
and surface approximation. Building on recent developments in geometric
separators, we present quasi-polynomial time algorithms for these problems with
improved approximation guarantees.Comment: 21 pages, 6 figure
Spanners for Geometric Intersection Graphs
Efficient algorithms are presented for constructing spanners in geometric
intersection graphs. For a unit ball graph in R^k, a (1+\epsilon)-spanner is
obtained using efficient partitioning of the space into hypercubes and solving
bichromatic closest pair problems. The spanner construction has almost
equivalent complexity to the construction of Euclidean minimum spanning trees.
The results are extended to arbitrary ball graphs with a sub-quadratic running
time.
For unit ball graphs, the spanners have a small separator decomposition which
can be used to obtain efficient algorithms for approximating proximity problems
like diameter and distance queries. The results on compressed quadtrees,
geometric graph separators, and diameter approximation might be of independent
interest.Comment: 16 pages, 5 figures, Late
Random Geometric Graphs
We analyse graphs in which each vertex is assigned random coordinates in a
geometric space of arbitrary dimensionality and only edges between adjacent
points are present. The critical connectivity is found numerically by examining
the size of the largest cluster. We derive an analytical expression for the
cluster coefficient which shows that the graphs are distinctly different from
standard random graphs, even for infinite dimensionality. Insights relevant for
graph bi-partitioning are included.Comment: 16 pages, 10 figures. Minor changes. Added reference
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