SLIDE 1
Gödel Hashing
matt.might.net @mattmight
Disclaimer
Gdel Hashing matt.might.net @mattmight Disclaimer simple, fun - - PowerPoint PPT Presentation
Gdel Hashing matt.might.net @mattmight Disclaimer simple, fun - - PowerPoint PPT Presentation
Gdel Hashing matt.might.net @mattmight Disclaimer simple, fun idea simple, fun idea works well in practice, simple, fun idea works well in practice, but theory says it will not. An old problem An
SLIDE 2
SLIDE 3
“simple, fun idea”
SLIDE 4
“simple, fun idea” “works well in practice,”
SLIDE 5
“simple, fun idea” “works well in practice,” “but theory says it will not.”
SLIDE 6
An old problem An older solution A big impact
SLIDE 7
An old problem
SLIDE 8
“CFA is slow!”
SLIDE 9
An older solution
SLIDE 10
SLIDE 11
functional monotonic perfect compact dynamic incremental
Gödel hashing
Inspired by a true theorem.
SLIDE 12
Word-level parallelism!
SLIDE 13
Great cache behavior!
SLIDE 14
A big impact
SLIDE 15
Minutes of work
SLIDE 16
2x
SLIDE 17
2x5x
SLIDE 18 2x5x8x
SLIDE 19 2x5x8x100x
SLIDE 20
Motivation
SLIDE 21
(f x)
SLIDE 22
f(x)
SLIDE 23
What is ?
f
SLIDE 24
Why not run the program?
SLIDE 25
e
SLIDE 26
e
SLIDE 27
e
SLIDE 28
e
What is f, here?
SLIDE 29
e
What is f, here?
SLIDE 30
...
e
SLIDE 31
...
e
AAM
SLIDE 32
e
...
SLIDE 33
e
SLIDE 34
e
SLIDE 35
Problem
SLIDE 36
SLIDE 37 ˆ ς1
SLIDE 38
SLIDE 39
SLIDE 40
v
SLIDE 41
v
SLIDE 42
v
ˆ ς1
ˆ ς2
SLIDE 43
ˆ ς1 = (e, ˆ
ρ, ˆ σ, ˆ κ)
ˆ σ
SLIDE 44
ˆ ς1
expression environment store stack
= (e, ˆ ρ, ˆ σ, ˆ κ)
ˆ σ
SLIDE 45
ˆ ς1
expression environment store stack
= (e, ˆ ρ, ˆ σ, ˆ κ)
ˆ σ
SLIDE 46
ˆ σ : [
Addr → P(\ Value) [ Addr (\ Value)
SLIDE 47
[ Addr (\ Value)
SLIDE 48
SLIDE 49
SLIDE 50 (\ Value)
[ Addr
SLIDE 51
SLIDE 52
SLIDE 53
SLIDE 54
SLIDE 55
SLIDE 56
SLIDE 57
SLIDE 58
SLIDE 59
SLIDE 60
First: Hash sets
SLIDE 61
Prime decomposition
SLIDE 62
SLIDE 63
SLIDE 64
Primes
p1 p2 p3 p4 SLIDE 65
Primes
p1 p2 p3 p4 SLIDE 66 p3 p4
SLIDE 67
p3 p4
SLIDE 68
p3 p4
×
SLIDE 69
{ } ,
SLIDE 70
{ } ,
SLIDE 71
⊆ A B
SLIDE 72
A B [ [ ] ] [ [ ] ] mod = 0
SLIDE 73
A B ∩
SLIDE 74
A B [ [ ] ] [ [ ] ] ( , ) gcd
SLIDE 75
A B [ [ ] ] [ [ ] ] lcm( , )
SLIDE 76
A B ∪
SLIDE 77
A B ∪ A
SLIDE 78
A B − A
SLIDE 79
A B [ [ ] ] [ [ ] ] ( , ) gcd A / [ [ ] ]
SLIDE 80
⊆
SLIDE 81
SLIDE 82
> ⊥
SLIDE 83
> ⊥
prime basis
SLIDE 84
n = p1 p2 p3
m1 m2 m3 . . .
