Space-efficient approximate Voronoi diagrams

@inproceedings{Arya2002SpaceefficientAV,
  title={Space-efficient approximate Voronoi diagrams},
  author={Sunil Arya and Theocharis Malamatos and David M. Mount},
  booktitle={STOC '02},
  year={2002}
}
(MATH) Given a set $S$ of $n$ points in $\IR^d$, a {\em $(t,\epsilon)$-approximate Voronoi diagram (AVD)} is a partition of space into constant complexity cells, where each cell $c$ is associated with $t$ representative points of $S$, such that for any point in $c$, one of the associated representatives approximates the nearest neighbor to within a factor of $(1+\epsilon)$. Like the Voronoi diagram, this structure defines a spatial subdivision. It also has the desirable properties of being easy… 
Robust Proximity Search for Balls Using Sublinear Space
TLDR
If k and ε are provided in advance, the data structure provides a data structure to answer such queries requiring O(n / k) space; that is, theData structure requires sublinear space if k is sufficiently large.
Approximating a voronoi cell
TLDR
It is shown that there exists a set of -approximate Voronoi neighbors with cardinality for ! and cardinality " # $ $ % '&)(#*,+.-0/2143 #$ $ for any fixed 6587 .
On Clustering Induced Voronoi Diagrams
TLDR
This paper investigates the general conditions for the influence function which ensure the existence of a small-size approximate CIVD for a set P of n points in ℝd for some fixed d and develops assignment algorithms to determine a proper site for each cell in the decomposition.
Approximating Voronoi Diagrams with Voronoi Diagrams
TLDR
A new method is presented that uses the Voronoi diagram of a superset of the input as an approximation to the true Voronoa diagram, and it supports approximate nearest neighbor queries in time O(log∆).
Space-Time Tradeoffs for Proximity Searching in Doubling Spaces
TLDR
The objective is to build a data structure so that given any query point q in the space, it is possible to efficiently determine a point of S whose distance from q is within a factor of (1 + i¾?) of the distance between q and its nearest neighbor in S.
The Geometric Stability of Voronoi Diagrams with Respect to Small Changes of the Sites
TLDR
The question is formalized precisely, and it is shown that the answer is positive in the case of Rd, or in (possibly infinite dimensional) uniformly convex normed spaces, assuming there is a common positive lower bound on the distance between the sites.
Voronoi Graph Traversal in High Dimensions with Applications to Topological Data Analysis and Piecewise Linear Interpolation
TLDR
A randomized approximation approach that mitigates the prohibitive cost of exact computation of Voronoi diagrams in high dimensions for machine learning applications and proposes an application of this approach to piecewise linear interpolation of high dimensional data that avoids explicit complete computation of an associated Delaunay triangulation.
Computational Geometry: Proximity and Location
TLDR
This chapter will present a number of geometric data structures that arise in the context of proximity and location, including Voronoi diagrams and Delaunay triangulations, and results on multidimensional nearest neighbor searching.
Approximate voronoi cell computation on spatial data streams
TLDR
This paper proposes AVC-SW, an approximate streaming algorithm that computes (1 + ε)-approximations to the actual exact Voronoi cell in O(κ) where κ is its sample size and reduces the expected memory requirements of the classic algorithm from O(w) to $$O(k \log (\frac{w}{k} + 1)$$ regardless of the distribution of the points in the 2-d space.
...
1
2
3
4
5
...

References

SHOWING 1-10 OF 17 REFERENCES
Approximate Nearest Neighbor Queries Revisited
TLDR
New methods to answer approximate nearest neighbor queries on a set of n points in d -dimensional Euclidean space are proposed and applications to various proximity problems are discussed.
Approximating Voronoi Diagrams of Convex Sites in any Dimension
TLDR
Given a set of disjoint convex sites in any dimension, a general algorithm is described that approximates their Voronoi diagram with arbitrary precision and the only primitive operation that is required is the computation of the distance from a point to a site.
Linear-size approximate voronoi diagrams
TLDR
It is shown that for a real parameter 2 ≤ γ ≤ 1/ε, it is possible to construct an AVD consisting of O(n) /ε(d) cells for T(i) = 1, and cells in these AVD are cubes or differences of two cubes.
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.
Approximate range searching
TLDR
It is shown that if one is willing to allow approximate ranges, then it is possible to do much better than current state-of-the-art results, and empirical evidence is given showing that allowing small relative errors can significantly improve query execution times.
Efficient search for approximate nearest neighbor in high dimensional spaces
TLDR
Significantly improving and extending recent results of Kleinberg, data structures whose size is polynomial in the size of the database and search algorithms that run in time nearly linear or nearly quadratic in the dimension are constructed.
An algorithm for approximate closest-point queries
TLDR
An algorithm for approximately solving the post office problem, given n points in d dimensions, build a data structure so that, given a query point, a closest site to a querying point can be found quickly.
Balanced aspect ratio trees: combining the advantages of k-d trees and octrees
TLDR
The balanced aspect ratio (BAR) tree is a binary space partition tree on S that has O(logn) depth in which every region is convex and “fat” (that is, has a bounded aspect ratio).
An optimal algorithm for approximate nearest neighbor searching fixed dimensions
TLDR
It is shown that it is possible to preprocess a set of data points in real D-dimensional space in O(kd) time and in additional space, so that given a query point q, the closest point of S to S to q can be reported quickly.
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.
...
1
2
...