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Bellman-Ford algorithm

Find all single-source shortest paths in a weighted directed graph, with detection of negative cycles.


Authors: Jean-Christophe Filliâtre / Yuto Takei

Topics: Graph Algorithms / Ghost code

Tools: Why3

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A proof of Bellman-Ford algorithm

By Yuto Takei (University of Tokyo, Japan) and Jean-Christophe FilliĆ¢tre (CNRS, France).

theory Graph

  use export list.List
  use export list.Append
  use export list.Length
  use export int.Int
  use export set.Fset

  (* the graph is defined by a set of vertices and a set of edges *)
  type vertex
  constant vertices: fset vertex
  constant edges: fset (vertex, vertex)

  predicate edge (x y: vertex) = mem (x,y) edges

  (* edges are well-formed *)
  axiom edges_def:
    forall x y: vertex.
    mem (x, y) edges -> mem x vertices /\ mem y vertices

  (* a single source vertex s is given *)
  val constant s: vertex
    ensures { mem result vertices }

  (* hence vertices is non empty *)
  lemma vertices_cardinal_pos: cardinal vertices > 0

  val constant nb_vertices: int
    ensures { result = cardinal vertices }

  (* paths *)
  clone export graph.IntPathWeight
     with type vertex = vertex, predicate edge = edge

  lemma path_in_vertices:
    forall v1 v2: vertex, l: list vertex.
    mem v1 vertices -> path v1 l v2 -> mem v2 vertices

  (* A key idea behind Bellman-Ford's correctness is that of simple path:
     if there is a path from s to v, there is a path with less than
     |V| edges. We formulate this in a slightly more precise way: if there
     a path from s to v with at least |V| edges, then there must be a duplicate
     vertex along this path. There is a subtlety here: since the last vertex
     of a path is not part of the vertex list, we distinguish the case where
     the duplicate vertex is the final vertex v.

     Proof: Assume [path s l v] with length l >= |V|.
       Consider the function f mapping i in {0..|V|} to the i-th element
       of the list l ++ [v]. Since all elements of the
       path (those in l and v) belong to V, then by the pigeon hole principle,
       f cannot be injective from 0..|V| to V. Thus there exist two distinct
       i and j in 0..|V| such that f(i)=f(j), which means that
              l ++ [v] = l1 ++ [u] ++ l2 ++ [u] ++ l3
       Two cases depending on l3=[] (and thus u=v) conclude the proof. Qed.
  *)

  clone pigeon.ListAndSet with type t = vertex

  predicate cyc_decomp (v: vertex) (l: list vertex)
                       (vi: vertex) (l1 l2 l3: list vertex) =
    l = l1 ++ l2 ++ l3 /\ length l2 > 0 /\
    path s l1 vi /\ path vi l2 vi /\ path vi l3 v

  lemma long_path_decomposition:
    forall l v. path s l v /\ length l >= cardinal vertices ->
      (exists vi l1 l2 l3. cyc_decomp v l vi l1 l2 l3)
      by exists n l1 l2 l3. Cons v l = l1 ++ Cons n (l2 ++ Cons n l3)
      so match l1 with
        | Nil -> cyc_decomp v l v l2 (Cons n l3) Nil
        | Cons v l1 -> cyc_decomp v l n l1 (Cons n l2) (Cons n l3)
        end

  lemma long_path_reduction:
    forall l v. path s l v /\ length l >= cardinal vertices ->
      exists l'. path s l' v /\ length l' < length l
      by exists vi l1 l2 l3. cyc_decomp v l vi l1 l2 l3 /\ l' = l1 ++ l3

  let lemma simple_path (v: vertex) (l: list vertex) : unit
    requires { path s l v }
    ensures  { exists l'. path s l' v /\ length l' < cardinal vertices }
  = let rec aux (n: int) : unit
      ensures { forall l. length l <= n /\ path s l v ->
          exists l'. path s l' v /\ length l' < cardinal vertices }
      variant { n }
    = if n > 0 then aux (n-1)
    in
    aux (length l)

  (* negative cycle [v] -> [v] reachable from [s] *)
  predicate negative_cycle (v: vertex) =
    mem v vertices /\
    (exists l1: list vertex. path s l1 v) /\
    (exists l2: list vertex. path v l2 v /\ path_weight l2 v < 0)

