| 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 |
|
|
