Nearest Neighbor Search and Metric Space Dimensions

Ken Clarkson

Bell Labs

Nearest Neighbor Search: the Problem

Given

Nearest Neighbor Search: Synonyms

Metric Space Dimension

Outline

Metric Spaces: Definition and Repairs

A metric space `(U,D)` has `D(x,y) ge 0` and `D(x,x)=0` for all `x,y in U`, and also:

New Metrics from Old

Start with uniform metric on finite set, or (RR, |x-y|)`;
Suppose `(U,D)`, and `(U_1,D_1)...(U_d,D_d)` are metric spaces.

New Metrics from Old: Transforms

The Biotope Transform

Given `a in U`, the biotope or Steinhaus transform
`hat D(x,y) := {: 2D(x,y):} / {:D(x,a) + D(y,a) + D(x,y):}
yields a metric.(How did I not know this?)
For `D(A,B)=mu(A Delta B)` and `a=O/`, get
`hat D(A,B) = {:mu(A Delta B):} / {:mu(A uu B):}`
Generalizations? Replacing `D(x,a) + D(y,a)` by `min_{a in T} D(x,a) + D(y,a)` seems to work, for `T subset U`.

Biotope Distance : a.k.a.

Packings, Coverings, Nets

Given `(U ,D)`, `epsilon > 0`, `P subset U` is an:

Gonzalez Construction Properties

Box Dimension

Box Dimension Equivalents

Hausdorff Dimension

Assouad Dimension

Doubling Constant

Metric Measure Spaces

Renyi dimension

Given `epsilon > 0`, let:

Renyi dimension, correlation dimension

Given `epsilon > 0`, let:

Renyi dimension and NN Search

Renyi and Information Dimensions

Information and Pointwise Dimensions

Pointwise Dimension and Others

Pointwise Dimension and NNs

Extremal Graphs

Dimensions and NN Data Structures

`Z=(U,D,mu)` a metric measure space.

Divide-and-Conquer

Several approaches could be sketched as:

Divide-and-Conquer Approaches