File size: 3,271 Bytes
97c3405 | 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 | import OAI.Combinatorics.ContingencyTables.Machines.CellLowerTestProgram
import OAI.Combinatorics.ContingencyTables.Machines.BinarySearchProgram
namespace OAI
/-! # Computing an attained cell maximum by binary feasibility search
A finite maximum is used only to prove the result of the executed binary
search. The program never enumerates feasible tables.
-/
namespace ContingencyTables.CellMaximumProgram
open MatchingFPRAS.TreeTyped CellLowerTestProgram
open scoped Classical
def bound (z : Context) : ℕ := ResidualGraphProgram.capacityEntry (z.1.2,z.2)
def boundRealizer : Realizer bound :=
composition (pair (composition first second) second) ResidualGraphProgram.capacityEntryRealizer
lemma polynomial_bound : PolynomialTime boundRealizer :=
polynomial_composition (polynomial_pair
(polynomial_composition polynomial_first polynomial_second) polynomial_second)
ResidualGraphProgram.polynomial_capacityEntry
def maximum (z : Context) : ℕ := BinarySearchProgram.search test (z,bound z)
def maximumRealizer : Realizer maximum :=
composition (f := fun z : Context=>(z,bound z))
(g := BinarySearchProgram.search test)
(pair identity boundRealizer) (BinarySearchProgram.searchRealizer test testRealizer)
lemma polynomial_maximum : PolynomialTime maximumRealizer :=
polynomial_composition (polynomial_pair polynomial_identity polynomial_bound)
(BinarySearchProgram.polynomial_search test testRealizer polynomial_test)
lemma bound_code {m n : ℕ} (r : Fin m→ℕ) (c : Fin n→ℕ) (b : Fin m→Fin n→ℕ)
(i : Fin m) (j : Fin n) :
bound (NetworkVertexCode.data r c b,(i.val,j.val))=b i j :=
ResidualGraphProgram.capacityEntry_code b i j
/-- For every feasible instance the computed value is attained, and it
bounds the same coordinate of every other feasible table. -/
theorem maximum_correct {m n : ℕ} (hEK : PublishedEdmondsKarp.EdmondsKarp1972Theorem1)
(r : Fin m→ℕ) (c : Fin n→ℕ) (b : Fin m→Fin n→ℕ)
(i : Fin m) (j : Fin n)
(hne : ∃ X,HasMargins X r c ∧ ∀ u v,X u v≤b u v) :
∃ X,HasMargins X r c ∧ (∀ u v,X u v≤b u v) ∧
X i j=maximum (NetworkVertexCode.data r c b,(i.val,j.val)) ∧
∀ Y,HasMargins Y r c → (∀ u v,Y u v≤b u v) → Y i j≤X i j := by
let S : Finset (BoundedTable r c b) := Finset.univ
have hS : S.Nonempty := by
obtain ⟨X,hm,hb⟩ := hne
exact ⟨⟨⟨X,hm⟩,hb⟩,Finset.mem_univ _⟩
obtain ⟨M,hM,hmax⟩ := Finset.exists_max_image S
(fun X : BoundedTable r c b=>X.val.val i j) hS
have ht (t : ℕ) :
test ((NetworkVertexCode.data r c b,(i.val,j.val)),t)=true ↔ t≤M.val.val i j := by
rw [CellLowerTestProgram.test_correct hEK]
constructor
· rintro ⟨X,hm,hb,hx⟩
exact hx.trans (hmax ⟨⟨X,hm⟩,hb⟩ (Finset.mem_univ _))
· intro hx
exact ⟨M.val.val,M.val.property,M.property,hx⟩
have he : maximum (NetworkVertexCode.data r c b,(i.val,j.val))=M.val.val i j := by
unfold maximum
rw [bound_code]
exact BinarySearchProgram.search_eq test _ _ _ ht (M.property i j)
exact ⟨M.val.val,M.val.property,M.property,he.symm,
fun Y hm hb=>hmax ⟨⟨Y,hm⟩,hb⟩ (Finset.mem_univ _)⟩
end ContingencyTables.CellMaximumProgram
end OAI
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