3,557 research outputs found
Approximating Longest Spanning Tree with Neighborhoods
We study the following maximization problem in the Euclidean plane: Given a collection of neighborhoods (polygonal regions) in the plane, the goal is to select a point in each neighborhood so that the longest spanning tree on selected points has maximum length. It is not known whether or not this problem is NP-hard. We present an approximation algorithm with ratio 0.548 for this problem. This improves the previous best known ratio of 0.511.
The presented algorithm takes linear time after computing a diameter. Even though our algorithm itself is fairly simple, its analysis is rather involved. In some part we deal with a minimization problem with multiple variables. We use a sequence of geometric transformations to reduce the number of variables and simplify the analysis
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
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