Space Complexity of 2-Dimensional Approximate Range Counting Zhewei - - PowerPoint PPT Presentation

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Space Complexity of 2-Dimensional Approximate Range Counting Zhewei Wei and Ke Yi Problem and Results Problem Definition

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SLIDE 1

Space Complexity of 2-Dimensional Approximate Range Counting

Zhewei Wei and Ke Yi

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SLIDE 2

Problem and Results

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SLIDE 3

Problem Definition

  • × ε
slide-4
SLIDE 4

Problem Definition

  • × ε
  • | ∩ | ± ε
slide-5
SLIDE 5

Problem Definition

  • × ε
  • | ∩ | ± ε
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SLIDE 6

1-Dimensional Case

  • (

ε log )

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SLIDE 7

1-Dimensional Case

  • (

ε log )

ε ε ε ε

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SLIDE 8

1-Dimensional Case

  • (

ε log )

  • Ω(

ε log )

ε ε ε ε

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SLIDE 9

1-Dimensional Case

  • (

ε log )

  • Ω(

ε log )

= ε

  • ε

ε ε ε

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SLIDE 10

1-Dimensional Case

  • (

ε log )

  • Ω(

ε log )

= ε

  • ε

ε ε ε

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SLIDE 11

1-Dimensional Case

  • (

ε log )

  • Ω(

ε log )

= ε

  • ε

ε ε ε ε

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SLIDE 12

1-Dimensional Case

  • (

ε log )

  • Ω(

ε log )

= ε

  • ε

ε ε ε ε

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SLIDE 13

1-Dimensional Case

  • (

ε log )

  • Ω(

ε log )

= ε

  • ε

ε ε ε ε

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SLIDE 14

1-Dimensional Case

  • (

ε log )

  • Ω(

ε log )

= ε

  • ε

ε ε ε ε

  • ε
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SLIDE 15

1-Dimensional Case

  • (

ε log )

  • Ω(

ε log )

= ε

  • ε

ε ε ε ε

  • ε

Ω(log

  • ε ) = Ω(

ε log )

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SLIDE 16

(Strong) Epsilon Approximation

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SLIDE 17

(Strong) Epsilon Approximation

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SLIDE 18

(Strong) Epsilon Approximation

∀,

  • | ∩ |

|| − | ∩ | ||

  • ≤ ε.

⇒

  • | ∩ |

|| · − | ∩ |

  • ≤ ε.
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SLIDE 19

(Strong) Epsilon Approximation

∀,

  • | ∩ |

|| − | ∩ | ||

  • ≤ ε.

⇒

  • | ∩ |

|| · − | ∩ |

  • ≤ ε.
  • (

ε log. ε) = ( ε log. ε log )

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SLIDE 20

(Weak) Epsilon Approximation

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SLIDE 21

(Weak) Epsilon Approximation

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SLIDE 22

(Weak) Epsilon Approximation

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SLIDE 23

(Weak) Epsilon Approximation

  • ∀,
  • | ∩ |

|| − | ∩ | ||

  • ≤ ε.

⇒

  • | ∩ |

|| · − | ∩ |

  • ≤ ε.
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SLIDE 24

(Weak) Epsilon Approximation

  • ∀,
  • | ∩ |

|| − | ∩ | ||

  • ≤ ε.

⇒

  • | ∩ |

|| · − | ∩ |

  • ≤ ε.
  • (

ε log. ε) = ( ε log. ε log )

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SLIDE 25

Our Results

  • ε
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SLIDE 26

Our Results

  • (

ε log log ε log ε log )

  • ε
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SLIDE 27

Our Results

  • (

ε log log ε log ε log )

log.

ε ε

  • ε
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SLIDE 28

Our Results

  • (

ε log log ε log ε log )

  • Ω( log ) log

log.

ε ε

  • ε
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SLIDE 29

Our Results

  • (

ε log log ε log ε log )

  • Ω( log ) log

Ω(

ε log ε log ) ε = log

log.

ε ε

  • ε
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SLIDE 30

Our Results

  • (

ε log log ε log ε log )

  • Ω( log ) log

Ω(

ε log ε log ) ε = log

log

  • log.

ε ε

  • ε
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SLIDE 31

Preliminaries

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SLIDE 32

Combinatorial Discrepancy

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SLIDE 33

Combinatorial Discrepancy

  • (, R) χ : →

{−, +}

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SLIDE 34

Combinatorial Discrepancy

  • (, R) χ : →

{−, +}

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SLIDE 35

Combinatorial Discrepancy

  • χ( ∩ )=
  • ∈∩

χ(); disc(, R)=min

χ max ∈R |χ( ∩ )| ;

disc(, R)=max

||= disc(, R).

