Approximate Nearest Line Search in High Dimensions Sepideh Mahabadi - - PowerPoint PPT Presentation

Approximate Nearest Line Search in High Dimensions

Sepideh Mahabadi

1

The NLS Problem

• Given: a set of 𝑂 lines 𝑀 in ℝ𝑒

2

The NLS Problem

• Given: a set of 𝑂 lines 𝑀 in ℝ𝑒
• Goal: build a data structure s.t.

– given a query 𝑟, find the closest line ℓ∗ to 𝑟

3

The NLS Problem

• Given: a set of 𝑂 lines 𝑀 in ℝ𝑒
• Goal: build a data structure s.t.

– given a query 𝑟, find the closest line ℓ∗ to 𝑟 – polynomial space – sub-linear query time

4

The NLS Problem

• Given: a set of 𝑂 lines 𝑀 in ℝ𝑒
• Goal: build a data structure s.t.

– given a query 𝑟, find the closest line ℓ∗ to 𝑟 – polynomial space – sub-linear query time

Approximation

• Finds an approximate closest line ℓ

𝑒𝑗𝑡𝑢 𝑟,ℓ ≤ 𝑒𝑗𝑡𝑢(𝑟, ℓ∗)(1 + 𝜗)

5

BACKGROUND

Nearest Neighbor Problems Motivation Previous Work Our result Notation

6

Nearest Neighbor Problem

NN: Given a set of 𝑂 points 𝑄, build a data structure s.t. given a query point 𝑟, finds the closest point 𝑞∗ to 𝑟.

7

Nearest Neighbor Problem

NN: Given a set of 𝑂 points 𝑄, build a data structure s.t. given a query point 𝑟, finds the closest point 𝑞∗ to 𝑟.

• Applications: database, information retrieval,

pattern recognition, computer vision

– Features: dimensions – Objects: points – Similarity: distance between points

8

Nearest Neighbor Problem

NN: Given a set of 𝑂 points 𝑄, build a data structure s.t. given a query point 𝑟, finds the closest point 𝑞∗ to 𝑟.

• Applications: database, information retrieval,

pattern recognition, computer vision

– Features: dimensions – Objects: points – Similarity: distance between points

• Current solutions suffer from “curse of

dimensionality”:

– Either space or query time is exponential in 𝑒 – Little improvement over linear search

9

Approximate Nearest Neighbor(ANN)

• ANN: Given a set of 𝑂 points 𝑄, build a data

structure s.t. given a query point 𝑟, finds an approximate closest point 𝑞 to 𝑟, i.e., 𝑒𝑗𝑡𝑢 𝑟,𝑞 ≤ 𝑒𝑗𝑡𝑢 𝑟, 𝑞∗ 1 + 𝜗

10

Approximate Nearest Neighbor(ANN)

• ANN: Given a set of 𝑂 points 𝑄, build a data

structure s.t. given a query point 𝑟, finds an approximate closest point 𝑞 to 𝑟, i.e., 𝑒𝑗𝑡𝑢 𝑟,𝑞 ≤ 𝑒𝑗𝑡𝑢 𝑟, 𝑞∗ 1 + 𝜗

• There exist data structures with different
• tradeoffs. Example:

– Space: 𝑒𝑂 𝑃

1 𝜗2

– Query time: 𝑒 log 𝑂

𝜗 𝑃 1

11

Motivation for NLS

One of the simplest generalizations of ANN: data items are represented by 𝑙- flats (affine subspace) instead of points

12

Motivation for NLS

One of the simplest generalizations of ANN: data items are represented by 𝑙- flats (affine subspace) instead of points

• Model data under linear variations
• Unknown or unimportant parameters in

database

13

Motivation for NLS

One of the simplest generalizations of ANN: data items are represented by 𝑙- flats (affine subspace) instead of points

• Model data under linear variations
• Unknown or unimportant parameters in

database

• Example:

– Varying light gain parameter of images – Each image/point becomes a line – Search for the closest line to the query image

14

Previous and Related Work

15

• Magen[02]: Nearest Subspace Search

– Query time is fast : 𝑒 + log 𝑂 +

1 𝜗 𝑃 1

– Space is super-polynomial : 2 log 𝑂 𝑃 1

Previous and Related Work

16

• Magen[02]: Nearest Subspace Search

– Query time is fast : 𝑒 + log 𝑂 +

1 𝜗 𝑃 1

– Space is super-polynomial : 2 log 𝑂 𝑃 1

Dual Problem: Database is a set of points, query is a 𝑙-flat

• [AIKN] for 1-flat: for any 𝑢 > 0

– Query time: 𝑃 𝑒3𝑂0.5+𝑢 – Space: 𝑒2𝑂𝑃

1 𝜗2+ 1 𝑢2

Previous and Related Work

17

• Magen[02]: Nearest Subspace Search

– Query time is fast : 𝑒 + log 𝑂 +

1 𝜗 𝑃 1

– Space is super-polynomial : 2 log 𝑂 𝑃 1

Dual Problem: Database is a set of points, query is a 𝑙-flat

• [AIKN] for 1-flat: for any 𝑢 > 0

– Query time: 𝑃 𝑒3𝑂0.5+𝑢 – Space: 𝑒2𝑂𝑃

1 𝜗2+ 1 𝑢2

• Very recently [MNSS] extended it for 𝑙-flats

– Query time 𝑃 𝑜

𝑙 𝑙+1−𝜍+𝑢

– Space: 𝑃(𝑜

1+

𝜏𝑙 𝑙+1−𝜍 + 𝑜 log𝑃 1 𝑢 𝑜)

