File size: 4,238 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 | import Mathlib
import OAI.Combinatorics.MatchingCount.Machines.Wrapper
import OAI.Combinatorics.MatchingCount.Complexity.OriginalCost
namespace OAI
namespace MatchingFPRAS.LiteralPhysical
open Polynomial PhysicalStream TreeTyped BitComplexity LiteralPacked
lemma polynomial_eval (p : Polynomial ℕ) : PolynomiallyBounded (fun x=>p.eval x) := by
refine Polynomial.induction_on' p ?_ ?_
· intro p q hp hq
simpa only [Polynomial.eval_add] using hp.add hq
· intro n a
simpa only [Polynomial.eval_monomial] using (PolynomiallyBounded.const a).mul (PolynomiallyBounded.id.pow n)
lemma polynomial_budget (w H : ℕ) {f g : ℕ→ℕ}
(hf : PolynomiallyBounded f) (hg : PolynomiallyBounded g) :
PolynomiallyBounded (fun x=>LiteralBlockMap.budget w H (f x) (g x)) :=
(hf.add (PolynomiallyBounded.const 1)).mul
((PolynomiallyBounded.const (3+w+H)).add ((PolynomiallyBounded.const 2).mul
(hg.add (hf.mul (PolynomiallyBounded.const (H+w))))))
lemma polynomial_packTime {f : ℕ→ℕ} (hf : PolynomiallyBounded f) :
PolynomiallyBounded (fun x=>packTime (f x)) :=
((PolynomiallyBounded.const 1).add (polynomial_budget 1 (18*width) hf (PolynomiallyBounded.const 0))).mul
(PolynomiallyBounded.const (MachineCost.programCost packCode))
lemma polynomial_postTime {f : ℕ→ℕ} (hf : PolynomiallyBounded f) :
PolynomiallyBounded (fun x=>postTime (f x)) :=
(PolynomiallyBounded.const eraseTime).add
(((PolynomiallyBounded.const 1).add (polynomial_budget (18*width) 1 hf (PolynomiallyBounded.const 0))).mul
(PolynomiallyBounded.const (MachineCost.programCost unpackCode)))
lemma polynomial_factor {A B : Type} [Codec A] [Codec B] {f : A→Experiment B}
(F : OnlineTree.Realizer f) {H n : ℕ→ℕ}
(hH : PolynomiallyBounded H) (hn : PolynomiallyBounded n) :
PolynomiallyBounded (fun x=>OnlineStack.Executes.factor F.program.code (H x) (n x)) := by
let a:=OnlineStack.accesses F.program.code
let o:=OnlineStack.overhead F.program.code
exact ((PolynomiallyBounded.const o).mul ((PolynomiallyBounded.const 1).add
((PolynomiallyBounded.const (2*a)).mul ((hH.add (hn.mul (PolynomiallyBounded.const a))).add
(PolynomiallyBounded.const 1))))).add (PolynomiallyBounded.const 1)
lemma mainTime_bound : ∃ C d : ℕ,0<C ∧ ∀ (G : GraphInput) (ε δ : ℚ),
0<ε→ε<1→0≤δ→wrapTime LiteralAlgorithm.algorithmRealizer (encodeInput G ε δ)≤timeBound C d G ε δ := by
obtain ⟨q,hq⟩:=LiteralAlgorithm.algorithm_bound
let F:=LiteralAlgorithm.algorithmRealizer
let bound:=fun x=>packTime x+q.eval x*OnlineStack.Executes.factor F.program.code (18*x+1) (q.eval x)+postTime (q.eval x)
have hb : PolynomiallyBounded bound :=
((polynomial_packTime PolynomiallyBounded.id).add ((polynomial_eval q).mul
(polynomial_factor F (((PolynomiallyBounded.const 18).mul PolynomiallyBounded.id).add
(PolynomiallyBounded.const 1)) (polynomial_eval q)))).add (polynomial_postTime (polynomial_eval q))
obtain ⟨C,d,hb⟩:=hb
refine ⟨C+1,d,by omega,?_⟩
intro G ε δ he he1 hd
let as:=encodeInput G ε δ
let x:=as.length+⌈ε⁻¹⌉₊+Nat.clog 2 ⌈δ⁻¹⌉₊+1
have hx : 1≤x := by dsimp only [x];omega
have hl : as.length≤x := by dsimp only [x];omega
have hc := hq G ε δ he he1 hd x hx hl (by dsimp only [x];omega) (by dsimp only [x];omega)
have hcost : F.cost as≤q.eval x := by dsimp only [F,as];omega
have hout : (LiteralAlgorithm.algorithm as).worst weight≤q.eval x := by dsimp only [as];omega
have hword : (LiteralAlgorithm.algorithm as).worst List.length≤q.eval x :=
Experiment.worst_le _ _ _ ((Experiment.all_worst _ weight).mono (fun bs hbs=>
(list_length_le_weight bs).trans (hbs.trans hout)))
have hweight : weight as≤18*x+1 := by rw [LiteralCodec.weight_word];omega
have hwrap : wrapTime F as≤bound x := by
unfold wrapTime OnlineTree.physicalCost
exact Nat.add_le_add (Nat.add_le_add (packTime_mono hl)
(Nat.mul_le_mul hcost (OnlineStack.Executes.factor_mono _ _ _ _ hweight hcost))) (postTime_mono hword)
apply hwrap.trans ((hb x hx).trans ?_)
change C*x^d≤(C+1)*x^d
exact Nat.mul_le_mul_right _ (by omega)
end MatchingFPRAS.LiteralPhysical
end OAI
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