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d5b7741 | 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 | import OAI.Combinatorics.ContingencyTables.Sampling.ExactPrefixFeasibility
import OAI.Combinatorics.ContingencyTables.Sampling.GreedyPublicProgram
namespace OAI
/-! # Uniform printing cost for every feasible exact-sampling output -/
namespace ContingencyTables.ExactPrintCost
open MatchingFPRAS MatchingFPRAS.TreeTyped MatchingFPRAS.PhysicalStream
open DensePrograms SmallGraphProfiles Polynomial
open scoped BigOperators Classical
noncomputable section
lemma weight_ofFn {A : Type} [Codec A] {n B : ℕ} (f : Fin n → A)
(h : ∀ i,weight (f i) ≤ B) : weight (List.ofFn f) ≤ n*(B+1)+1 := by
have hb := weight_map_bound (f := f) (as := List.finRange n) (fun i _ => h i)
simpa only [←List.ofFn_eq_map,List.length_finRange] using hb
def matrixSize {m n : ℕ} (r : Fin m → ℕ) : ℕ :=
m*(n*(4*weight (∑ i,r i)+2)+2)+1
lemma matrix_weight {m n : ℕ} (r : Fin m → ℕ) (c : Fin n → ℕ) (X : Table r c) :
weight (matrixCode X.val) ≤ matrixSize (n := n) r := by
have row (i : Fin m) : weight (List.ofFn (X.val i)) ≤ n*(4*weight (∑ i,r i)+2)+1 := by
apply weight_ofFn
intro j
exact weight_nat_small (entry_le_total X.property i j)
exact weight_ofFn (fun i => List.ofFn (X.val i)) row
/-- One fixed polynomial bounds both printing time and printed word length
uniformly over the feasible set. -/
theorem polynomial_print_bound : ∃ p : Polynomial ℕ,
∀ (m n : ℕ) (r : Fin m → ℕ) (c : Fin n → ℕ) (X : Table r c),
GreedyFeasibleTable.printMatrixRealizer.cost (matrixCode X.val)+
weight (GreedyFeasibleTable.printMatrix (matrixCode X.val)) ≤ p.eval (matrixSize (n := n) r) := by
obtain ⟨p,hp⟩ := GreedyFeasibleTable.polynomial_printMatrix
exact ⟨p,fun _ _ r c X => (hp _).trans
(polynomial_eval_monotone p (matrix_weight r c X))⟩
lemma prefix_print_bound {m n : ℕ} (r : Fin m → ℕ) (c : Fin n → ℕ)
(htotal : (∑ i,r i)=∑ j,c j) (B : ℕ)
(hB : ∀ X : Table r c,GreedyFeasibleTable.printMatrixRealizer.cost (matrixCode X.val) ≤ B)
(N : ℕ) :
(ExactCoreProgram.prefixExperiment (List.ofFn r,List.ofFn c) N).All
(fun o => o.elim 0 GreedyFeasibleTable.printMatrixRealizer.cost ≤ B) := by
exact (exactPrefix_feasible r c htotal N).mono (by
intro o ho
cases o with
| none => exact Nat.zero_le _
| some X =>
obtain ⟨T,rfl⟩ := ho
exact hB T)
end
end ContingencyTables.ExactPrintCost
end OAI
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