Our Result

We give a randomized algorithm that for any sufficiently small 𝜗 reports a 1 + 𝜗 -approximate solution with high probability

• Space: 𝑂 + 𝑒 𝑃

1 𝜗2

• Time : 𝑒 + log 𝑂 +

1 𝜗 𝑃 1

18

Our Result

We give a randomized algorithm that for any sufficiently small 𝜗 reports a 1 + 𝜗 -approximate solution with high probability

• Space: 𝑂 + 𝑒 𝑃

1 𝜗2

• Time : 𝑒 + log 𝑂 +

1 𝜗 𝑃 1

• Matches up to polynomials, the performance of best

algorithm for ANN. No exponential dependence on 𝑒

19

Our Result

We give a randomized algorithm that for any sufficiently small 𝜗 reports a 1 + 𝜗 -approximate solution with high probability

• Space: 𝑂 + 𝑒 𝑃

1 𝜗2

• Time : 𝑒 + log 𝑂 +

1 𝜗 𝑃 1

• Matches up to polynomials, the performance of best

algorithm for ANN. No exponential dependence on 𝑒

• The first algorithm with poly log query time and

polynomial space for objects other than points

20

Our Result

We give a randomized algorithm that for any sufficiently small 𝜗 reports a 1 + 𝜗 -approximate solution with high probability

• Space: 𝑂 + 𝑒 𝑃

1 𝜗2

• Time : 𝑒 + log 𝑂 +

1 𝜗 𝑃 1

• Matches up to polynomials, the performance of best

algorithm for ANN. No exponential dependence on 𝑒

• The first algorithm with poly log query time and

polynomial space for objects other than points

• Only uses reductions to ANN

21

Notation

• 𝑀 : the set of lines with size 𝑂
• q : the query point

22

Notation

• 𝑀 : the set of lines with size 𝑂
• q : the query point
• 𝐶(𝑑, 𝑠): ball of radius 𝑠 around 𝑑

23

Notation

• 𝑀 : the set of lines with size 𝑂
• q : the query point
• 𝐶(𝑑, 𝑠): ball of radius 𝑠 around 𝑑
• 𝑒𝑗𝑡𝑢: the Euclidean distance

between objects

24

Notation

• 𝑀 : the set of lines with size 𝑂
• q : the query point
• 𝐶(𝑑, 𝑠): ball of radius 𝑠 around 𝑑
• 𝑒𝑗𝑡𝑢: the Euclidean distance

between objects

• 𝑏𝑜𝑕𝑚𝑓: defined between lines

25

Notation

• 𝑀 : the set of lines with size 𝑂
• q : the query point
• 𝐶(𝑑, 𝑠): ball of radius 𝑠 around 𝑑
• 𝑒𝑗𝑡𝑢: the Euclidean distance

between objects

• 𝑏𝑜𝑕𝑚𝑓: defined between lines
• 𝜀-close: two lines ℓ , ℓ′ are 𝜀-close

if sin(𝑏𝑜𝑕𝑚𝑓 ℓ, ℓ′ ) ≤ 𝜀. Similarly we define 𝜀-far/ strictly 𝜀-close/ strictly 𝜀-far

26

Notation

• 𝑀 : the set of lines with size 𝑂
• q : the query point
• 𝐶(𝑑, 𝑠): ball of radius 𝑠 around 𝑑
• 𝑒𝑗𝑡𝑢: the Euclidean distance

between objects

• 𝑏𝑜𝑕𝑚𝑓: defined between lines
• 𝜀-close: two lines ℓ , ℓ′ are 𝜀-close

if sin(𝑏𝑜𝑕𝑚𝑓 ℓ, ℓ′ ) ≤ 𝜀. Similarly we define 𝜀-far/ strictly 𝜀-close/ strictly 𝜀-far

• 𝐷𝑄ℓ1→ℓ2: closest point on ℓ1 to ℓ2

27

MODULES

Unbounded Module Net Module Parallel Module

28

Unbounded Module - Intuition

• All lines in 𝑀 pass through the origin

𝑝

29

Unbounded Module - Intuition

• All lines in 𝑀 pass through the origin

𝑝

• Data structure:

– Project all lines onto any sphere 𝑇 𝑝,𝑠 to get point set 𝑄 – Build ANN data structure 𝐵𝑂𝑂(𝑄, 𝜗)

30

Unbounded Module - Intuition

• All lines in 𝑀 pass through the origin

𝑝

• Data structure:

– Project all lines onto any sphere 𝑇 𝑝,𝑠 to get point set 𝑄 – Build ANN data structure 𝐵𝑂𝑂(𝑄, 𝜗)

• Query Algorithm:

– Project the query on 𝑇(𝑝, 𝑠) to get 𝑟′ – Find the approximate closest point to 𝑟′, i.e., 𝑞 = 𝐵𝑂𝑂𝑄 𝑟′ – Return the corresponding line of 𝑞

31

Unbounded Module

• All lines in 𝑀 pass through a small ball

𝐶 𝑝, 𝑠

• Query is far enough, outside of 𝐶(𝑝, 𝑆)
• Use the same data structure and

query algorithm

32

Unbounded Module

• All lines in 𝑀 pass through a small ball

𝐶 𝑝, 𝑠

• Query is far enough, outside of 𝐶(𝑝, 𝑆)
• Use the same data structure and

query algorithm Lemma: if 𝑆 ≥ 𝑠

𝜗𝜀 , the returned line ℓ𝑞 is

• Either an approximate closest line
• Or is 𝜀-close to the closest line ℓ∗

33

Unbounded Module

• All lines in 𝑀 pass through a small ball

𝐶 𝑝, 𝑠

• Query is far enough, outside of 𝐶(𝑝, 𝑆)
• Use the same data structure and

query algorithm Lemma: if 𝑆 ≥ 𝑠

𝜗𝜀 , the returned line ℓ𝑞 is

• Either an approximate closest line
• Or is 𝜀-close to the closest line ℓ∗

This helps us further restrict our search to almost parallel lines to ℓ𝑞

34

Net Module

• Intuition: sampling points from each

line finely enough to get a set of points 𝑄, and building an 𝐵𝑂𝑂(𝑄, 𝜗) should suffice to find the approximate closest line.

38

Net Module

• Intuition: sampling points from each

line finely enough to get a set of points 𝑄, and building an 𝐵𝑂𝑂(𝑄, 𝜗) should suffice to find the approximate closest line. Lemma:

• Let 𝑦 be the separation parameter:

distance between two adjacent samples on a line

• Then

– Either the returned line ℓ𝑞 is an approximate closest line – Or 𝑒𝑗𝑡𝑢 𝑟, ℓ𝑞 ≤ 𝑦/𝜗

39

Parallel Module - Intuition

• All lines in 𝑀 are parallel

42

Parallel Module - Intuition

• All lines in 𝑀 are parallel
• Data structure:

– Project all lines onto any hyper-plane 𝑕 which is perpendicular to all the lines to get point set 𝑄 – Build ANN data structure 𝐵𝑂𝑂(𝑄, 𝜗)

43

Parallel Module - Intuition

• All lines in 𝑀 are parallel
• Data structure:

– Project all lines onto any hyper-plane 𝑕 which is perpendicular to all the lines to get point set 𝑄 – Build ANN data structure 𝐵𝑂𝑂(𝑄, 𝜗)

• Query algorithm:

– Project the query on 𝑕 to get 𝑟′ – Find the approximate closest point to 𝑟′, i.e., 𝑞 = 𝐵𝑂𝑂𝑄 𝑟′ – Return the corresponding line to 𝑞

44

Parallel Module

• All lines in 𝑀 are 𝜀-close to a base line ℓ𝑐
• Project the lines onto a hyper-plane 𝑕 which is

perpendicular to ℓ𝑐

• Query is close enough to 𝑕
• Use the same data structure and query algorithm

45

Parallel Module

• All lines in 𝑀 are 𝜀-close to a base line ℓ𝑐
• Project the lines onto a hyper-plane 𝑕 which is

perpendicular to ℓ𝑐

• Query is close enough to 𝑕
• Use the same data structure and query algorithm

Lemma: if 𝑒𝑗𝑡𝑢 𝑟, 𝑕 ≤

𝐸𝜗 𝜀 , then

• Either the returned line ℓ𝑞 is an approximate closest

line

• Or 𝑒𝑗𝑡𝑢 𝑟, ℓ𝑞

≤ 𝐸

46

Parallel Module

• All lines in 𝑀 are 𝜀-close to a base line ℓ𝑐
• Project the lines onto a hyper-plane 𝑕 which is

perpendicular to ℓ𝑐

• Query is close enough to 𝑕
• Use the same data structure and query algorithm

Lemma: if 𝑒𝑗𝑡𝑢 𝑟, 𝑕 ≤

𝐸𝜗 𝜀 , then

• Either the returned line ℓ𝑞 is an approximate closest

line

• Or 𝑒𝑗𝑡𝑢 𝑟, ℓ𝑞

≤ 𝐸 Thus, for a set of almost parallel lines, we can use a set

• f parallel modules to cover a bounded region.