  (* key lemma for existence of a negative cycle

     Proof: by induction on the (list) length of the shorter path l
       If length l < cardinal vertices, then it contradicts hypothesis 1
       thus length l >= cardinal vertices and thus the path contains a cycle
             s ----> n ----> n ----> v
       If the weight of the cycle n--->n is negative, we are done.
       Otherwise, we can remove this cycle from the path and we get
       an even shorter path, with a stricltly shorter (list) length,
       thus we can apply the induction hypothesis.                     Qed.
   *)
  predicate all_path_gt (v: vertex) (n: int) =
    forall l. path s l v /\ length l < cardinal vertices -> path_weight l v >= n

  let lemma key_lemma_1 (v: vertex) (l: list vertex) (n: int) : unit
    (* if any simple path has weight at least n *)
    requires { all_path_gt v n }
    (* and if there exists a shorter path *)
    requires { path s l v /\ path_weight l v < n }
    (* then there exists a negative cycle. *)
    ensures  { exists u. negative_cycle u }
  = let rec aux (m: int) : (_a: 'a)
      requires { forall u. not negative_cycle u }
      requires { exists l. path s l v /\ path_weight l v < n /\ length l <= m }
      ensures  { false }
      variant  { m }
    = assert { (exists l'. path s l' v /\ path_weight l' v < n /\ length l' < m)
        by exists l. path s l v /\ path_weight l v < n /\ length l <= m
        so exists vi l1 l2 l3. cyc_decomp v l vi l1 l2 l3
        so let res = l1 ++ l3 in
          path s res v /\ length res < length l /\ path_weight res v < n
          by path_weight l v =
            path_weight l1 vi + path_weight l2 vi + path_weight l3 v
          so path_weight l2 vi >= 0 by not negative_cycle vi
      };
      aux (m-1)
    in
    if pure { forall u. not negative_cycle u } then aux (length l)

end

module ImpmapNoDom

  use map.Map
  use map.Const

  type key

  type t 'a = abstract { mutable contents: map key 'a }

  val function create (x: 'a): t 'a
    ensures { result.contents = const x }

  val function ([]) (m: t 'a) (k: key): 'a
    ensures { result = m.contents[k] }

  val ghost function ([<-]) (m: t 'a) (k: key) (v: 'a): t 'a
    ensures { result.contents = m.contents[k <- v] }

  val ([]<-) (m: t 'a) (k: key) (v: 'a): unit
    writes { m }
    ensures { m = (old m)[k <- v] }

end

module BellmanFord

  use Graph
  use int.IntInf as D
  use ref.Ref
  clone set.SetImp as S with type elt = (vertex, vertex)
  clone ImpmapNoDom with type key = vertex

  type distmap = ImpmapNoDom.t D.t

  let initialize_single_source (s: vertex): distmap
    ensures { result = (create D.Infinite)[s <- D.Finite 0] }
  =
    let m = create D.Infinite in
    m[s] <- D.Finite 0;
    m

  (* [inv1 m pass via] means that we already performed [pass-1] steps
     of the main loop, and, in step [pass], we already processed edges
     in [via] *)
  predicate inv1 (m: distmap) (pass: int) (via: fset (vertex, vertex)) =
    forall v: vertex. mem v vertices ->
    match m[v] with
      | D.Finite n ->
        (* there exists a path of weight [n] *)
        (exists l: list vertex. path s l v /\ path_weight l v = n) /\
        (* there is no shorter path in less than [pass] steps *)
        (forall l: list vertex.
           path s l v -> length l < pass -> path_weight l v >= n) /\
        (* and no shorter path in i steps with last edge in [via] *)
        (forall u: vertex, l: list vertex.
           path s l u -> length l < pass -> mem (u,v) via ->
           path_weight l u + weight u v >= n)
      | D.Infinite ->
        (* any path has at least [pass] steps *)
        (forall l: list vertex. path s l v -> length l >= pass) /\
        (* [v] cannot be reached by [(u,v)] in [via] *)
        (forall u: vertex. mem (u,v) via -> (*m[u] = D.Infinite*)
          forall lu: list vertex. path s lu u -> length lu >= pass)
    end

  predicate inv2 (m: distmap) (via: fset (vertex, vertex)) =
    forall u v: vertex. mem (u, v) via ->
    D.le m[v] (D.add m[u] (D.Finite (weight u v)))

