Approximating Minimization Diagrams and Generalized Proximity Search

@inproceedings{HarPeled2013ApproximatingMD,
  title={Approximating Minimization Diagrams and Generalized Proximity Search},
  author={Sariel Har-Peled and Nirman Kumar},
  booktitle={FOCS},
  year={2013}
}
We investigate the classes of functions whose minimization diagrams can be approximated efficiently in Red. We present a general framework and a data-structure that can be used to approximate the minimization diagram of such functions. The resulting data-structure has near linear size and can answer queries in logarithmic time. Applications include approximating the Voronoi diagram of (additively or multiplicatively) weighted points. Our technique also works for more general distance functions… 

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References

SHOWING 1-10 OF 31 REFERENCES
Approximating Minimization Diagrams and Generalized Proximity Search
  • Sariel Har-Peled, N. Kumar
  • Computer Science, Mathematics
    2013 IEEE 54th Annual Symposium on Foundations of Computer Science
  • 2013
TLDR
A general framework and a data-structure are presented that can be used to approximate the minimization diagram of such functions as the Voronoi diagram of (additively or multiplicatively) weighted points and the nearest furthest-neighbor distance to a set of point sets.
Approximate nearest neighbors: towards removing the curse of dimensionality
TLDR
Two algorithms for the approximate nearest neighbor problem in high-dimensional spaces are presented, which require space that is only polynomial in n and d, while achieving query times that are sub-linear inn and polynometric in d.
Geometric Approximation Algorithms
TLDR
This book is the first to cover geometric approximation algorithms in detail, and topics covered include approximate nearest-neighbor search, shape approximation, coresets, dimension reduction, and embeddings.
A replacement for Voronoi diagrams of near linear size
  • Sariel Har-Peled
  • Computer Science
    Proceedings 2001 IEEE International Conference on Cluster Computing
  • 2001
TLDR
A new type of space decomposition that provides an /spl epsi/-approximation to the distance function associated with the Voronoi diagram of P, while being of near linear size, for d/spl ges/2.
Nearest-Neighbor Searching and Metric Space Dimensions
TLDR
Several measures of dimension can be estimated using nearest-neighbor searching, while others can be used to estimate the cost of that searching.
Approximate clustering via core-sets
TLDR
It is shown that for several clustering problems one can extract a small set of points, so that using those core-sets enable us to perform approximate clustering efficiently and are a substantial improvement over what was previously known.
Nearest-neighbor searching under uncertainty
TLDR
This work presents methods for computing an exact ENN or an ε-approximate ENN, for a given error parameter 0 < ε 0 < 1, under different distance functions and presents these methods as the first nontrivial methods for answering exact or ε -approximates ENN queries with provable performance guarantees.
Efficient partition trees
We prove a theorem on partitioning point sets inEd (d fixed) and give an efficient construction of partition trees based on it. This yields a simplex range searching structure with linear space,O(n
Approximate Nearest Neighbor: Towards Removing the Curse of Dimensionality
TLDR
Two algorithms for the approximate nearest neighbor problem in high dimensional spaces for data sets of size n living in IR are presented, achieving query times that are sub-linear in n and polynomial in d.
Constructing approximate shortest path maps in three dimensions
TLDR
An algorithm is presented that computes a subdivision of R 3, which can be used to answer eciently approximate shortest path queries, and a distanceO;s(t) that "-approximates the length of a shortest path from s to t that avoids the interiors of the obstacles".
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