# 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… Expand

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#### References

SHOWING 1-10 OF 31 REFERENCES

Approximating Minimization Diagrams and Generalized Proximity Search

- Mathematics, Computer Science
- 2013 IEEE 54th Annual Symposium on Foundations of Computer Science
- 2013

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. Expand

Approximate nearest neighbors: towards removing the curse of dimensionality

- Mathematics, Computer Science
- STOC '98
- 1998

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. Expand

Geometric Approximation Algorithms

- Mathematics
- 2011

Exact algorithms for dealing with geometric objects are complicated, hard to implement in practice, and slow. Over the last 20 years a theory of geometric approximation algorithms has emerged. These… Expand

A replacement for Voronoi diagrams of near linear size

- Computer Science
- Proceedings 2001 IEEE International Conference on Cluster Computing
- 2001

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. Expand

Nearest-Neighbor Searching and Metric Space Dimensions

- Mathematics
- 2005

Given a set S of points in a metric space with distance function D, the nearest-neighbor searching problem is to build a data structure for S so that for an input query point q, the point s 2 S that… Expand

Approximate clustering via core-sets

- Mathematics, Computer Science
- STOC '02
- 2002

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. Expand

Nearest-neighbor searching under uncertainty

- Computer Science
- PODS '12
- 2012

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. Expand

Efficient partition trees

- Computer Science, Mathematics
- SCG '91
- 1991

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… Expand

Approximate Nearest Neighbor: Towards Removing the Curse of Dimensionality

- Computer Science, Mathematics
- Theory Comput.
- 2012

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. Expand

Constructing approximate shortest path maps in three dimensions

- Computer Science, Mathematics
- SCG '98
- 1998

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". Expand