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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 48 49 50 51 52 53 54 55 56 57 58 | import OAI.Combinatorics.EditApproximation.Binary.IntegerBinaryAccuracy
import OAI.Combinatorics.EditApproximation.Binary.IntegerBinarySpace
namespace OAI
/-! An edit-distance approximation scheme conditional on finite scalar
Bernstein concentration (Tropp, ACM217, corrected 2023, Theorem 4.16).
The same uniform binary algorithm is used in all four clauses. -/
namespace EditApproximation
open MeasureTheory Filter
/-- For each fixed binary accuracy in `(0,1)`, the concrete uniform algorithm
has success probability at least `2/3`, deterministic zero on equal inputs,
worst-case expected almost-linear time including preprocessing and sampling,
and polynomial space on every outcome. Alphabet bounds are analysis parameters,
not inputs to the algorithm. The finite initial range uses exact suffix DP. -/
theorem almostLinearEditDistance_conditional (epsilon : BinaryFraction)
(hepsilon : 0 < epsilon.value) (_hepsilonOne : epsilon.value < 1)
(bernstein : ∀ N, ComputedBernsteinInputs N) :
(∀ (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
refine ⟨?_, ?_, ?_, ?_⟩
· intro source target N hsize
exact integerBinaryAnswer_conditional_accuracy source target N epsilon hepsilon hsize (bernstein N)
· intro source target N heq outcome
exact integerBinaryAnswer_equal source target N epsilon heq outcome
· intro C eta heta
exact integerBinaryWork_expected_near_linear C epsilon hepsilon eta heta
· intro C
exact integerBinarySpace_polynomial C epsilon hepsilon
/-- The cutoff selects an exact answer, rather than leaving the small-input
regime outside the uniform algorithm's correctness statement. -/
theorem integerBinaryAnswer_fallback (source target : List ℤ) (N : ℕ) (epsilon : BinaryFraction)
(hsmall : ¬ physicalLargeInput N epsilon) (outcome) :
integerBinaryAnswer source target N epsilon outcome = EditDistortion.edit source target := by
simp only [integerBinaryAnswer, hsmall, ite_false, exactSuffixAnswer_eq_edit, integerSymbols_edit]
end EditApproximation
end OAI
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