  • (, R) χ : →

{−, +}

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SLIDE 36

Combinatorial Discrepancy

  • χ( ∩ )=
  • ∈∩

χ(); disc(, R)=min

χ max ∈R |χ( ∩ )| ;

disc(, R)=max

||= disc(, R).

  • (, R) χ : →

{−, +}

  • (log. ) Ω(log )
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SLIDE 37

Lebesgue Discrepancy

  • (, R) [, )
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SLIDE 38

Lebesgue Discrepancy

  • (, R) [, )

(, R) = sup∈R

  • | ∩ | −
  • ∩ [, )
  • .
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SLIDE 39

Lebesgue Discrepancy

  • (, R) [, )

(, R) = sup||= (, R). (, R) = sup∈R

  • | ∩ | −
  • ∩ [, )
  • .
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SLIDE 40

Lebesgue Discrepancy

  • (, R) [, )
  • Θ(log )

(, R) = sup||= (, R). (, R) = sup∈R

  • | ∩ | −
  • ∩ [, )
  • .
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SLIDE 41

(Strong) Epsilon Net

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SLIDE 42

(Strong) Epsilon Net

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SLIDE 43

(Strong) Epsilon Net

  • | ∩ | ≥ ε

| ∩ | ≥

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SLIDE 44

(Strong) Epsilon Net

  • | ∩ | ≥ ε

| ∩ | ≥

  • Θ(

ε log log ε)

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SLIDE 45

(Weak) Epsilon Net

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SLIDE 46

(Weak) Epsilon Net

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SLIDE 47

(Weak) Epsilon Net

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SLIDE 48

(Weak) Epsilon Net

  • | ∩ | ≥ ε

| ∩ | ≥

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SLIDE 49

(Weak) Epsilon Net

  • | ∩ | ≥ ε

| ∩ | ≥

  • (

ε log log ε)

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SLIDE 50

Upper Bound

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SLIDE 51

Data Structure

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SLIDE 52

Data Structure

  • ε (

ε log log ε)

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SLIDE 53

Data Structure

  • ε (

ε log log ε)

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SLIDE 54

Data Structure

  • ε (

ε log log ε)

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SLIDE 55

Data Structure

  • ε (

ε log log ε)

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SLIDE 56

Data Structure

  • ε (

ε log log ε)

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SLIDE 57

Data Structure

  • ε (

ε log log ε)

≥ ε ≥ ε

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SLIDE 58

Data Structure

  • ε (

ε log log ε)

  • ≥ ε

≥ ε

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SLIDE 59

Data Structure

  • ε (

ε log log ε)

ε (log

ε)

  • ≥ ε

≥ ε

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SLIDE 60

Data Structure

  • ε (

ε log log ε)

ε (log

ε)

  • (

ε log log ε(log + log ε))

  • ≥ ε

≥ ε

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SLIDE 61

Data Structure

  • ε (

ε log log ε)

ε (log

ε)

  • (

ε log log ε(log + log ε))

  • ≥ ε

≥ ε

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SLIDE 62

Data Structure

  • ε (

ε log log ε)

ε (log

ε)

  • (

ε log log ε(log + log ε))

  • ≥ ε

≥ ε = ε

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SLIDE 63

Data Structure

  • ε (

ε log log ε)

ε (log

ε)

  • (

ε log log ε(log + log ε))

ε

  • ≥ ε

≥ ε = ε

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SLIDE 64

Data Structure

  • ε (

ε log log ε)

ε (log

ε)

  • (

ε log log ε(log + log ε))

ε ε log

ε

  • ≥ ε

≥ ε = ε

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SLIDE 65

Data Structure

  • ε (

ε log log ε)

ε (log

ε)

  • (

ε log log ε(log + log ε))

ε ε log

ε

ε =

ε log

ε ⇒ (

ε log log ε log ε log )

  • ≥ ε

≥ ε = ε

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SLIDE 66

Lower Bound

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SLIDE 67

Data Structure to CD

  • Ω( log ) log
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SLIDE 68

Data Structure to CD

  • Ω( log ) log
  • P∗ Ω( log )

∀ , ∈ P∗ disc( ∪ , R) ≥ log

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SLIDE 69

Data Structure to CD

  • Ω( log ) log
  • P∗ Ω( log )

∀ , ∈ P∗ disc( ∪ , R) ≥ log min

χ max ∈R |χ(( ∪ ) ∩ )|≥ log

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SLIDE 70

Data Structure to CD

  • Ω( log ) log

χ() =

  • ∈ ;

− ∈ .

