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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 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 | import Mathlib
import OAI.Combinatorics.MatchingCount.Complexity.OnlineInitialDataCost
import OAI.Combinatorics.MatchingCount.Algorithm.Algorithm
import OAI.Combinatorics.MatchingCount.Complexity.OnlineRunCost
import OAI.Combinatorics.MatchingCount.Machines.OnlineBounds
namespace OAI
namespace MatchingFPRAS.LiteralAlgorithm
open TreeTyped TreeFraction PhysicalStream Polynomial OnlineSourceBounds LiteralInput
lemma initialize_materialize {G : GraphInput} [Nontrivial (Fin G.n)]
(hn : 2≤G.n) (hev : Even G.n) (ε δ : ℚ) (he : 0<ε) (he1 : ε<1) (hd : 0≤δ) :
OnlineInitial.initializeInput (materialize (parseInput (encodeInput G ε δ)))=
(2*Amplification.halfRepetitions δ+1,
OnlineTrial.initial (reorder G) (FiniteTrial.parameters G hn hev ε he he1)) := by
rw [materialize_source,OnlineGraph.initialize_eq,
OnlineInitial.initializeInput_source hn hev ε δ he he1 hd]
rfl
lemma numeric_materialize (G : GraphInput) (ε δ : ℚ) :
OnlineInitial.configNumeric (OnlineInitial.config (materialize (parseInput (encodeInput G ε δ))))=
OnlineInitial.configNumeric (OnlineInitial.config (OnlineInitial.sourceSeed (G:=G) ε δ)) := by
rw [materialize_source]
rfl
lemma materialize_vertices (G : GraphInput) (ε δ : ℚ) :
(materialize (parseInput (encodeInput G ε δ))).1.1.length=G.n := by
rw [LiteralInput.parseInput_source];simp only [materialize,List.length_range]
lemma materialize_epsilon (G : GraphInput) (ε δ : ℚ) :
(materialize (parseInput (encodeInput G ε δ))).2.1=TreeFraction.repr ε := by
rw [LiteralInput.parseInput_source];rfl
lemma graph_weight_materialize (G : GraphInput) (ε δ : ℚ) :
weight (OnlineGraph.graph (reorder G) (ExecutableAdaptive.graphRep G))≤
weight (materialize (parseInput (encodeInput G ε δ))) := by
rw [materialize_source]
change weight _≤weight (OnlineGraph.graph (reorder G) (ExecutableAdaptive.graphRep G),_)
rw [weight_prod (OnlineGraph.graph (reorder G) (ExecutableAdaptive.graphRep G))]
omega
lemma compute_cost (a : Parsed) : computeRealizer.cost a=
materializeRealizer.cost a+OnlineInitial.initializeInputRealizer.cost (materialize a)+
OnlineAmplification.runRealizer.cost (OnlineInitial.initializeInput (materialize a)) := rfl
lemma compute_bound : ∃ q : Polynomial ℕ,∀ (G : GraphInput) [Nontrivial (Fin G.n)]
(_hn : 2≤G.n) (_hev : Even G.n) (ε δ : ℚ) (_he : 0<ε) (_he1 : ε<1) (_hd : 0≤δ)
(x : ℕ),1≤x→G.n≤x→⌈ε⁻¹⌉₊≤x→Nat.clog 2 ⌈δ⁻¹⌉₊≤x→
weight (parseInput (encodeInput G ε δ))≤x→
computeRealizer.cost (parseInput (encodeInput G ε δ))+
(compute (parseInput (encodeInput G ε δ))).worst weight≤q.eval x := by
obtain ⟨m,hm⟩:=work_materialize
obtain ⟨i,hi⟩:=OnlineInitial.work_initializeInput
obtain ⟨r,hr⟩:=OnlineAmplification.run_bound
obtain ⟨b,hb⟩:=polynomial_bound normBound_polynomial
let M:=m.comp (2*X)
let S:Polynomial ℕ:=2*X+b+M
let R:Polynomial ℕ:=11*X+11+b+M
refine ⟨M+i.comp S+r.comp R,?_⟩
intro G inst hn hev ε δ he he1 hd x hx hnx hex hdx hwx
let a:=parseInput (encodeInput G ε δ)
let seed:=materialize a
let p:=FiniteTrial.parameters G hn hev ε he he1
have hn2:G.n≤2*x := by omega
have ha:a.1=G.n := by dsimp only [a];rw [LiteralInput.parseInput_source]
have hmat:materializeRealizer.cost a+weight seed≤M.eval x := by
apply (hm a).trans
apply (polynomial_eval_monotone m (show a.1+weight a≤2*x by dsimp only [a] at *;omega)).trans
simp only [M,Polynomial.eval_comp,Polynomial.eval_mul,Polynomial.eval_ofNat,Polynomial.eval_X];rfl
have hseed:OnlineInitial.seedSize seed≤S.eval x := by
have hnum:OnlineInitial.configNumeric (OnlineInitial.config seed)≤b.eval x := by
dsimp only [seed,a]
rw [numeric_materialize]
exact (numeric_le hn hev ε δ he he1 x hn2 hex).trans (hb x hx)
have hlen:(seed.1.1).length=G.n:=materialize_vertices G ε δ
have hep:natCeil (inverse seed.2.1)=⌈ε⁻¹⌉₊ := by
rw [show seed.2.1=TreeFraction.repr ε from materialize_epsilon G ε δ,inverse_repr ε he.le,natCeil_repr _ (inv_nonneg.mpr he.le)]
simp only [OnlineInitial.seedSize,hlen,hep,S,Polynomial.eval_add,Polynomial.eval_mul,
Polynomial.eval_ofNat,Polynomial.eval_X]
omega
have hini:OnlineInitial.initializeInputRealizer.cost seed ≤ i.eval (S.eval x) :=
(Nat.le_trans (Nat.le_add_right _ _) (hi seed)).trans (polynomial_eval_monotone i hseed)
have hnorm:OnlineTrial.sizeNorm p (OnlineGraph.graph (reorder G) (ExecutableAdaptive.graphRep G))≤b.eval x+M.eval x := by
apply (norm_le hn hev ε he he1 x hn2 hex _).trans
have hg:=graph_weight_materialize G ε δ
have hnb:=hb x hx
dsimp only [seed,a] at hmat
omega
have hrep:2*Amplification.halfRepetitions δ+1≤10*x+11:=by
rw [Amplification.repetitions_source];omega
have harg:2*Amplification.halfRepetitions δ+1+
OnlineTrial.sizeNorm p (OnlineGraph.graph (reorder G) (ExecutableAdaptive.graphRep G))≤R.eval x := by
simp only [R,Polynomial.eval_add,Polynomial.eval_mul,Polynomial.eval_ofNat,Polynomial.eval_X]
omega
have hrun:= (hr G (reorder G) (reorder_mem G) p (2*Amplification.halfRepetitions δ+1)).trans
(polynomial_eval_monotone r harg)
have hinput:=initialize_materialize hn hev ε δ he he1 hd
rw [compute_cost]
unfold compute
rw [hinput]
simp only [Polynomial.eval_add,Polynomial.eval_comp]
dsimp only [a,seed,p] at hini hmat hrun
omega
end MatchingFPRAS.LiteralAlgorithm
end OAI
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