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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
16 dimensions. 512 pivots
L2MaxDist
0
100
200
300
400
500
600
700
0 1 2 3 4 5 6 7 8
128 dimensions. 32 pivots
L2MaxDist
0
100
200
300
400
500
600
700
800
0 1 2 3 4 5 6 7 8
128 dimensions. 512 pivots
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
Distancecomputations
[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
Distancecomputations
[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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