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ea76aac | 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 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 | import OAI.Computability.MatchingCount.RecursiveEval
namespace OAI
section
namespace BinaryMatching.DenseOracle
open Finset UnitFlow CutAlgebra OddCutParity
variable {n : ℕ}
def total (c : Fin n → Fin n → ℕ) : ℕ := ∑ i, ∑ j, c i j
lemma entry_le_total (c : Fin n → Fin n → ℕ) (i j : Fin n) : c i j≤total c :=
(single_le_sum (fun _ _ => Nat.zero_le _) (mem_univ j)).trans
(single_le_sum (fun x _ => Nat.zero_le (∑ y, c x y)) (mem_univ i))
lemma source_le_total (c : Fin n → Fin n → ℕ) (a : Fin n) :
UnitMincut.sourceCapacity c a≤total c := single_le_sum (fun _ _ => Nat.zero_le _) (mem_univ a)
lemma push_total_le (c : Fin n → Fin n → ℕ) (f : Fin n → Fin n) :
total (push c f)≤total c := by
calc
_ ≤ ∑ i,∑ j,∑ u,∑ v,if f u=i ∧ f v=j then c u v else 0 := by
apply sum_le_sum; intro i hi
apply sum_le_sum; intro j hj
unfold push
split_ifs
· exact Nat.zero_le _
· rfl
_ = ∑ u,∑ v,∑ i,∑ j,if f u=i ∧ f v=j then c u v else 0 := sum_four_swap _
_ = total c := by simp [total,ite_and,Finset.sum_ite_irrel,eq_comm]
lemma contract_total_le (c : Matrix n ℕ) (f : Fin n → Fin n) :
total (view (contract c f))≤total (view c) := by
rw [contract_view]
exact push_total_le _ _
lemma feasible_abs_bound (c : Fin n → Fin n → ℕ) (a z : Fin n) (b : UnitFlow.Matrix (Fin n))
(hb : Feasible c a z b) (i j : Fin n) : (b i j).natAbs≤total c := by
have hu : b i j≤(total c:ℤ) := (hb.capacity i j).trans (by exact_mod_cast entry_le_total c i j)
have hl : b j i≤(total c:ℤ) := (hb.capacity j i).trans (by exact_mod_cast entry_le_total c j i)
rw [hb.antisymm j i] at hl
have hh : |b i j|≤(total c:ℤ) := abs_le.mpr ⟨by omega,hu⟩
rw [←Int.natCast_natAbs] at hh
exact_mod_cast hh
lemma flow_abs_bound (c : Matrix n ℕ) (a z : Fin n) (k : ℕ) (i j : Fin n) :
(view (iterate c a z k) i j).natAbs≤total (view c) := by
rw [iterate_view]
exact feasible_abs_bound _ _ _ _ (UnitMincut.iterate_feasible _ _ _ _) _ _
lemma path_length_bound (c : Matrix n ℕ) (b : Matrix n ℤ) (a : Fin n) (k : ℕ)
(v : Fin n) (p : List (Fin n)) (hp : (search c b a k)[v]=some p) : p.length≤n := by
have hs := congrFun (search_view c b a k) v
change (search c b a k)[v]=_ at hs
rw [hs] at hp
have hh := (ResidualSearch.good_search (UnitMincut.residual (view c) (view b)) a k).simple v p hp
simpa only [Fintype.card_fin] using hh.length_le_card
def calls : ℕ → Matrix n ℕ → Finset (Fin n) → ℕ
| 0,_,_ => 1
| k+1,c,U =>
let A := terminalCut c U
if OddSide U A then 1 else
match OddMincut.outside A, OddMincut.outside Aᶜ with
| some r,some q =>
1+calls k (contract c (contractMap A r)) (A∩U)+
calls k (contract c (contractMap Aᶜ q)) (Aᶜ∩U)
| _,_ => 1
lemma calls_bound (fuel : ℕ) (c : Matrix n ℕ) (U : Finset (Fin n))
(he : Even U.card) (hU : 2≤U.card) (hbound : U.card≤fuel) :
calls fuel c U≤U.card-1 := by
induction fuel generalizing c U with
| zero => omega
| succ k ih =>
let A := terminalCut c U
obtain ⟨hsep,_⟩ := TerminalMincut.terminalCut_correct (view c) U hU
rw [←terminalCut_eq] at hsep
change TerminalMincut.Separates U A at hsep
by_cases hodd : OddSide U A
· have hcall : calls (k+1) c U=1 := by
rw [calls]
change (if OddSide U A then 1 else _)=1
exact ite_eq_left hodd
rw [hcall]; omega
· have heA : Even (A∩U).card := by
simp only [OddSide,Nat.odd_iff] at hodd
rw [Nat.even_iff]; omega
have heB := even_other U A he heA
obtain ⟨hposA,hposB⟩ := separate_cards U A hsep
have hpart := terminal_partition U A
have htwoA : 2≤(A∩U).card := by rw [Nat.even_iff] at heA; omega
have htwoB : 2≤(Aᶜ∩U).card := by rw [Nat.even_iff] at heB; omega
have hboundA : (A∩U).card≤k := by omega
have hboundB : (Aᶜ∩U).card≤k := by omega
rw [calls]
change (if OddSide U A then 1 else _ )≤_
rw [ite_eq_right hodd]
split
· rename_i r q hr hq
have h₁ := ih (contract c (contractMap A r)) (A∩U) heA htwoA hboundA
have h₂ := ih (contract c (contractMap Aᶜ q)) (Aᶜ∩U) heB htwoB hboundB
change 1+calls k (contract c (contractMap A r)) (A∩U)+
calls k (contract c (contractMap Aᶜ q)) (Aᶜ∩U)≤U.card-1
omega
· omega
lemma initial_total_bound (G : Input) (j : ℕ) :
total (Algorithm.capacities (Algorithm.iterate G j).numerators)≤G.n^2*Algorithm.D₀ G := by
have hh (i k : Fin G.n) : Algorithm.capacities (Algorithm.iterate G j).numerators i k≤Algorithm.D₀ G := by
unfold Algorithm.capacities
split_ifs with hik hki
· have hb := (Algorithm.iterate_box G j ⟨(i,k),hik⟩).2
exact Int.toNat_le.mpr (by simpa only [Algorithm.entry] using hb)
· have hb := (Algorithm.iterate_box G j ⟨(k,i),hki⟩).2
exact Int.toNat_le.mpr (by simpa only [Algorithm.entry] using hb)
· exact Nat.zero_le _
calc
_ ≤ ∑ _i : Fin G.n,∑ _k : Fin G.n,Algorithm.D₀ G :=
sum_le_sum fun i _ => sum_le_sum fun k _ => hh i k
_ = _ := by simp; ring
end BinaryMatching.DenseOracle
end
end OAI
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