47

ALGORITHMS

General Case

• Input lines can have any configuration
• Divergent Case
• Input lines are 𝑃(𝜗)-far from each other
• Almost Parallel Case
• Input lines are all 𝑃(𝜗)-close to each other

51

Outline of the Algorithms

• Input: a set of 𝑜 lines 𝑇

52

Outline of the Algorithms

• Input: a set of 𝑜 lines 𝑇
• Randomly choose a subset of 𝑜/2 lines 𝑈

53

Outline of the Algorithms

• Input: a set of 𝑜 lines 𝑇
• Randomly choose a subset of 𝑜/2 lines 𝑈
• Solve the problem over 𝑈 to get a line ℓ𝑞

54

Outline of the Algorithms

• Input: a set of 𝑜 lines 𝑇
• Randomly choose a subset of 𝑜/2 lines 𝑈
• Solve the problem over 𝑈 to get a line ℓ𝑞
• For log𝑜 iterations

– Use ℓ𝑞 to find a much closer line ℓ𝑞′ – Update ℓ𝑞 with ℓ𝑞

′

55

Improvement step

Outline of the Algorithms

• Input: a set of 𝑜 lines 𝑇
• Randomly choose a subset of 𝑜/2 lines 𝑈
• Solve the problem over 𝑈 to get a line ℓ𝑞
• For log𝑜 iterations

– Use ℓ𝑞 to find a much closer line ℓ𝑞′ – Update ℓ𝑞 with ℓ𝑞

′

56

Improvement step

Outline of the Algorithms

• Input: a set of 𝑜 lines 𝑇
• Randomly choose a subset of 𝑜/2 lines 𝑈
• Solve the problem over 𝑈 to get a line ℓ𝑞
• For log𝑜 iterations

– Use ℓ𝑞 to find a much closer line ℓ𝑞′ – Update ℓ𝑞 with ℓ𝑞

′

Why?

57

Improvement step

Outline of the Algorithms

• Input: a set of 𝑜 lines 𝑇
• Randomly choose a subset of 𝑜/2 lines 𝑈
• Solve the problem over 𝑈 to get a line ℓ𝑞
• For log𝑜 iterations

– Use ℓ𝑞 to find a much closer line ℓ𝑞′ – Update ℓ𝑞 with ℓ𝑞

′

Let ℓ1, … , ℓlog 𝑜 be the log 𝑜 closest lines to 𝑟 in the set 𝑇

58

Improvement step

Outline of the Algorithms

• Input: a set of 𝑜 lines 𝑇
• Randomly choose a subset of 𝑜/2 lines 𝑈
• Solve the problem over 𝑈 to get a line ℓ𝑞
• For log𝑜 iterations

– Use ℓ𝑞 to find a much closer line ℓ𝑞′ – Update ℓ𝑞 with ℓ𝑞

′

Let ℓ1, … , ℓlog 𝑜 be the log 𝑜 closest lines to 𝑟 in the set 𝑇 With high probability at least one of {ℓ1, … , ℓlog 𝑜} are sampled in 𝑈

– 𝑒𝑗𝑡𝑢 𝑟, ℓ𝑞 ≤ 𝑒𝑗𝑡𝑢 𝑟, ℓlog 𝑜 (1 + 𝜗) – log 𝑜 improvement steps suffices to find an approximate closest line

59

Improvement step

Improvement Step

Given a line ℓ, how to improve it, i.e., find a closer line?

60

Improvement Step

Given a line ℓ, how to improve it, i.e., find a closer line?

• Data structure
• Query Processing Algorithm

61

General Case

• Search among all lines that are 𝜗-far from

current line using Divergent Case

62

General Case

• Search among all lines that are 𝜗-far from

current line using Divergent Case

• Search among the lines that are almost

parallel to line found in previous step using Almost Parallel Case

63

Divergent Case

Assume any two lines are 𝜗-far; they diverge quickly.

64

Divergent Case

Assume any two lines are 𝜗-far; they diverge quickly.

• Let ℓ be the current line, and ℓ∗ be the closest

line to 𝑟

• Let 𝑦 = 𝑒𝑗𝑡𝑢(𝑟, ℓ)
• 𝑒𝑗𝑡𝑢 𝑟, ℓ∗ ≤ 𝑦

65

Divergent Case

Assume any two lines are 𝜗-far; they diverge quickly.

• Let ℓ be the current line, and ℓ∗ be the closest

line to 𝑟

• Let 𝑦 = 𝑒𝑗𝑡𝑢(𝑟, ℓ)
• 𝑒𝑗𝑡𝑢 𝑟, ℓ∗ ≤ 𝑦

– All potential ℓ∗ intersect 𝐶(𝑟, 𝑦)

66

Divergent Case

Assume any two lines are 𝜗-far; they diverge quickly.

• Let ℓ be the current line, and ℓ∗ be the closest

line to 𝑟

• Let 𝑦 = 𝑒𝑗𝑡𝑢(𝑟, ℓ)
• 𝑒𝑗𝑡𝑢 𝑟, ℓ∗ ≤ 𝑦

– All potential ℓ∗ intersect 𝐶(𝑟, 𝑦) – Good news: we can build a net module inside 𝐶 𝑟, 𝑦 with separation parameter 𝑦ϵ2 to improve over ℓ

67

Divergent Case

Assume any two lines are 𝜗-far; they diverge quickly.

