| import Mathlib |
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| namespace OAI |
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| section |
| section |
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| noncomputable section |
| namespace HyperbolicObstruction.PlanarRecursion |
| open SimpleGraph |
| attribute [local instance] Classical.propDecidable |
| variable {V : Type*} |
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| lemma exists_distance_parent (G : SimpleGraph V) (hG : G.Connected) |
| (root v : V) (hv : v ≠ root) : |
| ∃ w, G.Adj v w ∧ G.dist root w + 1 = G.dist root v := by |
| obtain ⟨p,hp⟩ := hG.exists_walk_length_eq_dist v root |
| cases p with |
| | nil => exact (hv rfl).elim |
| | @cons v w root h p => |
| refine ⟨w,h,?_⟩ |
| have hh := dist_le p |
| have hd := h.diff_dist_adj (u := root) |
| rw [G.dist_comm (u := w) (v := root)] at hh |
| rw [G.dist_comm (u := v) (v := root)] at hp |
| simp only [Walk.length_cons] at hp |
| omega |
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| variable [Fintype V] |
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|
| /-- A literal shortest-path spanning tree of a finite connected graph. |
| Root distances are preserved exactly, not merely bounded by the number of |
| vertices. This is the radius-control input for the planar separator. -/ |
| theorem exists_bfs_tree (G : SimpleGraph V) (hG : G.Connected) (root : V) : |
| ∃ T : SimpleGraph V, T ≤ G ∧ T.IsTree ∧ ∀ v, T.dist root v = G.dist root v := by |
| classical |
| have : Nonempty V := ⟨root⟩ |
| let parent : V → V := fun v => if h : v = root then root else |
| (exists_distance_parent G hG root v h).choose |
| have hp : ∀ v, v ≠ root → G.Adj v (parent v) ∧ |
| G.dist root (parent v)+1 = G.dist root v := by |
| intro v hv |
| dsimp only [parent] |
| rw [dite_eq_right hv] |
| exact (exists_distance_parent G hG root v hv).choose_spec |
| let T : SimpleGraph V := { |
| Adj := fun v w => (v ≠ root ∧ parent v = w) ∨ (w ≠ root ∧ parent w = v) |
| symm := ⟨by intro v w h; exact h.elim Or.inr Or.inl⟩ |
| loopless := ⟨by |
| intro v h |
| rcases h with ⟨hv,he⟩ | ⟨hv,he⟩ <;> |
| exact (hp v hv).1.ne he.symm⟩ } |
| have hle : T ≤ G := by |
| intro v w h |
| rcases h with ⟨hv,rfl⟩ | ⟨hw,rfl⟩ |
| · exact (hp v hv).1 |
| · exact (hp w hw).1.symm |
| have hpaths : ∀ v, ∃ p : T.Walk v root, p.length = G.dist root v := by |
| intro v |
| generalize hn : G.dist root v = n |
| induction n using Nat.strong_induction_on generalizing v with |
| | h n ih => |
| by_cases hv : v = root |
| · subst v |
| exact ⟨.nil,by simpa only [Walk.length_nil,SimpleGraph.dist_self] using hn⟩ |
| · have hd := (hp v hv).2 |
| obtain ⟨p,hp'⟩ := ih (G.dist root (parent v)) (by omega) (parent v) rfl |
| exact ⟨p.cons (Or.inl ⟨hv,rfl⟩),by simp only [Walk.length_cons,hp']; omega⟩ |
| have hconn : T.Connected := by |
| apply Connected.mk |
| intro v w |
| exact (show T.Reachable v root from ⟨(hpaths v).choose⟩).trans |
| (show T.Reachable w root from ⟨(hpaths w).choose⟩).symm |
| have hcard : Nat.card T.edgeSet + 1 ≤ Nat.card V := by |
| let E := (Finset.univ.erase root).image (fun v => s(v,parent v)) |
| have hsub : T.edgeFinset ⊆ E := by |
| intro e he |
| induction e using Sym2.inductionOn with |
| | hf v w => |
| have hadj : T.Adj v w := by simpa using he |
| rcases hadj with ⟨hv,rfl⟩ | ⟨hw,rfl⟩ |
| · exact Finset.mem_image.mpr ⟨v,Finset.mem_erase.mpr ⟨hv,Finset.mem_univ _⟩,rfl⟩ |
| · exact Finset.mem_image.mpr ⟨w,Finset.mem_erase.mpr ⟨hw,Finset.mem_univ _⟩,Sym2.eq_swap⟩ |
| have hh := (Finset.card_le_card hsub).trans (Finset.card_image_le (s := Finset.univ.erase root) |
| (f := fun v => s(v,parent v))) |
| have he : (Finset.univ.erase root).card + 1 = Fintype.card V := by |
| have hh : 0 < Fintype.card V := Fintype.card_pos |
| simp only [Finset.card_erase_of_mem (Finset.mem_univ root),Finset.card_univ] |
| omega |
| rw [edgeFinset_card] at hh |
| simp only [Nat.card_eq_fintype_card] |
| omega |
| have hT : T.IsTree := by |
| apply isTree_iff_connected_and_card.mpr |
| exact ⟨hconn,Nat.le_antisymm hcard hconn.card_vert_le_card_edgeSet_add_one⟩ |
| refine ⟨T,hle,hT,?_⟩ |
| intro v |
| apply le_antisymm |
| · obtain ⟨p,hp'⟩ := hpaths v |
| simpa only [T.dist_comm (u := v) (v := root),hp'] using dist_le p |
| · exact (hconn root v).dist_anti hle |
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| end HyperbolicObstruction.PlanarRecursion |
| end |
| end |
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| section |
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| end |
| end |
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| end OAI |
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