File size: 2,878 Bytes
5d5048e | 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 | import OAI.Combinatorics.LowSpaceSubsetSum.Compression.CompressionPreparedCells
import OAI.Combinatorics.LowSpaceSubsetSum.Storage.ResidueLoopCells
import OAI.Combinatorics.LowSpaceSubsetSum.Compression.BudgetedCompressionSchedule
namespace OAI
/-! # Reusing one compression workspace throughout the residue loop -/
namespace LowSpaceSubsetSum.CompressionScheduleCells
open PreparedCompressionPass (Arrays)
variable {n : ℕ}
def bucket (I : Instance n) (C B : ℕ) (A : Arrays) (P : ℕ) (hP : 0 < P)
(z budget : ℕ) :=
if z < P then
CompressionPreparedCells.run I C B A P I.target z (by exact_mod_cast hP) budget
else ((some (false,budget),0),8)
theorem bucket_value (I : Instance n) (C B : ℕ) (A : Arrays) (P : ℕ) (hP : 0 < P)
(z budget : ℕ) : (bucket I C B A P hP z budget).1 =
BudgetedCompressionSchedule.bucket I C B A P hP z budget := by
unfold bucket BudgetedCompressionSchedule.bucket
split
· exact CompressionPreparedCells.value I C B A P I.target z _ budget
· rfl
def run (I : Instance n) (C B : ℕ) (A : Arrays) (P : ℕ) (hP : 0 < P) (budget : ℕ) :=
let out := ResidueLoopCells.run (bucket I C B A P hP) 0 P budget
(out.1,out.2+4*PreparedCompressionPass.size A+32)
theorem value (I : Instance n) (C B : ℕ) (A : Arrays) (P : ℕ) (hP : 0 < P) (budget : ℕ) :
(run I C B A P hP budget).1 = BudgetedCompressionSchedule.run I C B A P hP budget := by
rw [run,ResidueLoopCells.value]
have he : (fun z b => (bucket I C B A P hP z b).1) =
BudgetedCompressionSchedule.bucket I C B A P hP := by
funext z b
exact bucket_value I C B A P hP z b
rw [he]
rfl
theorem cells (I : Instance n) (hn : 10000 ≤ n) (C B : ℕ) (A : Arrays)
(P : ℕ) (hP : 0 < P) (budget : ℕ) (hB : B ≤ C)
(hL : ∀ i, (A.first i).size ≤ C) (hR : ∀ i, (A.second i).size ≤ C) (hC : C ≤ 2^n) :
(run I C B A P hP budget).2 ≤ inputScale I^9*(C+1) := by
let u := inputScale I
have hu : 10000 ≤ u := by unfold u inputScale; omega
have hlarge : 100 ≤ u^8 := by
have hp : u ≤ u^8 := by
simpa only [pow_one] using Nat.pow_le_pow_right (show 1 ≤ u by omega) (by decide : 1 ≤ 8)
omega
have hb (z b : ℕ) : (bucket I C B A P hP z b).2 ≤ u^8*(C+1) := by
unfold bucket
split
· exact CompressionPreparedCells.cells I hn C B A P I.target z _ b hB hL hR hC
· dsimp only; nlinarith only [hlarge]
have h := ResidueLoopCells.cells (bucket I C B A P hP) (u^8*(C+1)) hb 0 P budget
have hs := CompressionPreparedCells.arrays_size A C hL hR
have hm : 2*u^8*(C+1) ≤ u^9*(C+1) := by
apply Nat.mul_le_mul_right
calc
_ ≤ u*u^8 := Nat.mul_le_mul_right _ (by omega)
_ = _ := (pow_succ' _ _).symm
change _+4*PreparedCompressionPass.size A+32 ≤ _
nlinarith only [h,hs,hlarge,hm]
end LowSpaceSubsetSum.CompressionScheduleCells
end OAI
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