• Let ℓ be the current line, and ℓ∗ be the closest

line to 𝑟

• Let 𝑦 = 𝑒𝑗𝑡𝑢(𝑟, ℓ)
• 𝑒𝑗𝑡𝑢 𝑟, ℓ∗ ≤ 𝑦

– All potential ℓ∗ intersect 𝐶(𝑟, 𝑦) – Good news: we can build a net module inside 𝐶 𝑟, 𝑦 with separation parameter 𝑦ϵ2 to improve over ℓ – Bad news: we don’t know this ball in advance

68

Divergent Case contd.

What we know:

• 𝑒𝑗𝑡𝑢 ℓ,ℓ∗ ≤ 2𝑦
• Let 𝑟′ be the projection of 𝑟 on ℓ

69

Divergent Case contd.

What we know:

• 𝑒𝑗𝑡𝑢 ℓ,ℓ∗ ≤ 2𝑦
• Let 𝑟′ be the projection of 𝑟 on ℓ

70

Divergent Case contd.

What we know:

• 𝑒𝑗𝑡𝑢 ℓ,ℓ∗ ≤ 2𝑦
• Let 𝑟′ be the projection of 𝑟 on ℓ

– 𝐷𝑄ℓ→ℓ∗ is not farther than

𝑦 𝜗 from 𝑟′

since they are 𝜗-far

71

Divergent Case contd.

What we know:

• 𝑒𝑗𝑡𝑢 ℓ,ℓ∗ ≤ 2𝑦
• Let 𝑟′ be the projection of 𝑟 on ℓ

– 𝐷𝑄ℓ→ℓ∗ is not farther than

𝑦 𝜗 from 𝑟′

since they are 𝜗-far – 𝑪(𝒓′, 𝑷

𝒚 𝝑 ) touches all such lines

72

Data Structure

For each ℓ ∈ 𝑇

• Sort all lines ℓ′ according to their distance from ℓ

73

Data Structure

For each ℓ ∈ 𝑇

• Sort all lines ℓ′ according to their distance from ℓ
• For all 1 ≤ 𝑗 ≤ 𝑜, let 𝑇𝑗 be the 𝑗𝑢ℎ closest lines

74

Data Structure

For each ℓ ∈ 𝑇

• Sort all lines ℓ′ according to their distance from ℓ
• For all 1 ≤ 𝑗 ≤ 𝑜, let 𝑇𝑗 be the 𝑗𝑢ℎ closest lines

– Sort all lines in 𝑇𝑗 such as ℓ′ according to the position of 𝐷𝑄ℓ→ℓ′

75

Data Structure

For each ℓ ∈ 𝑇

• Sort all lines ℓ′ according to their distance from ℓ
• For all 1 ≤ 𝑗 ≤ 𝑜, let 𝑇𝑗 be the 𝑗𝑢ℎ closest lines

– Sort all lines in 𝑇𝑗 such as ℓ′ according to the position of 𝐷𝑄ℓ→ℓ′ – For each interval of lines 𝐵 in sorted 𝑇𝑗

76

Data Structure

For each ℓ ∈ 𝑇

• Sort all lines ℓ′ according to their distance from ℓ
• For all 1 ≤ 𝑗 ≤ 𝑜, let 𝑇𝑗 be the 𝑗𝑢ℎ closest lines

– Sort all lines in 𝑇𝑗 such as ℓ′ according to the position of 𝐷𝑄ℓ→ℓ′ – For each interval of lines 𝐵 in sorted 𝑇𝑗

• Find smallest ball 𝐶𝐵(oA, rA) with its

center on ℓ which intersects all lines in 𝐵

• > (𝑠

𝐵 ≤ 𝑃( 𝑦 𝜗))

77

Data Structure

For each ℓ ∈ 𝑇

• Sort all lines ℓ′ according to their distance from ℓ
• For all 1 ≤ 𝑗 ≤ 𝑜, let 𝑇𝑗 be the 𝑗𝑢ℎ closest lines

– Sort all lines in 𝑇𝑗 such as ℓ′ according to the position of 𝐷𝑄ℓ→ℓ′ – For each interval of lines 𝐵 in sorted 𝑇𝑗

• Find smallest ball 𝐶𝐵(oA, rA) with its

center on ℓ which intersects all lines in 𝐵

• > (𝑠

𝐵 ≤ 𝑃( 𝑦 𝜗))

• Construct a net module inside of the ball
• f 𝐶(𝑝𝐵, 𝑠

𝐵/𝜗2) with separation 𝑠 𝐵𝜗3

(#samples = O(𝑜 𝑠

𝐵/(𝜗2𝑠 𝐵𝜗3)) = 𝑃(𝑜/𝜗5))

78

Data Structure

For each ℓ ∈ 𝑇

• Sort all lines ℓ′ according to their distance from ℓ
• For all 1 ≤ 𝑗 ≤ 𝑜, let 𝑇𝑗 be the 𝑗𝑢ℎ closest lines

