File size: 3,697 Bytes
e5a97c6 | 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 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 | import Mathlib
import OAI.Combinatorics.MatchingCount.Tapes.GuardBits
namespace OAI
namespace MatchingFPRAS.FiniteTrial
open BitComplexity Amplification
variable {G : GraphInput} [Nontrivial (Fin G.n)]
lemma oneBits_le_trial (p : Trial.Parameters G) : oneBits p ≤ p.K*stageBits p := by
have hk : 1≤p.K := p.K_pos
have hs : 1≤p.S := p.S_pos
unfold stageBits
have h := Nat.mul_le_mul hk hs
have hh := Nat.mul_le_mul_right (oneBits p) h
simpa only [one_mul, Nat.mul_assoc] using hh
lemma guardedBits_le (p : Trial.Parameters G) : guardedBits p≤17*(p.K*stageBits p) := by
unfold guardedBits guardBits
have h := Nat.mul_le_mul_left 16 (oneBits_le_trial p)
omega
lemma guardedAmplifiedBits_le (p : Trial.Parameters G) (δ : ℚ) :
guardedAmplifiedBits p δ≤17*amplifiedBits p δ := by
have h := Nat.mul_le_mul_left (2*halfRepetitions δ+1) (guardedBits_le p)
simpa only [guardedAmplifiedBits, amplifiedBits, Nat.mul_left_comm, Nat.mul_assoc] using h
end MatchingFPRAS.FiniteTrial
namespace MatchingFPRAS.BitComplexity
open FiniteTrial Trial
theorem polynomial_guarded_bits : ∃ C d : ℕ, ∀ (G : GraphInput) [Nontrivial (Fin G.n)]
(hn : 2 ≤ G.n) (hev : Even G.n) (e δ : ℚ) (he : 0<e) (he1 : e<1), G.n ≤ 2*G.edges.card →
guardedAmplifiedBits (parameters G hn hev e he he1) δ ≤ timeBound C d G e δ := by
obtain ⟨C,d,h⟩:=polynomial_fair_bits
refine ⟨17*C,d,fun G _ hn hev e δ he he1 hs=>?_⟩
apply (guardedAmplifiedBits_le _ δ).trans
have hh := Nat.mul_le_mul_left 17 (h G hn hev e δ he he1 hs)
simpa only [timeBound,Nat.mul_assoc] using hh
end MatchingFPRAS.BitComplexity
namespace MatchingFPRAS.FiniteTrial
open BitComplexity Variation Finset
open scoped Classical
lemma padded_guarded_success {G : GraphInput} [Nontrivial (Fin G.n)]
(p : Trial.Parameters G) (δ : ℚ) (B : ℕ) (ht : guardedAmplifiedBits p δ≤B)
(hz : 0<Z G) (hd : 0<δ) :
((univ.filter (fun b=>Accurate p (padSampler ht (guardedAmplifiedBitOutput p δ) b))).card:ℚ)/
(2:ℚ)^B≥1-δ := by
rw [padSampler_success]
exact guardedAmplifiedBitOutput_success p hz δ hd
theorem guarded_scheduled_fair_tapes : ∃ C d : ℕ, 0<C ∧
∀ (G : GraphInput) [Nontrivial (Fin G.n)] (_ : 2≤G.n) (_ : Even G.n)
(e δ : ℚ) (_ : 0<e) (_ : e<1), G.n≤2*G.edges.card →
∃ f : (Fin (timeBound C d G e δ) → Bool) → ℚ,
(∀ b,0≤f b) ∧ (Z G=0 → ∀ b,f b=0) ∧ (0<δ →
((univ.filter (fun b=>(1-e)*(Z G:ℚ)≤f b ∧ f b≤(1+e)*(Z G:ℚ))).card:ℚ) /
(2:ℚ)^(timeBound C d G e δ)≥1-δ) := by
obtain ⟨C,d,h⟩:=polynomial_guarded_bits
refine ⟨C+1,d,by omega,fun G _ hn hev e δ he he1 hs=>?_⟩
let p:=parameters G hn hev e he he1
have ht : guardedAmplifiedBits p δ ≤ timeBound (C+1) d G e δ := by
exact (h G hn hev e δ he he1 hs).trans (Nat.mul_le_mul_right _ (by omega))
let f:=padSampler ht (guardedAmplifiedBitOutput p δ)
have hz : Z G=0 → ∀ b,f b=0 := fun hZ b => guardedAmplifiedBitOutput_zero p hZ δ _
refine ⟨f,fun b=>guardedAmplifiedBitOutput_nonneg p δ _,hz,fun hd=>?_⟩
by_cases hZ : Z G=0
· have hf : (univ.filter (fun b=>(1-e)*(Z G:ℚ)≤f b ∧ f b≤(1+e)*(Z G:ℚ)))=univ := by
ext b
simp [hZ,hz hZ b]
rw [hf,card_univ]
simp only [Fintype.card_fun,Fintype.card_bool,Fintype.card_fin,Nat.cast_pow,Nat.cast_ofNat]
rw [div_self (by positivity : (2:ℚ)^timeBound (C+1) d G e δ≠0)]
linarith
· convert padded_guarded_success p δ _ ht (Nat.pos_of_ne_zero hZ) hd using 3
congr 1
ext b
simp only [mem_filter,mem_univ,true_and]
rfl
end MatchingFPRAS.FiniteTrial
end OAI
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