  • P∗ Ω( log )

∀ , ∈ P∗ disc( ∪ , R) ≥ log min

χ max ∈R |χ(( ∪ ) ∩ )|≥ log

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SLIDE 71

Data Structure to CD

  • Ω( log ) log

χ() =

  • ∈ ;

− ∈ .

  • P∗ Ω( log )

∀ , ∈ P∗ disc( ∪ , R) ≥ log min

χ max ∈R |χ(( ∪ ) ∩ )|≥ log

  • || ∩ | − | ∩ || ≥ log
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SLIDE 72

Point Sets

  • P∗ Ω( log )

∀ , ∈ P∗ disc( ∪ , R) ≥ log

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SLIDE 73

Point Sets

  • P∗ Ω( log )

∀ , ∈ P∗ disc( ∪ , R) ≥ log

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SLIDE 74

Point Sets

  • P∗ Ω( log )

∀ , ∈ P∗ disc( ∪ , R) ≥ log

(, R) = (√ log log )

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SLIDE 75

Point Sets

  • P∗ Ω( log )

∀ , ∈ P∗ disc( ∪ , R) ≥ log

(, R) = (√ log log ) ε = Ω(

  • log log
  • )
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SLIDE 76

Point Sets

  • P∗ Ω( log )

∀ , ∈ P∗ disc( ∪ , R) ≥ log

  • {, }

(, R) = (√ log log ) ε = Ω(

  • log log
  • )
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SLIDE 77

Point Sets

  • P∗ Ω( log )

∀ , ∈ P∗ disc( ∪ , R) ≥ log

  • {, }

() (, R) = (√ log log ) ε = Ω(

  • log log
  • )
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SLIDE 78

Binary Nets

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SLIDE 79

Binary Nets

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SLIDE 80

Binary Nets

(, R) = (log )

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SLIDE 81

Binary Nets

(, R) = (log )

log

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SLIDE 82

Binary Nets

(, R) = (log )

log

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SLIDE 83

Binary Nets

(, R) = (log ) disc(, R) = Ω(log )

log

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SLIDE 84

Canonical Cells

=

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SLIDE 85

Binary Nets

  • =
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SLIDE 86

Binary Nets

  • =
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SLIDE 87

Binary Nets

  • =
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SLIDE 88

Binary Nets

  • =
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SLIDE 89

Binary Nets

  • =
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SLIDE 90

Binary Nets

  • =
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SLIDE 91

Binary Nets

(, R) = (log ) disc(, R) = Ω(log )

log

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SLIDE 92

Binary Nets

(, R) = (log ) disc(, R) = Ω(log )

log

  • log
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SLIDE 93

Number of Binary Nets

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SLIDE 94

Number of Binary Nets

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SLIDE 95

Number of Binary Nets

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SLIDE 96

Number of Binary Nets

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SLIDE 97

Number of Binary Nets

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SLIDE 98

Number of Binary Nets

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SLIDE 99

Number of Binary Nets

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SLIDE 100

Number of Binary Nets

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SLIDE 101

Number of Binary Nets

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SLIDE 102

Number of Binary Nets

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SLIDE 103

Number of Binary Nets

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SLIDE 104

Number of Binary Nets

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SLIDE 105

Number of Binary Nets

slide-106
SLIDE 106

Number of Binary Nets

slide-107
SLIDE 107

Number of Binary Nets

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SLIDE 108

Number of Binary Nets

slide-109
SLIDE 109

Number of Binary Nets

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SLIDE 110

Number of Binary Nets

  • log ⇒
  • log
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SLIDE 111

Binary Nets

(, R) = (log )

log

disc(, R) = Ω(log )

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SLIDE 112

Corner Volume

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SLIDE 113

Corner Volume

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SLIDE 114

Corner Volume

slide-115
SLIDE 115

Corner Volume Distance

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SLIDE 116

Corner Volume Distance

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SLIDE 117

Corner Volume Distance

  • log
slide-118
SLIDE 118

Corner Volume

≥ log disc() = Ω(log )

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SLIDE 119

Corner Volume

Ω() ≥ log disc() = Ω(log )

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SLIDE 120

Corner Volume

Ω() ≥ log disc() = Ω(log )

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SLIDE 121

Corner Volume

  • Ω()

≥ log disc() = Ω(log )

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SLIDE 122

Corner Volume

  • Ω()

≥ log disc() = Ω(log )

