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Approximate Voronoi Diagrams: Techniques, tools, and applications to k th ANN search Nirman Kumar University of California, Santa-Barbara January 13th, 2016 Similarity Search ? Need similarity search to make sense of the world! When an

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Approximate Voronoi Diagrams: Techniques, tools, and applications to kth ANN search

Nirman Kumar

University of California, Santa-Barbara

January 13th, 2016

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Similarity Search

?

Need similarity search to make sense of the world!

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When an appropriate metric is defined

Similarity search reduces to NN search

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Nearest neighbor search

Set of points P: find quickly for a query q, the closest point to q in P

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Nearest neighbor search

Also important in other domains

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Approximate nearest neighbor search (ANN)

Find any point x with d(q, x) ≤ (1 + ε)d1(q, P)

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Space partitioning

Most data structures for NN (or ANN) search partition space

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Space partitioning

In low dimensions this is an explicit paritioning

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Space partitioning

In high dimensions the partitioning is implicit (via hash functions)

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Voronoi diagrams

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Voronoi diagrams

Very efficient in dimensions d ≤ 2

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Voronoi diagrams

Performance degrades sharply - bad even for d = 3

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This talk

◮ Construction of Approximate Voronoi

Diagrams

◮ Tools used - Quadtrees, WSPD ◮ Construction of AVD for kth ANN ◮ Some open problems

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Approximate Voronoi Diagrams (AVD)

A space partition as before

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Approximate Voronoi Diagrams (AVD)

With each region is associated 1 rep (a point of P)

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Approximate Voronoi Diagrams (AVD)

This rep is a valid ANN for any q in region

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Main ideas behind ANN search and AVDs

◮ If the query point is “far” any point is a

good ANN

◮ A region can be approximated well by cubes ◮ Point location can be done in a set of cubes

efficiently

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Tool 1: Quadtrees

A quadtree - intuitively

[0, 1] × [0, 1]

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Tool 1: Quadtrees

A quadtree on points

a b c d e f g h i a c b d i g h e f

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Tool 1: Quadtrees

The compressed version

a b c d e f g h i a c b d i g h e f

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Tool 1: Quadtrees

Point Location ≡ find leaf node containing a point

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Tool 1: Quadtrees

Height h: O(log h) time - O(log log n) for balanced tree!

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Tool 1: Quadtrees

But height not bounded as function of n

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Tool 1: Quadtrees

Use compressed quadtree - height bounded by O(n)

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Tool 2: Well separated pairs decomposition

How many distances among points - Ω(n2)

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Tool 2: Well separated pairs decomposition

What if distances within (1 ± ε) are considered the same?

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Tool 2: Well separated pairs decomposition

About O(n/εd) different distinct distances upto (1 ± ε)

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Tool 2: Well separated pairs decomposition

◮ How can we represent them? ◮ Given a pair of points, which

bucket does it belong to?

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Tool 2: Well separated pairs decomposition

The WSPD data structure captures this

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Tool 2: Well separated pairs decomposition More formally

◮ A collection of pairs Ai, Bi ⊂ P ◮ Ai ∩ Bi = ∅ ◮ Every pair of points is separated by some Ai, Bi ◮ Each pair Ai, Bi is well separated

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Tool 2: Well separated pairs decomposition A well separated pair is a dumbbell

ℓ ≥ 1/ε max{r1, r2}

r1 r2

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Tool 2: Well separated pairs decomposition WSPD example

a a b c d e f b c d f e a b c d f e

A1 = {a, b, c}, B1 = {e} A1 = {a}, B1 = {b, c} . . .

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Tool 2: Well separated pairs decomposition

Main result about WSPDs

There is a ε−1-WSPD of size O(nε−d) - It can be constructed in O(n log n + nε−d) time

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AVD results

The main result

◮ O(n/εd) cells ◮ Query time - O(log(n/ε))

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The AVD algorithm

Construct a 8-WSPD for the point set

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The AVD algorithm

Let (Ai, Bi) for i = 1, . . . , m be the pairs

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The AVD algorithm

For each pair do some processing

  • output some cells
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The AVD algorithm

Preprocess them for point location

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The AVD algorithm

So what is the processing per pair?

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The AVD algorithm Consider a WSPD dumbbell

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The AVD algorithm Concentric balls increasing radii - r/4 to ≈ r/ε

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The AVD algorithm Tile each ball (rad x) by cubes of size ≈ εx

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The AVD algorithm

Store the ε/c ANN for some point in each cell

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So why does it work?

Every pair of competing points is resolved

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So why does it work?

p1, p2 resolved by the WSPD pair separating them

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So why does it work?

p1 p2

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So why does it work?

q

p1 p2

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So why does it work?

q

p1 p2

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So why does it work?

q

p1 p2

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Bounding the AVD complexity

The shown method gives O(n/εd log 1/ε) cubes

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Bounding the AVD complexity

This can be improved to O(n/εd)

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kth ANN search Given q output a point u ∈ P such that: (1 − ε)dk(q, P) ≤ d(q, u) ≤ (1 + ε)dk(q, P)

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Applications of kth ANN search

◮ Density estimation ◮ Functions of the form : F(q) = k

i=1 f(di(q, P))

◮ kth ANN on balls

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Applications of kth ANN search Density estimation

density ≈ #points

area

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The result

AVD for kth ANN

◮

O((n/k)ε−d log 1/ε) cells

◮ Query time - O(log(n/(kε)))

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Quorum clustering

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Quorum clustering

Find smallest ball containing k points

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Quorum clustering

Find smallest ball containing k points

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Quorum clustering

Remove points and repeat

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Quorum clustering

Remove points and repeat

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Quorum clustering

Remove points and repeat

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Quorum clustering

Remove points and repeat

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Quorum clustering

Remove points and repeat

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Quorum clustering

Remove points and repeat

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Quorum clustering

A way to summarize points

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Quorum clustering

Has properties favorable for kth ANN problem

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Quorum clustering

Quorum clustering too expensive to compute

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Quorum clustering

Can compute approximate quorum clustering

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Quorum clustering

◮ Computed in: O(n logd n) time in I

Rd [Carmi, Dolev, Har-Peled, Katz and Segal, 2005]

◮ Computed in: O(n log n) time in I

Rd [Har-Peled and K., 2012]

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Why is quorum clustering useful

c1

c2 c3 r1 r2 r3 q

◮ x = dk(q, P) ◮ r1 ≤ x ◮ x + r1 ≥ d(q, c1) =

⇒ d(q, c1) ≤ 2x

◮ x ≤ d(q, c1) + r1 ≤ 3x

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Refining the approximation

Just as in AVDs generate a list of cells

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Refining the approximation

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Refining the approximation

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Refining the approximation

For closest ball use ANN data structure in I Rd+1

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Refining the approximation

b = b(c, r) → (c, r) ∈ I Rd+1

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Refining the approximation

Some cells generated by AVD for ball centers

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Refining the approximation

Store some info with each cell

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Refining the approximation

A kth ANN, and approximate closest ball

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Open problems

◮ In high dimensions, is there a data structure for kth NN

whose space requirement is f(n/k)?

◮ There is an AVD for weighted ANN similar to AVD as

shown - is there an extension to weighted kth ANN?