| import Mathlib |
| import OAI.Combinatorics.MatchingCount.Algorithm.OnlineStage |
| import OAI.Combinatorics.MatchingCount.Algorithm.OnlineScheduleNat |
|
|
| namespace OAI |
|
|
| namespace MatchingFPRAS.OnlineSchedule |
| open TreeTyped TreeFraction Transport ExecutableSource NumericLadder ExecutableScaling |
| open TreeTyped.Numbered Enlargement UnitRefresh ExecutableAdaptive |
| open ExecutableBottleneck ExecutableHierarchy ExecutableRefresh |
|
|
| attribute [local instance] NumberedHierarchy.labelOrder |
| attribute [local instance] Transport.sym2Order Transport.bagEdgeOrder Transport.bagEdgeFintype |
| attribute [local instance] UnitRefresh.Computable.allowedDec Transport.prodOrder Transport.sumOrder |
|
|
| def lengthAfter {A B : Type} [Codec A] [Codec B] {f : A→List B} (F : Realizer f) : |
| Realizer (fun a=>(f a).length) := composition F listLength |
| lemma polynomial_lengthAfter {A B : Type} [Codec A] [Codec B] {f : A→List B} {F : Realizer f} |
| (hF : PolynomialTime F) : PolynomialTime (lengthAfter F) := |
| polynomial_composition hF polynomial_listLength |
|
|
| def rangeAfter {A : Type} [Codec A] {f : A→ℕ} (F : Realizer f) : |
| Realizer (fun a=>List.range (f a)) := composition F range |
|
|
| def listIndices {A : Type} [Codec A] : Realizer (fun xs : List A=>List.range xs.length) := |
| composition listLength range |
| lemma polynomial_listIndices {A : Type} [Codec A] : PolynomialTime (listIndices (A:=A)) := |
| polynomial_boundedRange polynomial_listLength Polynomial.X (fun xs=>by simpa using list_length_le_weight xs) |
|
|
| def indicesAfter {A B : Type} [Codec A] [Codec B] {f : A→List B} (F : Realizer f) : |
| Realizer (fun a=>List.range (f a).length) := composition F listIndices |
| lemma polynomial_indicesAfter {A B : Type} [Codec A] [Codec B] {f : A→List B} {F : Realizer f} |
| (hF : PolynomialTime F) : PolynomialTime (indicesAfter F) := |
| polynomial_composition hF polynomial_listIndices |
|
|
| abbrev PrecisionData := ScanInput×(ℕ×ℕ) |
|
|
| def scanError (x : PrecisionData) : ℕ := |
| 2*(1+(x.1.1.2.2*(ladderQ x.1.1.1.1 x.1.1.1.2)^2+ladderQ x.1.1.1.1 x.1.1.1.2))+ |
| 4*(8*x.2.1^2+4*x.2.1+3)+2*x.1.1.1.1^2*x.2.2 |
|
|
| def scanErrorRealizer : Realizer scanError := |
| ((((constant 2).mulNat ((constant 1).addNat |
| ((first.fst.snd.snd.mulNat (composition (composition first.fst.fst qRealizer) (fixedPower 2))).addNat |
| (composition first.fst.fst qRealizer)))).addNat |
| ((constant 4).mulNat ((((constant 8).mulNat (composition second.fst (fixedPower 2))).addNat |
| ((constant 4).mulNat second.fst)).addNat (constant 3)))).addNat |
| (((constant 2).mulNat (composition first.fst.fst.fst (fixedPower 2))).mulNat second.snd)).congr (fun _=>rfl) |
| lemma polynomial_scanError : PolynomialTime scanErrorRealizer := by |
| unfold scanErrorRealizer |
| apply polynomial_congr |
| dsimp only [Realizer.fst, Realizer.snd] |
| have := polynomial_q |
| have := polynomial_fixedPower 2 |
| simp only [Realizer.addNat,Realizer.mulNat] |
| tree_cost |
|
|
| def precisionPlan (x : PrecisionData) : ((ℕ×ℕ)×ℕ)×ℕ := |
| (((Nat.clog 2 (discreteLadderSteps x.1.1.1.1 x.1.1.1.2 x.1.1.2.1 x.1.1.2.2 x.1.2 *scanError x*x.1.2), |
| x.1.1.1.1),discreteLadderSteps x.1.1.1.1 x.1.1.1.2 x.1.1.2.1 x.1.1.2.2 x.1.2), |
| Nat.clog 2 (6*x.2.1*x.1.2)) |
| def precisionPlanRealizer : Realizer precisionPlan := |
| (pair (pair (pair |
| (composition (((composition first scansRealizer).mulNat scanErrorRealizer).mulNat first.snd) clogTwo) |
| first.fst.fst.fst) (composition first scansRealizer)) |
| (composition (((constant 6).mulNat second.fst).mulNat first.snd) clogTwo)).congr (fun _=>rfl) |