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SLIDE 123

Large Corner Volume

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SLIDE 124

Large Corner Volume

slide-125
SLIDE 125

Large Corner Volume

  • ≥

∗

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SLIDE 126

Large Corner Volume

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SLIDE 127

Large Corner Volume

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SLIDE 128

Large Corner Volume

  • ≥

∗

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SLIDE 129

Large Corner Volume

  • ≥

∗ ≥ ∗

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SLIDE 130

Large Corner Volume

  • ≥
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SLIDE 131

Large Corner Volume

  • log ⇒ ≥ log
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SLIDE 132

P∗ Ω( log ) ∀ , ∈ P∗ disc( ∪ , R) ≥ log

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SLIDE 133

CD of Union of 2 Binary Sets

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SLIDE 134

CD of Union of 2 Binary Sets

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SLIDE 135

CD of Union of 2 Binary Sets

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SLIDE 136

Corner Volume Distance

slide-137
SLIDE 137

Corner Volume Distance

slide-138
SLIDE 138

Corner Volume Distance

slide-139
SLIDE 139

Corner Volume Distance

slide-140
SLIDE 140

Corner Volume Distance

slide-141
SLIDE 141

Corner Volume Distance

≥ log disc(∪ ) = Ω(log )

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SLIDE 142

P∗ Ω( log ) ∀ , ∈ P∗ ∆(, ) ≥ log

slide-143
SLIDE 143

Binary Nets with Large CVD

slide-144
SLIDE 144

Binary Nets with Large CVD

slide-145
SLIDE 145

Binary Nets with Large CVD

slide-146
SLIDE 146

Binary Nets with Large CVD

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SLIDE 147

Binary Nets with Large CVD

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SLIDE 148

Binary Nets with Large CVD

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SLIDE 149

Binary Nets with Large CVD

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SLIDE 150

Binary Nets with Large CVD

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SLIDE 151

Binary Nets with Large CVD

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SLIDE 152

Binary Nets with Large CVD

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SLIDE 153

Binary Nets with Large CVD

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SLIDE 154

Binary Nets with Large CVD

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SLIDE 155

Binary Nets with Large CVD

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SLIDE 156

Binary Nets with Large CVD

∆ ≥

  • =
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SLIDE 157

Binary Nets with Large CVD

∆ ≥

  • =
  • · · ·

−

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SLIDE 158

Binary Nets with Large CVD

∆ ≥

  • =
  • ∆

≥

  • =
  • · · ·

−

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SLIDE 159

Binary Nets with Large CVD

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SLIDE 160

Binary Nets with Large CVD

slide-161
SLIDE 161

Binary Nets with Large CVD

slide-162
SLIDE 162

Binary Nets with Large CVD

  • · · ·
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SLIDE 163

Binary Nets with Large CVD

  • · · ·
slide-164
SLIDE 164

Binary Nets with Large CVD

  • · · ·
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SLIDE 165

Binary Nets with Large CVD

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SLIDE 166

Binary Nets with Large CVD

  • + =
  • ∆

+ ≥

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SLIDE 167

Binary Nets with Large CVD

  • · · ·

(, ) ( − , −

)

+ =

  • ∆

+ ≥

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SLIDE 168

Binary Nets with Large CVD

  • · · ·

(, ) ( − , −

)

+ =

  • ∆

+ ≥

  • +

=

  • ∆

+ ≥

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SLIDE 169

Binary Nets with Large CVD

∆ ≥

  • =
  • · · ·

· log

  • =
  • ∆

· log

  • ≥
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SLIDE 170

Binary Nets with Large CVD

∆ ≥

  • =
  • · · ·

· log

  • =
  • ∆

· log

  • ≥
  • Z = (, . . . , log

)

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SLIDE 171

Binary Nets with Large CVD

∆ ≥

  • =
  • · · ·

· log

  • =
  • ∆

· log

  • ≥
  • Z = (, . . . , log

)

Z

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SLIDE 172

Binary Nets with Large CVD

∆ ≥

  • =
  • · · ·

· log

  • =
  • ∆

· log

  • ≥
  • {, }

log

  • |P| =

log

  • Z = (, . . . , log

)

Z

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SLIDE 173

Binary Nets with Large CVD

∆ ≥

  • =
  • · · ·

(Z(), Z()) ≥ log ∆(, ) ≥

log

· log

  • =
  • ∆

· log

  • ≥
  • {, }

log

  • |P| =

log

  • Z = (, . . . , log

)

Z

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SLIDE 174

Binary Nets with Large CVD

∆ ≥

  • =
  • · · ·

(Z(), Z()) ≥ log ∆(, ) ≥

log

· log

  • =
  • ∆

· log

  • ≥
  • {, }

log

  • |P| =

log

  • {, }

log

  • Z = (, . . . , log

)

