Dynamic programming using histomorphisms Jevgeni Kabanov Viinistu, - - PowerPoint PPT Presentation

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Dynamic programming using histomorphisms Jevgeni Kabanov Viinistu, - - PowerPoint PPT Presentation

Oct 22, 2022 435 likes •911 views

Dynamic programming using histomorphisms Jevgeni Kabanov Viinistu, 2005 Catamorphism (fold) Structural recursion combinator Generic foldr (Haskell) Eats (folds) trees from bottom-up, producing combined result Similar to Visitor

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SLIDE 1

Dynamic programming using histomorphisms

Jevgeni Kabanov Viinistu, 2005

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SLIDE 2

Catamorphism (fold)

  • Structural recursion combinator
  • Generic foldr (Haskell)
  • Eats (folds) trees from bottom-up, producing

combined result

  • Similar to Visitor pattern in OOP, but doesn’t

update structures

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SLIDE 3

Sum fold animation

  • 1
  • 5
  • 3
  • 7

Let’s count this tree sum...

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SLIDE 4

Sum fold animation

  • 1
  • 5
  • 3
  • 7
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SLIDE 5

Sum fold animation

  • 1
  • 5
  • 3
  • 7
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SLIDE 6

Sum fold animation

  • 1

5

  • 3
  • 7
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SLIDE 7

Sum fold animation

  • 1
  • 5
  • 3
  • 7
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SLIDE 8

Sum fold animation

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SLIDE 9

Sum fold animation

  • 6
  • 3
  • 7
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SLIDE 10

Sum fold animation

  • 6
  • 3
  • 7
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SLIDE 11

Sum fold animation

  • 6
  • 3
  • 7
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SLIDE 12

Sum fold animation

  • 6
  • 3

7

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SLIDE 13

Sum fold animation

  • 6
  • 3
  • 7
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SLIDE 14

Sum fold animation

  • 6
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SLIDE 15

Sum fold animation

  • 6
  • 10
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SLIDE 16

Sum fold animation

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SLIDE 17

Sum fold animation

  • 16

And the result is 16 = 1 + 5 + 3 + 7

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SLIDE 18

Histomorphism

  • Introduced by Varmo & Tarmo in 1999
  • Course-of-value structural recursion combinator
  • Inspired by dynamic programming technique
  • Moves bottom-up annotating the tree with results
  • Allows to reuse sub(-sub)* node results
  • Finally collapses the tree producing the end result
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SLIDE 19

Funny sum histo animation

  • 1
  • 5
  • 3
  • 7

Let’s count this tree (funny) sum...

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SLIDE 20

Funny sum histo animation

  • 1
  • 5
  • 3
  • 7
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SLIDE 21

Funny sum histo animation

  • 1
  • 5
  • 3
  • 7
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SLIDE 22

Funny sum histo animation

  • 1

5

  • 3
  • 7
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SLIDE 23

Funny sum histo animation

  • 1
  • 5
  • 3
  • 7
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SLIDE 24

Funny sum histo animation

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SLIDE 25

Funny sum histo animation

  • 6
  • 1
  • 5
  • 3
  • 7
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SLIDE 26

Funny sum histo animation

  • 6
  • 1
  • 5

3

  • 7
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SLIDE 27

Funny sum histo animation

  • 6
  • 1
  • 5
  • 3
  • 7
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SLIDE 28

Funny sum histo animation

  • 6
  • 1
  • 5
  • 3

7

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SLIDE 29

Funny sum histo animation

  • 6
  • 1
  • 5
  • 3
  • 7
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SLIDE 30

Funny sum histo animation

  • 6
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SLIDE 31

Funny sum histo animation

  • 6
  • 10
  • 1
  • 5
  • 3
  • 7
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SLIDE 32

Funny sum histo animation

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SLIDE 33

Funny sum histo animation

  • 24
  • 6
  • 10
  • 1
  • 5
  • 3
  • 7
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SLIDE 34

