| 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 |
|
|
