File size: 3,147 Bytes
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 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 | import OAI.Combinatorics.ContingencyTables.Transport.ResidualAugmentation
namespace OAI
/-!
# The bounded deterministic integral max-flow algorithm
Conditional on the augmentation-count bound, concrete residual searches,
bottlenecks and flow updates preserve feasibility and yield a stopping
certificate.
-/
namespace ContingencyTables.EdmondsKarpAlgorithm
open PublishedEdmondsKarp ResidualAugmentation
variable {n : ℕ}
def advance (c : Capacity n) (s t : Fin n) (f : Flow n) : Flow n :=
(step c s t f).getD f
def run (c : Capacity n) (s t : Fin n) : ℕ → Flow n
| 0 => 0
| k+1 => advance c s t (run c s t k)
def maxFlow (c : Capacity n) (s t : Fin n) : Flow n := run c s t (n^3+1)
lemma zero_feasible (c : Capacity n) (s t : Fin n) : Feasible c s t 0 := by
refine ⟨?_,?_,?_⟩
· intro u v; simp
· intro u v
change (0:ℤ)≤(c u v:ℤ)
exact_mod_cast Nat.zero_le (c u v)
· intro u _ _; simp
lemma run_feasible (c : Capacity n) (s t : Fin n) (hst : s≠t) (k : ℕ) :
Feasible c s t (run c s t k) := by
induction k with
| zero => exact zero_feasible c s t
| succ k ih =>
cases hs : step c s t (run c s t k) with
| none => simpa only [run,advance,hs,Option.getD_none] using ih
| some g =>
simpa only [run,advance,hs,Option.getD_some] using
step_feasible c s t hst (run c s t k) g ih hs
lemma run_step (c : Capacity n) (s t : Fin n) (k : ℕ) (g : Flow n)
(h : step c s t (run c s t k)=some g) : run c s t (k+1)=g := by
simp only [run,advance,h,Option.getD_some]
lemma run_stops (c : Capacity n) (s t : Fin n) (k : ℕ)
(h : step c s t (run c s t k)=none) (j : ℕ) :
run c s t (k+j)=run c s t k := by
induction j with
| zero => simp
| succ j ih =>
rw [Nat.add_succ,run,ih,advance,h,Option.getD_none]
theorem termination (hEK : EdmondsKarp1972Theorem1)
(c : Capacity n) (hc : ∀ u,c u u=0) (s t : Fin n) (hst : s≠t) :
∃ i≤n^3,step c s t (run c s t i)=none := by
by_contra hn
have hsome : ∀ i≤n^3,∃ g,step c s t (run c s t i)=some g := by
intro i hi
cases hs : step c s t (run c s t i) with
| none => exact False.elim (hn ⟨i,hi,hs⟩)
| some g => exact ⟨g,rfl⟩
let F : Fin (n^3+2) → Flow n := fun i=>run c s t i.val
have hF : AugmentationTrace c s t F := by
refine ⟨rfl,?_,?_⟩
· intro i
exact run_feasible c s t hst i.val
· intro i
obtain ⟨g,hg⟩ := hsome i.val (by have hi:=i.isLt; omega)
have he : F i.succ=g := run_step c s t i.val g hg
rw [he]
exact step_augmentation c s t hst (F i.castSucc) g hg
exact no_long_trace hEK c hc s t hst F hF
theorem maxFlow_correct (hEK : EdmondsKarp1972Theorem1)
(c : Capacity n) (hc : ∀ u,c u u=0) (s t : Fin n) (hst : s≠t) :
IsMaximum c s t (maxFlow c s t) := by
obtain ⟨i,hi,hs⟩ := termination hEK c hc s t hst
have hstop := run_stops c s t i hs (n^3+1-i)
have he : i+(n^3+1-i)=n^3+1 := by omega
rw [he] at hstop
unfold maxFlow
rw [hstop]
exact step_none_maximum c s t (run c s t i) (run_feasible c s t hst i) hs
end ContingencyTables.EdmondsKarpAlgorithm
end OAI
|