Funny sum histo animation

  • 24

And the result is 24 = 1 × 2 + 5 + 3 + 7 × 2

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SLIDE 35

Generic hylomorphism

  • General recursion combinator
  • 2 stages:
  • 1. Build an intermediate structure using unfold
  • 2. Collapse the intermediate structure using fold
  • The intermediate structure corresponds to the

implicit call tree

  • The intermediate structure does not really have to be

built

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SLIDE 36

Dynamic hylomorphism

  • Dynamic recursion combinator
  • The fold is replaced by the histomorphism

FA

F[ (ϕ) ]

  • A

ϕ

  • [

(ϕ) ]

  • f
  • FµF

in

  • F[

({ | ψ | },in−1) ]

  • µF

{ | ψ | }

  • F(Fν(B))

ψ

B

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SLIDE 37

Challenges

  • Histomorphism expressive power
  • Dynamic hylomorphism expressive power
  • Properties of transformation to dynamic recursion
  • Deriving dynamic definition
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SLIDE 38

Case study

  • Fibonacci numbers
  • Binary partition number
  • Levenshtein (Edit) distance
  • Longest common subsequence
  • Only first two can be defined as pure histomorphisms
  • General recursion is needed
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SLIDE 39

Inspiration

Fibonacci dependency tree n

  • n − 1
  • n − 2
  • n − 2

n − 3 n − 3 n − 4 Collapsed dependency graph n

  • n − 1
  • n − 2
  • n − 3

· · ·

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SLIDE 40

Inspiration (2)

Levenshtein (Edit) distance dependency tree

Dij

  • Di−1j
  • Di−1j−1
  • Dij−1
  • Di−2j Di−2j−1 Di−1j−1 Di−2j−1 Di−2j−2 Di−1j−2 Di−1j−1 Di−1j−2 Dij−2
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SLIDE 41

Inspiration (3)

Levenshtein (Edit) distance collapsed dependency graph Dij

  • Di−1j
  • Dij−1
  • Di−2j
  • Di−1j−1
  • Dij−2
  • Di−3j

Di−2j−1 Di−1j−2 Dij−3

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SLIDE 42

Transformation

  • Original definition: f = ψ ◦ Tf ◦ ϕ
  • Dynamic definition: f = ψ ◦ σ ◦ T′[

(f, in−1) ] ◦ ϕ′ – ϕ′ generates more compact intermediate structure – T′ defines the structure recursive pattern – σ restores one level of the old structure – σ and T′ are uniquely determined by ϕ′

  • The consumer (algebra) part is preserved
  • The producer (coalgebra) part is consistently

updated

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SLIDE 43

Dependency algebra

Let

  • Original dependency producers: hi : A → A
  • Dynamic dependency producers: h′

j : A → A

  • Projections: πi : Tν(C) → Tν(C),

πi = [in, outi ◦ outr] ◦ in−1

  • Deep projections:

π∗

i = outl ◦π′ kl ◦ π′ kl−1 ◦ · · · ◦ π′ k2 ◦ outk1

  • Induction indicator: p : A → Bool
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SLIDE 44

Dependency algebra (2)

Then

  • ϕ = (id + id, h1, h2, . . . , hn) ◦ p?
  • ϕ′ = (id + id, h′

1, h′ 2, . . . , h′ m) ◦ p′?

  • σ = [inl, (out0 +out0, π∗

1, π∗ 2, . . . , π∗ n) ◦ (p ◦ out0)?]

And ϕ′ has to satisfy following for each i ∈ I, each s ∈ S: P(s, i) = k1, k2, . . . , kl ∈ J∗

  • utl ◦πi ◦ [

(id, ϕ) ] = outl ◦π′

kl ◦ π′ kl−1 ◦ · · · ◦ π′ k1 ◦ [

(id, ϕ′) ] hi(s) = h′

kl ◦ h′ kl−1 ◦ · · · ◦ h′ k1(s)

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SLIDE 45

Future work

  • More categorical approach to transformation
  • A solid proof for dependency algebra
  • (Semi)-automatical derivation for restricted cases