Z

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SLIDE 175

Binary Nets with Large CVD

  • =

log ∃Z ⊂ {, } |Z| = Ω()

∀Z, Z ∈ Z (Z, Z) = Ω()

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SLIDE 176

Binary Nets with Large CVD

Z Z Z

(Z, Z) <

  • Z
  • =

log ∃Z ⊂ {, } |Z| = Ω()

∀Z, Z ∈ Z (Z, Z) = Ω()

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SLIDE 177

Binary Nets with Large CVD

Z Z Z

(Z, Z) <

  • Z
  • =

log ∃Z ⊂ {, } |Z| = Ω()

∀Z, Z ∈ Z (Z, Z) = Ω()

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SLIDE 178

Binary Nets with Large CVD

Ω() Z Z Z

(Z, Z) <

  • Z
  • =

log ∃Z ⊂ {, } |Z| = Ω()

∀Z, Z ∈ Z (Z, Z) = Ω()

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SLIDE 179

Binary Nets with Large CVD

Z Z Ω() Z Z Z

(Z, Z) <

  • Z
  • =

log ∃Z ⊂ {, } |Z| = Ω()

∀Z, Z ∈ Z (Z, Z) = Ω()

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SLIDE 180

Binary Nets with Large CVD

Z Z (Z, Z) Ω() Z Z Z

(Z, Z) <

  • Z
  • =

log ∃Z ⊂ {, } |Z| = Ω()

∀Z, Z ∈ Z (Z, Z) = Ω()

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SLIDE 181

Binary Nets with Large CVD

Z Z (Z, Z) Pr[(Z, Z) <

] ≤ −

≤ −

Ω() Z Z Z

(Z, Z) <

  • Z
  • =

log ∃Z ⊂ {, } |Z| = Ω()

∀Z, Z ∈ Z (Z, Z) = Ω()

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SLIDE 182

Binary Nets with Large CVD

Z Z (Z, Z) Pr[(Z, Z) <

] ≤ −

≤ −

Ω() (Z) ≤

  • ⇒
  • Z

Z Z

(Z, Z) <

  • Z
  • =

log ∃Z ⊂ {, } |Z| = Ω()

∀Z, Z ∈ Z (Z, Z) = Ω()

slide-183
SLIDE 183
  • ∃P∗ ⊂ P Ω( log ) ∀ , ∈

P∗ ∆(, ) ≥ log

slide-184
SLIDE 184

Conclusion and Open Problems

slide-185
SLIDE 185

Conclusion and Open Problems

  • (

ε log log ε log ε log )

slide-186
SLIDE 186

Conclusion and Open Problems

  • (

ε log log ε log ε log )

log.

ε ε

slide-187
SLIDE 187

Conclusion and Open Problems

  • (

ε log log ε log ε log )

  • Ω( log ) log

log.

ε ε

slide-188
SLIDE 188

Conclusion and Open Problems

  • (

ε log log ε log ε log )

  • Ω( log ) log

Ω(

ε log ε log ) ε = log

log.

ε ε

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SLIDE 189

Conclusion and Open Problems

  • (

ε log log ε log ε log )

  • Ω( log ) log

Ω(

ε log ε log ) ε = log

log

  • log.

ε ε

slide-190
SLIDE 190

Conclusion and Open Problems

  • (

ε log log ε log ε log )

  • Ω( log ) log

Ω(

ε log ε log ) ε = log

log

  • log.

ε ε

slide-191
SLIDE 191

Conclusion and Open Problems

slide-192
SLIDE 192

Conclusion and Open Problems

  • ε (/ε)
slide-193
SLIDE 193

Conclusion and Open Problems

  • ε (/ε)

Ω(

ε log log ε)

ε

slide-194
SLIDE 194

Conclusion and Open Problems

  • ε (/ε)

Ω(

ε log log ε)

ε ε (/ε)

slide-195
SLIDE 195

Conclusion and Open Problems

  • ε (/ε)

Ω(

ε log log ε)

ε ε (/ε) ε Ω(

ε log log ε)

slide-196
SLIDE 196

Conclusion and Open Problems

  • ε (/ε)

Ω(

ε log log ε)

ε ε (/ε) ε Ω(

ε log log ε)

slide-197
SLIDE 197

Conclusion and Open Problems

  • ε (/ε)

Ω(

ε log log ε)

ε ε (/ε) ε Ω(

ε log log ε)

slide-198
SLIDE 198

Conclusion and Open Problems

  • ε (/ε)

Ω(

ε log log ε)

ε ε (/ε) ε Ω(

ε log log ε)

(

ε log ε)

slide-199
SLIDE 199

Thank you!