Gdel Hashing matt.might.net @mattmight Disclaimer simple, fun - PowerPoint PPT Presentation

Gödel 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 older solution A big impact

An old problem

“CFA is slow!”

An older solution

Gödel hashing functional monotonic compact dynamic incremental perfect Inspired by a true theorem.

Word-level parallelism!

Great cache behavior!

A big impact

Minutes of work

2x

2x 5x

2x 5x 8x

2x 5x 8x 100x

Motivation

(f x)

f(x)

What is ? f

Wh y not run the program?

e

e

e

e What is f , here ?

e What is f , here ?

e ...

e AAM ...

e ...

e

e

Problem

� � � � � � � � � � � � � � � � � � � � � � � � � � � � ˆ � � ς 1 � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �

� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �

� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �

v

v

ˆ ˆ ς 2 ς 1 v

ς 1 = ( e, ˆ ρ , ˆ σ , ˆ κ ) ˆ ˆ σ

expression store = ( e, ˆ ρ , ˆ σ , ˆ κ ) ˆ ˆ σ ς 1 environment stack

expression store = ( e, ˆ ρ , ˆ σ , ˆ κ ) ˆ ˆ σ ς 1 environment stack

σ : [ Addr → P ( \ [ ( \ ˆ Value ) Addr Value )

( \ Value ) Addr [

[ ( \ Addr Value )

First: Hash sets

Prime decomposition

Primes p 1 p 3 p 2 p 4

Primes p 1 p 3 p 2 p 4

p 3 p 4

p 3 p 4

p 3 p 4 ×

} { ,

} { ,

⊆ A B

[ [ ] ] [ [ ] ] = 0 mod A B

A B ∩

( [ [ ] ] [ [ ] ] ) gcd A B ,

lcm ( [ [ ] ] [ [ ] ] ) A B ,

A B ∪

A A B ∪

A A B −

[ [ A / ] ] ( [ [ ] ] [ [ ] ] ) gcd A B ,

⊆

> ⊥

> prime basis ⊥

m 1 m 2 m 3 . . . n = p 1 p 2 p 3

G } { n = , ,

⊥

⊥

⊥

⊥

⊥

⊥

⊥

p 1 p 3 p 4

p 1 p 3 p 4

t y x

lcm ( [ [ ] ] [ [ ] ] ) y x ,

v y x

[ [ ] ] [ [ ] ] = 0 mod x y

u x y