| import Mathlib |
| import OAI.Combinatorics.MatchingCount.Algorithm.SortedList |
| import OAI.Combinatorics.MatchingCount.Machines.TreeFastLookup |
|
|
| namespace OAI |
|
|
| namespace MatchingFPRAS.OnlineMedian |
| open TreeTyped TreeFraction |
|
|
| def relation (a b : Fraction) : Prop := a.1*b.2≤b.1*a.2 |
| instance : DecidableRel relation := fun _ _=>inferInstanceAs (Decidable (_≤_)) |
| def compareRealizer : TreeTyped.Realizer (fun x : Fraction×Fraction=>decide (relation x.1 x.2)) := |
| (composition (composition (pair second first) lessRealizer) boolNot).congr (by |
| intro x |
| change (!(decide (x.2.1*x.1.2<x.1.1*x.2.2)))=decide (x.1.1*x.2.2≤x.2.1*x.1.2) |
| simp only [←decide_not,not_lt]) |
| lemma polynomial_compare : PolynomialTime compareRealizer := by |
| unfold compareRealizer; apply polynomial_congr |
| exact polynomial_composition (polynomial_composition (polynomial_pair polynomial_second polynomial_first) polynomial_less) polynomial_boolNot |
| |
| lemma relation_repr (a b : ℚ) (ha : 0≤a) (hb : 0≤b) : relation (TreeFraction.repr a) (TreeFraction.repr b) ↔ a≤b := by |
| have hl:=less_val (TreeFraction.repr b) (TreeFraction.repr a) b.pos a.pos |
| rw [val_repr ha,val_repr hb] at hl |
| have he : relation (TreeFraction.repr a) (TreeFraction.repr b) ↔ ¬b<a := by |
| unfold relation |
| have hlt : (TreeFraction.repr b).1*(TreeFraction.repr a).2 < (TreeFraction.repr a).1*(TreeFraction.repr b).2 ↔ b<a := by |
| have := congrArg (fun b : Bool=>b=true) hl |
| exact Iff.of_eq (by simpa only [TreeFraction.less,decide_eq_true_eq] using this) |
| exact (not_lt).symm.trans (not_congr hlt) |
| exact he.trans not_lt |
| |
| def sorted (xs : List Fraction) : List Fraction := xs.insertionSort relation |
| def sortedRealizer : TreeTyped.Realizer sorted := insertionSort relation compareRealizer |
| lemma polynomial_sorted : PolynomialTime sortedRealizer := polynomial_insertionSort relation polynomial_compare |
| |
| def median (xs : List Fraction) : Fraction := (sorted xs).getD (xs.length/2) (0,1) |
| def medianRealizer : TreeTyped.Realizer median := |
| composition (pair sortedRealizer (composition (pair listLength (constant 2)) quotient)) (getDFast (0,1)) |
| lemma polynomial_median : PolynomialTime medianRealizer := |
| polynomial_composition (polynomial_pair polynomial_sorted |
| (polynomial_composition (polynomial_pair polynomial_listLength (polynomial_constant _)) polynomial_quotient)) (polynomial_getDFast _) |
| |
| lemma sorted_repr (xs : List ℚ) (hs : ∀ q∈xs,0≤q) : |
| sorted (xs.map TreeFraction.repr)=(xs.mergeSort (·≤·)).map TreeFraction.repr := by |
| rw [List.mergeSort_eq_insertionSort (·≤·)] |
| symm |
| apply List.map_insertionSort (·≤·) relation TreeFraction.repr xs |
| intro a ha b hb |
| exact (relation_repr a b (hs a ha) (hs b hb)).symm |
| |
| lemma median_repr {k : ℕ} (x : Fin (2*k+1)→ℚ) (hx : ∀ i,0≤x i) : |
| median (List.ofFn (fun i=>TreeFraction.repr (x i)))=TreeFraction.repr (Amplification.median x) := by |
| rw [show List.ofFn (fun i=>TreeFraction.repr (x i))=(List.ofFn x).map TreeFraction.repr by rw [List.map_ofFn]; rfl] |
| unfold median |
| rw [sorted_repr _ (by intro q hq; obtain ⟨i,rfl⟩:=List.mem_ofFn.mp hq; exact hx i)] |
| rw [List.length_map,List.length_ofFn,show (2*k+1)/2=k by omega] |
| change (List.map TreeFraction.repr ((List.ofFn x).mergeSort (·≤·))).getD k (TreeFraction.repr 0)=_ |
| rw [List.getD_map _ 0 TreeFraction.repr] |
| rw [List.getD_eq_getElem _ _ (by simp; omega)] |
| rfl |
| end MatchingFPRAS.OnlineMedian |
| |
| end OAI |
| |