SLIDE 85
n = { } , , G
SLIDE 86
⊥
SLIDE 87
⊥
SLIDE 88
⊥
SLIDE 89
⊥
SLIDE 90
⊥
SLIDE 91
⊥
SLIDE 92
⊥
SLIDE 93
SLIDE 94 p1 p3 p4
SLIDE 95 p1 p3 p4
SLIDE 96
t x y
SLIDE 97
[ [ ] ] [ [ ] ] lcm( , ) x y
SLIDE 98
v x y
SLIDE 99
[ [ ] ] [ [ ] ] mod = 0 x y
SLIDE 100
u x y
SLIDE 101
[ [ ] ] [ [ ] ] ( , ) gcd x y
SLIDE 102
But, does it work for CFA?
SLIDE 103
ˆ σ : [
Addr → P(\ Value) [ Addr (\ Value)
SLIDE 104
[ Addr (\ Value)
SLIDE 105
{ } { } ˆ a1 ˆ a2 , , ˆ v1 ˆ v2 ˆ a1 ˆ a2 ˆ v1 ˆ v2 { } { }
SLIDE 106
{ } { } ˆ a1 ˆ a2 ˆ v1 ˆ v2 ˆ a1 ˆ a2 ˆ v1 ˆ v2 { } { } [ [ [ [ ] ] ] ] 7! 7! 7! 7!
SLIDE 107
has a prime basis.
L1 L2 has a prime basis.
SLIDE 108
has a prime basis.
L1 L2 has a prime basis. ×
SLIDE 109
has a prime basis.
L1 L2 has a prime basis. ×
SLIDE 110
has a prime basis.
L2 has a prime basis. X →
SLIDE 111
What else?
SLIDE 112
{ [ [ ] ] } a b ,
n m
[ [ ] ]
SLIDE 113
[ [ ] ] a b
n m
[ [ ] ]
SLIDE 114
⊆ A B
SLIDE 115
A B [ [ ] ] [ [ ] ] mod = 0
SLIDE 116
A B ∪
SLIDE 117
A B [ [ ] ] [ [ ] ] ×
SLIDE 118
h i [ [ [ [ ] ] ] ] a b c , , [ [ ] ]
SLIDE 119
a
b c
[ [ ]
]
[ [ ]
]
[ [ ]
]
p1 p2 p3
SLIDE 120
Wait a minute...
SLIDE 121
gcd is O(n2)
SLIDE 122
is O(n2)
mod
SLIDE 123
How is this more efficient?
SLIDE 124
SLIDE 125
SLIDE 126
are sparse. Flow sets
SLIDE 127
99% of flow sets: < 5 values
SLIDE 128
Median flow set: 2 values
SLIDE 129
Primes are dense.
SLIDE 130
U
SLIDE 131
U lnU
SLIDE 132
1,000,000 abstract values?
SLIDE 133
23 bit prime
SLIDE 134
Most flow sets fit in a word. Most
SLIDE 135
Most of the time, . Most
n = 1
SLIDE 136
If not, great locality.
SLIDE 137
4-6%
SLIDE 138 2x5x8x100x
SLIDE 139
Programming is about making choices.
SLIDE 140
E 3 ‘s E
SLIDE 141
Elegance Efficacy Efficiency E E E
SLIDE 142
Pick any two Programmers: two
SLIDE 143
Functional Programmers: Pick any two three
SLIDE 144
Questions?
SLIDE 145
Algebraic data types?
SLIDE 146
deriving (Hashable)
SLIDE 147
SLIDE 148
p1 p2 p3 p4 p5
SLIDE 149
p1 p2 p3 p4 p5
SLIDE 150
SLIDE 151