– Sort all lines in 𝑇𝑗 such as ℓ′ according to the position of 𝐷𝑄ℓ→ℓ′ – For each interval of lines 𝐵 in sorted 𝑇𝑗

• Find smallest ball 𝐶𝐵(oA, rA) with its

center on ℓ which intersects all lines in 𝐵

• > (𝑠

𝐵 ≤ 𝑃( 𝑦 𝜗))

• Construct a net module inside of the ball
• f 𝐶(𝑝𝐵, 𝑠

𝐵/𝜗2) with separation 𝑠 𝐵𝜗3

(#samples = O(𝑜 𝑠

𝐵/(𝜗2𝑠 𝐵𝜗3)) = 𝑃(𝑜/𝜗5))

• Construct an unbounded module outside
• f 𝐶𝐵 𝑝𝐵,

1 𝜗2 𝑠 𝐵

79

Query Processing Algorithm

Given query point 𝑟

80

Query Processing Algorithm

Given query point 𝑟 – Project 𝑟 on ℓ to get 𝑟′ – Use binary search to find the set 𝐵 of all lines ℓ′ that are within distance 2𝑦 of ℓ, and that 𝐷𝑄ℓ→ℓ′ is within distance 2𝑦/𝜗 of 𝑟′

81

Query Processing Algorithm

Given query point 𝑟 – Project 𝑟 on ℓ to get 𝑟′ – Use binary search to find the set 𝐵 of all lines ℓ′ that are within distance 2𝑦 of ℓ, and that 𝐷𝑄ℓ→ℓ′ is within distance 2𝑦/𝜗 of 𝑟′

82

Query Processing Algorithm

Given query point 𝑟 – Project 𝑟 on ℓ to get 𝑟′ – Use binary search to find the set 𝐵 of all lines ℓ′ that are within distance 2𝑦 of ℓ, and that 𝐷𝑄ℓ→ℓ′ is within distance 2𝑦/𝜗 of 𝑟′ – Let 𝐶𝐵(𝑝𝐵, 𝑠

𝐵) be the corresponding ball

83

Query Processing Algorithm

Given query point 𝑟 – Project 𝑟 on ℓ to get 𝑟′ – Use binary search to find the set 𝐵 of all lines ℓ′ that are within distance 2𝑦 of ℓ, and that 𝐷𝑄ℓ→ℓ′ is within distance 2𝑦/𝜗 of 𝑟′ – Let 𝐶𝐵(𝑝𝐵, 𝑠

𝐵) be the corresponding ball

– If 𝑦 ∈ 𝐶𝐵(𝑝𝐵,

𝑠𝐵 𝜗2) use net module:

• Find approximate closest line -> done!
• Or find a line with distance at most

𝑠

𝐵𝜗2 ≤ 𝑦𝜗 (𝑠 𝐵 ≤ 𝑦/𝜗) -> we improved

84

Query Processing Algorithm

Given query point 𝑟 – Project 𝑟 on ℓ to get 𝑟′ – Use binary search to find the set 𝐵 of all lines ℓ′ that are within distance 2𝑦 of ℓ, and that 𝐷𝑄ℓ→ℓ′ is within distance 2𝑦/𝜗 of 𝑟′ – Let 𝐶𝐵(𝑝𝐵, 𝑠

𝐵) be the corresponding ball

– If 𝑦 ∈ 𝐶𝐵(𝑝𝐵,

𝑠𝐵 𝜗2) use net module:

• Find approximate closest line -> done!
• Or find a line with distance at most

𝑠

𝐵𝜗2 ≤ 𝑦𝜗 (𝑠 𝐵 ≤ 𝑦/𝜗) -> we improved

– Otherwise use unbounded module to find the approximate closest line -> done!

85

Almost Parallel

All lines are 2𝜗-close to each other. For each line ℓ

• Partition the space into slabs using

perpendicular hyperplanes to ℓ s.t. for any pair of lines ℓ1, ℓ2:

86

slab

Almost Parallel

All lines are 2𝜗-close to each other. For each line ℓ

• Partition the space into slabs using

perpendicular hyperplanes to ℓ s.t. for any pair of lines ℓ1, ℓ2:

– In each slab the relative order of dist𝐼 ℓ,𝑝 ℓ, ℓ1 and 𝑒𝑗𝑡𝑢𝐼 ℓ,𝑝 (ℓ, ℓ2) on the hyper-plane remains the same as we move 𝑝 on ℓ in the slab There is a unique ordering of the lines

87

Almost Parallel

All lines are 2𝜗-close to each other. For each line ℓ

• Partition the space into slabs using

perpendicular hyperplanes to ℓ s.t. for any pair of lines ℓ1, ℓ2:

