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-

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L

L

L

L1 2

6

u10

u13u5

u4

u11

u2

u12

u3

u7

u1

u15

u14

u6

u8

u9

q

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u13

u4

u11

u2

u12

u3

u7

u1

u15

u14

u6

u8

u10

u5

u9

u11

2 3 4 5 6

u2 u9 u8u3 u5 u12 u13u1 u10u4 u6 u7 u14 u15q

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BKT u11

2 3 4 5 6

u2 u9 u8u1 u10u7

3 4

u6

u5

5 6 7

u13 u3 u12u14

2 3

u15 u4

0

u11

4

2

u7

3

2

u14

4

u10

3

6

2

u15

3

2

u4

5

2

u6

5

u1

3

3

4

u5

0

4

u13

5

4

u3

6

4

u12

7

5

u2

7

5

u9

7

6

u8

7

3 6543 7 7

FQHT

0 2 3 4 5 6

u5

u11

u11

u5

2 3 4 5 6

u2 u9 u8u1 u10

3 4 5

u7 u15 u14 u4 u6

FQT

0

4

u12u3u13

7650

u5

u11

7650

FQA

u14 u4 u6 u2 u9 u8u7 u15u11 u12u3u13u5u10u1

-

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u13

u4

u11

u2

u12

u3

u7

u1

u15

u14

u6

u8

u9

u11 VPT

u7 u9

u15

> 2.9 > 4.0

> 3.1

u14 u4

u6

u10 u1

u8

u13 u2

u3

u12 u6

u5

u10

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u2 u5

u4 u6 u12 u10 u9 u8 u3 u7 u11 u15 u14 u1 u13

u10

u13u5

u2

u12

u3

u7

u1

u15

u6

u8

u9u14

q

u11

u4

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u10

u13u5

u4

u2

u12

u3

u7

u1

u15

u14

u8

u9

u2 u5 u3 u9

u10 u12 u6 u14 u13 u8u4u7 u11 u15 u1

u11

u6

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-

u13

u4

u2

u12

u3

u7

u15

u6

u8

u9u14

u11

u1q

u5

u10

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u10

u13u5

u4

u11

u2

u12

u7

u1

u15

u14

u6

u8

u9

q

u3

-

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Index

Equivalence classes

Data

Query q

Traverse index

(internal

complexity)

q

Search in candidate classes

(external complexity)

Indexing Querying

-

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D([x],[y])

y

d(x,y)

x

p

[y]

[x]

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u13

u4

u11

u2

u12

u3

u7

u1

u15

u14

u6

u8

u8

u11

u3

u7u4

u8

u9

u10

u11

u6

u15

q

u5

u5

q

u10

u9u13

u1

u14

u2

u12

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u1

u2

q

q

a1

b1

a2

b2

d(x,u1)

d(x,u2)

a2

b1a1

b2

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u10

u13c1

u4

c3

u12

c2

u7

u1

u15

u14c4

u6

u8

q1

u11

q2

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2r

d(p,q)

2r

d(p,q)

d(p,x) d(p,x)

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0

100

200

300

400

500

600

700

800

900

1000

0 50 100 150 200 250 300

Distanc

eComputations

[pivots] 100,000 elements. Radius captures 0.01% of N

8 [ext]8 [in+ext]

8 [in]

0

10000

20000

30000

40000

50000

60000

70000

80000

90000

100000

4 6 8 10 12 14 16 18 20

Distancecomputations

[dimension] 100,000 elements. Radius captures 0.01% of N

k = 4 pivotsk = 8 pivots

k = 16 pivotsk = 32 pivotsk = 64 pivots

k = 128 pivotsk = 256 pivots

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0

50

100

150

200

250

0 1 2 3 4 5 6 7 8

16 dimensions. 32 pivots

L2MaxDist

0

50

100

150

200

250

0 1 2 3 4 5 6 7 8


L2MaxDist

0

100

200

300

400

500

600

700

0 1 2 3 4 5 6 7 8


L2MaxDist

0

100

200

300

400

500

600

700

800

0 1 2 3 4 5 6 7 8


L2MaxDist

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0

10000

20000

30000

40000

50000

60000

70000

80000

90000

100000

4 6 8 10 12 14 16 18 20

DistanceComputations

[dimension] 100,000 elements. Radius captures 0.01% of N

m = 2m = 4m = 8m = 16m = 32m = 64m = 128m = 256

0

10000

20000

30000

40000

50000

60000

70000

80000

90000

100000

4 6 8 10 12 14 16 18 20


[dimension] 100,000 elements. Radius captures 0.01% of n

k=4 pivotsk=16 pivotsk=64 pivots

Voronoi m=128

0

10000

20000

30000

40000

50000

60000

70000

80000

90000

100000

4 6 8 10 12 14 16 18 20


[dimension] 100,000 elements. Radius captures 0.10% of n

k=4 pivotsk=16 pivotsk=64 pivots

Voronoi m=128

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-

-

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BKT

Coarsified

Arrays

FHQAFMVPA

FHQT

FQT

FMVPT

VPFGHT Covering Radius

BST VT MT

SAT

Voronoi Type Pivot-Based

Trees

MVPT

LAESA-like

AESA

Fixed Height

Scope Coarsened

Indexing Algorithms

GNATHyperplane

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u1

u2

un

p3p2p1

d(ui,pj)

Bucket CoarseningRange Coarsening

of cell values

Restricted domain

Original

Last coordinates

not always computed

Pivots have

only local scope

Scope Coarsening

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Uniform percentilesUniform width

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examen ed2 2

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