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Approximate Range Searching: The Absolute Model
TLDR
It is shown how idempotence can be used to improve not only approximate, but also exact halfspace range searching, because its data structures are much simpler than both their exact and relative model counterparts, and so are amenable to efficient implementation.
Near-Optimal ε-Kernel Construction and Related Problems
TLDR
New algorithms for fundamental problems in geometric approximation are described, achieving significant improvements to the exponent of the ε-dependency in their running times, and it is shown that it is possible to obtain near-optimal preprocessing time for the most efficient data structures to approximately answer queries for nearest-neighbor searching, directional width, and polytope membership.
On the Combinatorial Complexity of Approximating Polytopes
TLDR
It is shown that there exists an approximating polytope whose total combinatorial complexity is O(1/\varepsilon ^{(d-1)/2}), where O~ conceals a polylogarithmic factor in 1/ε, a significant improvement upon the best known bound.
Approximate polytope membership queries
TLDR
A reduction from approximate nearest neighbor searching to approximate polytope membership queries is introduced and it is shown that this tradeoff provides significant improvements to the best known space-time tradeoffs for approximate nearest neighbors searching.
Linear-Time Approximation Algorithms for Unit Disk Graphs
TLDR
This work proposes a method to obtain linear-time approximation algorithms for unit disk graph problems and presents an alternative linear- time approximation scheme for the minimum vertex cover, which could be obtained by an indirect application of the method.
Greedy and Local Search Heuristics to Build Area-Optimal Polygons
TLDR
These heuristic solutions to the problems of finding the maximum and minimum area polygons with a given set of vertices are presented, based mostly on two simple algorithmic paradigms: greedy and local search.
Optimal Approximate Polytope Membership
TLDR
A data structure is presented that achieves logarithmic query time with storage of only O(1/e(d−1)/2), which matches the worst-case lower bound on the complexity of any e-approximating polytope.
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