– In each slab the relative order of dist𝐼 ℓ,𝑝 ℓ, ℓ1 and 𝑒𝑗𝑡𝑢𝐼 ℓ,𝑝 (ℓ, ℓ2) on the hyper-plane remains the same as we move 𝑝 on ℓ in the slab There is a unique ordering of the lines – 𝑒𝑗𝑡𝑢𝐼 ℓ,𝑝 ℓ1,ℓ2 on the hyper-plane is monotone

88

Almost Parallel

All lines are 2𝜗-close to each other. For each line ℓ

• Partition the space into slabs using

perpendicular hyperplanes to ℓ s.t. for any pair of lines ℓ1, ℓ2:

– In each slab the relative order of dist𝐼 ℓ,𝑝 ℓ, ℓ1 and 𝑒𝑗𝑡𝑢𝐼 ℓ,𝑝 (ℓ, ℓ2) on the hyper-plane remains the same as we move 𝑝 on ℓ in the slab There is a unique ordering of the lines – 𝑒𝑗𝑡𝑢𝐼 ℓ,𝑝 ℓ1,ℓ2 on the hyper-plane is monotone The minimum ball intersecting any prefix of lines have its center on the boundary of slab

89

Almost Parallel

All lines are 2𝜗-close to each other. For each line ℓ

• Partition the space into slabs using

perpendicular hyperplanes to ℓ s.t. for any pair of lines ℓ1, ℓ2:

– In each slab the relative order of dist𝐼 ℓ,𝑝 ℓ, ℓ1 and 𝑒𝑗𝑡𝑢𝐼 ℓ,𝑝 (ℓ, ℓ2) on the hyper-plane remains the same as we move 𝑝 on ℓ in the slab There is a unique ordering of the lines – 𝑒𝑗𝑡𝑢𝐼 ℓ,𝑝 ℓ1,ℓ2 on the hyper-plane is monotone The minimum ball intersecting any prefix of lines have its center on the boundary of slab.

• 𝑃 𝑜2 slabs suffices

90

Data Structure in Each Slab

• For each 𝑗, let 𝐶(𝑝, 𝑠) be the smallest ball

touching the closest 𝑗𝑢ℎ lines s.t. 𝑝 ∈ ℓ. We know 𝑝 would be on the boundary of slab.

91

Data Structure in Each Slab

• For each 𝑗, let 𝐶(𝑝, 𝑠) be the smallest ball

touching the closest 𝑗𝑢ℎ lines s.t. 𝑝 ∈ ℓ. We know 𝑝 would be on the boundary of slab.

92

Data Structure in Each Slab

• For each 𝑗, let 𝐶(𝑝, 𝑠) be the smallest ball

touching the closest 𝑗𝑢ℎ lines s.t. 𝑝 ∈ ℓ. We know 𝑝 would be on the boundary of slab.

93

Data Structure in Each Slab

• For each 𝑗, let 𝐶(𝑝, 𝑠) be the smallest ball

touching the closest 𝑗𝑢ℎ lines s.t. 𝑝 ∈ ℓ. We know 𝑝 would be on the boundary of slab.

• Let 𝜀0 > ⋯ > 𝜀𝑢 be all pairwise angles
• Let 𝑆0 =

𝑠 𝜗𝜀0 , … , 𝑆𝑢 = 𝑠 𝜗𝜀𝑢

• Consider the balls 𝐶 𝑝, 𝑆0 , … , 𝐶 𝑝, 𝑆𝑢

94

Data Structure in Each Slab

• For each 𝑗, let 𝐶(𝑝, 𝑠) be the smallest ball

touching the closest 𝑗𝑢ℎ lines s.t. 𝑝 ∈ ℓ. We know 𝑝 would be on the boundary of slab.

• Let 𝜀0 > ⋯ , > be all pairwise angles
• Let 𝑆0 =

𝑠 𝜗𝜀0 , … , 𝑆𝑢 = 𝑠 𝜗𝜀𝑢

• Consider the balls 𝐶 𝑝, 𝑆0 , … , 𝐶 𝑝, 𝑆𝑢
• Build net module inside 𝐶 𝑝, 𝑆0

95

Data Structure in Each Slab

• For each 𝑗, let 𝐶(𝑝, 𝑠) be the smallest ball

touching the closest 𝑗𝑢ℎ lines s.t. 𝑝 ∈ ℓ. We know 𝑝 would be on the boundary of slab.

• Let 𝜀0 > ⋯ > 𝜀𝑢 be all pairwise angles
• Let 𝑆0 =

𝑠 𝜗𝜀0 , … , 𝑆𝑢 = 𝑠 𝜗𝜀𝑢

• Consider the balls 𝐶 𝑝, 𝑆0 , … , 𝐶 𝑝, 𝑆𝑢
• Build net module inside 𝐶 𝑝, 𝑆0

96

Data Structure in Each Slab

• For each 𝑗, let 𝐶(𝑝, 𝑠) be the smallest ball

touching the closest 𝑗𝑢ℎ lines s.t. 𝑝 ∈ ℓ. We know 𝑝 would be on the boundary of slab.