| lemma polynomial_precisionPlan : PolynomialTime precisionPlanRealizer := by |
| unfold precisionPlanRealizer |
| apply polynomial_congr |
| dsimp only [Realizer.fst, Realizer.snd] |
| have := polynomial_scanError |
| have := polynomial_scans |
| have := polynomial_clogTwo |
| simp only [Realizer.mulNat] |
| tree_cost |
|
|
| abbrev RequestInput := SparseStageData×ℕ |
|
|
| def sourcePrecision (x : RequestInput) : PrecisionData := |
| let b:=sparseBagInput x.1 |
| (((((sourceVertices b).length,x.1.1.2.1.2),(x.1.1.2.1.1.2.length/2,x.1.1.2.2.1.length)), |
| 1000*(x.1.1.2.1.1.2.length/2)*x.2), |
| ((sourceTables b).1.length,(sourceParents b).length)) |
| def requestBagRealizer : Realizer (fun x : RequestInput=>sparseBagInput x.1) := |
| composition first sparseBagInputRealizer |
| def requestKRealizer : Realizer (fun x : RequestInput=>x.1.1.2.1.1.2.length/2) := |
| composition (pair (composition first.fst.snd.fst.fst.snd listLength) (constant 2)) quotient |
| def requestNDRealizer : Realizer (fun x : RequestInput=>((sourceVertices (sparseBagInput x.1)).length,x.1.1.2.1.2)) := |
| pair (composition (composition (g:=sourceVertices) requestBagRealizer sourceVerticesRealizer) listLength) first.fst.snd.fst.snd |
| def requestKTRealizer : Realizer (fun x : RequestInput=>(x.1.1.2.1.1.2.length/2,x.1.1.2.2.1.length)) := |
| pair requestKRealizer (composition first.fst.snd.snd.fst listLength) |
| def requestMRealizer : Realizer (fun x : RequestInput=>1000*(x.1.1.2.1.1.2.length/2)*x.2) := |
| ((constant 1000).mulNat requestKRealizer).mulNat second |
| def requestScanRealizer : Realizer (fun x : RequestInput=> |
| ((((sourceVertices (sparseBagInput x.1)).length,x.1.1.2.1.2), |
| (x.1.1.2.1.1.2.length/2,x.1.1.2.2.1.length)),1000*(x.1.1.2.1.1.2.length/2)*x.2)) := |
| pair (pair requestNDRealizer requestKTRealizer) requestMRealizer |
| def requestTablesRealizer : Realizer (fun x : RequestInput=>sourceTables (sparseBagInput x.1)) := |
| composition (f:=fun x : RequestInput=>sparseBagInput x.1) (g:=sourceTables) |
| requestBagRealizer sourceTablesRealizer |
| def requestCatalogRealizer : Realizer (fun x : RequestInput=>(sourceTables (sparseBagInput x.1)).1) := |
| Realizer.fst (f:=fun x : RequestInput=>sourceTables (sparseBagInput x.1)) requestTablesRealizer |
| def requestEdgesRealizer : Realizer (fun x : RequestInput=>(sourceTables (sparseBagInput x.1)).1.length) := |
| lengthAfter (f:=fun x : RequestInput=>(sourceTables (sparseBagInput x.1)).1) requestCatalogRealizer |
| def requestLabelsRealizer : Realizer (fun x : RequestInput=>(sourceParents (sparseBagInput x.1)).length) := |
| composition (composition (g:=sourceParents) requestBagRealizer sourceParentsRealizer) listLength |
| def sourcePrecisionRealizer : Realizer sourcePrecision := |
| pair requestScanRealizer (pair requestEdgesRealizer requestLabelsRealizer) |
| lemma polynomial_requestBag : PolynomialTime requestBagRealizer := |
| polynomial_composition polynomial_first polynomial_sparseBagInput |
| lemma polynomial_requestK : PolynomialTime requestKRealizer := by |
| unfold requestKRealizer |
| dsimp only [Realizer.fst, Realizer.snd] |
| have := polynomial_quotient |
| tree_cost |
| lemma polynomial_requestND : PolynomialTime requestNDRealizer := by |
| unfold requestNDRealizer |
| dsimp only [Realizer.fst, Realizer.snd] |
| have := polynomial_requestBag |
| have := polynomial_sourceVertices |
| tree_cost |
| lemma polynomial_requestKT : PolynomialTime requestKTRealizer := by |
| unfold requestKTRealizer |
| dsimp only [Realizer.fst, Realizer.snd] |
| have := polynomial_requestK |
| tree_cost |
| lemma polynomial_requestM : PolynomialTime requestMRealizer := |
