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e63eb7f | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 | import OAI.Combinatorics.EditApproximation.AlmostLinearConditional
import OAI.Combinatorics.EditApproximation.Probability.BernsteinFinite
namespace OAI
/-! A uniform binary approximation scheme for edit distance.
The finite Bernstein inequality supplies the concentration bounds for the
initial estimator and the refinement samples. -/
namespace EditApproximation
open MeasureTheory Filter
/-- Finite product-measure concentration for the initial estimator and refinement. -/
theorem computedBernsteinInputs (N : ℕ) : ComputedBernsteinInputs N := by
constructor
· exact finiteBernsteinInput _
· intro b h
exact finiteBernsteinInput _
/-- A uniform almost-linear approximation scheme. The same binary algorithm
satisfies the approximation, equality, expected-work and space bounds. -/
theorem almostLinearEditDistance (epsilon : BinaryFraction)
(hepsilon : 0 < epsilon.value) (_hepsilonOne : epsilon.value < 1)
:
(∀ (source target : List ℤ) (N : ℕ), source.length + target.length ≤ N →
(2 / 3 : ℝ) ≤ (computedFullLaw (source.map integerSymbolCode) (target.map integerSymbolCode) N epsilon).real
{outcome | EditDistortion.edit source target ≤ integerBinaryAnswer source target N epsilon outcome ∧
(integerBinaryAnswer source target N epsilon outcome : ℝ) ≤
(1 + (epsilon.value : ℝ)) * EditDistortion.edit source target}) ∧
(∀ (source target : List ℤ) (N : ℕ), source = target → ∀ outcome,
integerBinaryAnswer source target N epsilon outcome = 0) ∧
(∀ (C : ℕ) (eta : ℝ), 0 < eta → ∀ᶠ N : ℕ in atTop,
∀ source target : List ℤ, source.length + target.length ≤ N →
(∀ z ∈ source ++ target, z.natAbs ≤ (N + 2) ^ C) →
(∫ outcome, integerBinaryWork source target N epsilon outcome
∂computedFullLaw (source.map integerSymbolCode) (target.map integerSymbolCode) N epsilon) ≤
(N + 2 : ℝ) ^ (1 + eta)) ∧
(∀ C : ℕ, ∀ᶠ N : ℕ in atTop, ∀ source target : List ℤ,
source.length + target.length ≤ N →
(∀ z ∈ source ++ target, z.natAbs ≤ (N + 2) ^ C) →
∀ outcome, integerBinarySpace source target N epsilon outcome ≤ (N + 2) ^ (C + 65564)) := by
exact almostLinearEditDistance_conditional epsilon hepsilon _hepsilonOne computedBernsteinInputs
end EditApproximation
end OAI
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