• Let 𝜀0 > ⋯ > 𝜀𝑢 be all pairwise angles
• Let 𝑆0 =

𝑠 𝜗𝜀0 , … , 𝑆𝑢 = 𝑠 𝜗𝜀𝑢

• Consider the balls 𝐶 𝑝, 𝑆0 , … , 𝐶 𝑝, 𝑆𝑢
• Build net module inside 𝐶 𝑝, 𝑆0
• For each ball 𝐶(𝑝, 𝑆𝑗)

– Build unbounded module on it

97

Data Structure in Each Slab

• For each 𝑗, let 𝐶(𝑝, 𝑠) be the smallest ball

touching the closest 𝑗𝑢ℎ lines s.t. 𝑝 ∈ ℓ. We know 𝑝 would be on the boundary of slab.

• Let 𝜀0 > ⋯ > 𝜀𝑢 be all pairwise angles
• Let 𝑆0 =

𝑠 𝜗𝜀0 , … , 𝑆𝑢 = 𝑠 𝜗𝜀𝑢

• Consider the balls 𝐶 𝑝, 𝑆0 , … , 𝐶 𝑝, 𝑆𝑢
• Build net module inside 𝐶 𝑝, 𝑆0
• For each ball 𝐶(𝑝, 𝑆𝑗)

– Build unbounded module on it – For each line ℓ𝑐

• Build a set of parallel modules with ℓ𝑐

as their base line for all the lines that are 𝜀𝑗-close to ℓ𝑐 , so that they cover the space between 𝐶(𝑝, 𝑆𝑗) and 𝐶(𝑝, 𝑆𝑗+1) with separation 𝑆𝑗+1𝜗

98

Query Processing Algorithm

• Given 𝑟, find the right slab, and

retrieve all candidate lines

• Using binary search find 𝑠

99

Query Processing Algorithm

• Given 𝑟, find the right slab, and

retrieve all candidate lines

• Using binary search find 𝑠
• Find largest 𝑗 such that 𝑟 ∉ 𝐶(𝑝, 𝑆𝑗)

100

Query Processing Algorithm

• Given 𝑟, find the right slab, and

retrieve all candidate lines

• Using binary search find 𝑠
• Find largest 𝑗 such that 𝑟 ∉ 𝐶(𝑝, 𝑆𝑗)
• Use the unbounded module of

𝐶(𝑝, 𝑆𝑗) to find a line ℓ′, we know – Either ℓ′ is an approximate closest line -> done – It is 𝜀𝑗+1-close to ℓ∗

101

Query Processing Algorithm

• Given 𝑟, find the right slab, and

retrieve all candidate lines

• Using binary search find 𝑠
• Find largest 𝑗 such that 𝑟 ∉ 𝐶(𝑝, 𝑆𝑗)
• Use the unbounded module of

𝐶(𝑝, 𝑆𝑗) to find a line ℓ′, we know – Either ℓ′ is an approximate closest line -> done – It is 𝜀𝑗+1-close to ℓ∗

• Use the parallel modules of ℓ′ to

find an approximate closest line. -> done

102

Query Processing Algorithm

• Given 𝑟, find the right slab, and

retrieve all candidate lines

• Using binary search find 𝑠
• Find largest 𝑗 such that 𝑟 ∉ 𝐶(𝑝, 𝑆𝑗)
• Use the unbounded module of

𝐶(𝑝, 𝑆𝑗) to find a line ℓ′, we know – Either ℓ′ is an approximate closest line -> done – It is 𝜀𝑗+1-close to ℓ∗

• Use the parallel modules of ℓ′ to

find an approximate closest line. -> done

103

Summary

• Nearest Line Search Problem

104

Summary

• Nearest Line Search Problem
• Modules: unbounded, net, parallel

105

Summary

• Nearest Line Search Problem
• Modules: unbounded, net, parallel
• Use of random sampling

106

Summary

• Nearest Line Search Problem
• Modules: unbounded, net, parallel
• Use of random sampling
• How to improve given a line

107

Summary

• Nearest Line Search Problem
• Modules: unbounded, net, parallel
• Use of random sampling
• How to improve given a line
• Bounds of our algorithm

– Polynomial Space:

𝑒𝑂 𝜗 𝑃 1

× 𝒯

𝑂 𝜗 𝑃 1

, 𝜗 = 𝑃 𝑂 + 𝑒 𝑃

1 𝜗2

– Poly-logarithmic query time :

𝑒 log 𝑂 𝑃 1 × 𝒰(

𝑂 𝜗 𝑃 1

, 𝜗) = 𝑒 + log 𝑂 +

1 𝜗 𝑃 1

108

Future Work

• The current result is not good in practice

– Large exponents – Algorithm is complicated Can we get a simpler algorithms?

109

Future Work

• The current result is not good in practice

– Large exponents – Algorithm is complicated Can we get a simpler algorithms?

• Generalization to higher dimensional flats

110

Future Work

• The current result is not good in practice

– Large exponents – Algorithm is complicated Can we get a simpler algorithms?

• Generalization to higher dimensional flats
• Generalization to other objects, e.g. balls

111

THANK YOU!

112