| polynomial_mulNat (polynomial_mulNat (polynomial_constant _) polynomial_requestK) polynomial_second |
| lemma polynomial_requestScan : PolynomialTime requestScanRealizer := |
| TreeTyped.polynomial_pair (TreeTyped.polynomial_pair polynomial_requestND polynomial_requestKT) polynomial_requestM |
| lemma polynomial_requestTables : PolynomialTime requestTablesRealizer := |
| polynomial_composition polynomial_requestBag polynomial_sourceTables |
| lemma polynomial_requestCatalog : PolynomialTime requestCatalogRealizer := |
| polynomial_fst polynomial_requestTables |
| lemma polynomial_requestEdges : PolynomialTime requestEdgesRealizer := |
| polynomial_lengthAfter polynomial_requestCatalog |
| lemma polynomial_requestLabels : PolynomialTime requestLabelsRealizer := |
| polynomial_composition (polynomial_composition polynomial_requestBag |
| polynomial_sourceParents) polynomial_listLength |
| lemma polynomial_sourcePrecision : PolynomialTime sourcePrecisionRealizer := |
| TreeTyped.polynomial_pair polynomial_requestScan (TreeTyped.polynomial_pair polynomial_requestEdges polynomial_requestLabels) |
|
|
| def sourceRequest (x : RequestInput) : OnlineSample.Request := |
| let plan:=precisionPlan (sourcePrecision x) |
| (plan.1,(plan.2,List.range ((sourceTables (sparseBagInput x.1)).1.length))) |
| def requestPlanRealizer : Realizer (fun x : RequestInput=>precisionPlan (sourcePrecision x)) := |
| composition (g:=precisionPlan) sourcePrecisionRealizer precisionPlanRealizer |
| def requestIndicesRealizer : Realizer (fun x : RequestInput=>List.range (sourceTables (sparseBagInput x.1)).1.length) := |
| indicesAfter (f:=fun x : RequestInput=>(sourceTables (sparseBagInput x.1)).1) requestCatalogRealizer |
| def sourceRequestRealizer : Realizer sourceRequest := |
| pair requestPlanRealizer.fst (pair requestPlanRealizer.snd requestIndicesRealizer) |
| lemma polynomial_requestPlan : PolynomialTime requestPlanRealizer := |
| polynomial_composition polynomial_sourcePrecision polynomial_precisionPlan |
| lemma polynomial_requestIndices : PolynomialTime requestIndicesRealizer := |
| polynomial_indicesAfter polynomial_requestCatalog |
| lemma polynomial_sourceRequest : PolynomialTime sourceRequestRealizer := |
| TreeTyped.polynomial_pair (polynomial_fst polynomial_requestPlan) |
| (TreeTyped.polynomial_pair (polynomial_snd polynomial_requestPlan) polynomial_requestIndices) |
|
|
| lemma sparseBagInput_source {G : GraphInput} {j : ℕ} (s : FiniteTrial.ScaleState G j) |
| (K D : ℕ) (ft fq : List Bool) : |
| sparseBagInput (OnlineStage.data s K D ft fq)= |
| (((matrixRep s.activity,List.replicate (2*K+2) false),List.range G.n),List.replicate (2*K) false) := by |
| have h:=sparseUpdateData_source s K D ft fq |
| exact congrArg (fun x=>x.1.1.1.1) h |
|
|
| lemma sourcePrecision_source {G : GraphInput} [Nontrivial (Fin G.n)] (hn : 2≤G.n) |
| {j : ℕ} (s : FiniteTrial.ScaleState G j) (K D S : ℕ) (ft fq : List Bool) : |
| sourcePrecision (OnlineStage.data s K D ft fq,S)= |
| (((((Fintype.card (SourceVertex (Fin G.n) K)),D),(K,ft.length)),1000*K*S), |
| (Fintype.card (BagEdge (RationalSource.bags s.activity K)),Fintype.card (RationalSource.Label s.activity K))) := by |
| unfold sourcePrecision |
| rw [sparseBagInput_source] |
| dsimp only [OnlineStage.data] |
| rw [sourceVertices_length,sourceTables_catalog hn s.activity |
| (fun i j=>(s.activity_pos i j).le) K,sourceParents_source hn s.activity |
| (fun i j=>(s.activity_pos i j).le) K] |
| simp only [List.length_replicate,Nat.mul_div_cancel_left _ (by decide : 0<2),sourceCatalog,List.length_ofFn, |
| ExecutableDP.parentRep_length] |
|
|
| end MatchingFPRAS.OnlineSchedule |
|
|
| end OAI |
|
|