  let rec lemma inv2_path (m: distmap) (y z: vertex) (l:list vertex) : unit
    requires { inv2 m edges }
    requires { path y l z }
    ensures  { D.le m[z] (D.add m[y] (D.Finite (path_weight l z))) }
    variant  { length l }
  = match l with
    | Nil -> ()
    | Cons _ q ->
      let hd = match q with
        | Nil -> z
        | Cons h _ ->
          assert { path_weight l z = weight y h + path_weight q z };
          assert { D.le m[h] (D.add m[y] (D.Finite (weight y h))) };
          h
        end in
      inv2_path m hd z q
    end

  (* key lemma for non-existence of a negative cycle

     Proof: let us assume a negative cycle reachable from s, that is
         s --...--> x0 --w1--> x1 --w2--> x2 ... xn-1 --wn--> x0
       with w1+w2+...+wn < 0.
       Let di be the distance from s to xi as given by map m.
       By [inv2 m edges] we have di-1 + wi >= di for all i.
       Summing all such inequalities over the cycle, we get
            w1+w2+...+wn >= 0
       hence a contradiction.                                  Qed. *)
  lemma key_lemma_2:
     forall m: distmap. inv1 m (cardinal vertices) empty -> inv2 m edges ->
     forall v: vertex. not (negative_cycle v
       so exists l1. path s l1 v
       so exists l2. path v l2 v /\ path_weight l2 v < 0
       so D.le m[v] (D.add m[v] (D.Finite (path_weight l2 v)))
     )

  let relax (m: distmap) (u v: vertex) (pass: int)
            (ghost via: fset (vertex, vertex))
    requires { 1 <= pass /\ mem (u, v) edges /\ not (mem (u, v) via) }
    requires { inv1 m pass via }
    ensures  { inv1 m pass (add (u, v) via) }
  = label I in
    let n = D.add m[u] (D.Finite (weight u v)) in
    if D.lt n m[v]
    then begin
      m[v] <- n;
      assert { match (m at I)[u] with
        | D.Infinite -> false
        | D.Finite w0 ->
          let w1 = w0 + weight u v in
          n = D.Finite w1
          && (exists l. path s l v /\ path_weight l v = w1
            by exists l2.
              path s l2 u /\ path_weight l2 u = w0 /\ l = l2 ++ Cons u Nil
          ) && (forall l. path s l v /\ length l < pass -> path_weight l v >= w1
            by match (m at I)[v] with
            | D.Infinite -> false
            | D.Finite w2 -> path_weight l v >= w2
            end
          ) && (forall z l. path s l z /\ length l < pass ->
            mem (z,v) (add (u,v) via) ->
            path_weight l z + weight z v >= w1
            by if z = u then path_weight l z >= w0
            else match (m at I)[v] with
              | D.Infinite -> false
              | D.Finite w2 -> path_weight l z + weight z v >= w2
              end
          )
        end
      }
    end else begin
      assert { forall l w1. path s l u /\ length l < pass /\ m[v] = D.Finite w1 ->
        path_weight l u + weight u v >= w1
        by match m[u] with
        | D.Infinite -> false
        | D.Finite w0 -> path_weight l u >= w0
        end
      }
    end

  val get_edges (): S.set
    ensures { result = edges }

  exception NegativeCycle

  let bellman_ford ()
    ensures { forall v: vertex. mem v vertices ->
      match result[v] with
        | D.Finite n ->
            (exists l: list vertex. path s l v /\ path_weight l v = n) /\
            (forall l: list vertex. path s l v -> path_weight l v >= n
             by all_path_gt v n)
        | D.Infinite ->
            (forall l: list vertex. not (path s l v))
      end }
    raises { NegativeCycle -> exists v: vertex. negative_cycle v }
  = let m = initialize_single_source s in
    for i = 1 to nb_vertices - 1 do
      invariant { inv1 m i empty }
      let es = get_edges () in
      while not (S.is_empty es) do
        invariant { subset es.S.to_fset edges /\ inv1 m i (diff edges es.S.to_fset) }
        variant   { S.cardinal es }
        let ghost via = diff edges es.S.to_fset in
        let (u, v) = S.choose_and_remove es in
        relax m u v i via
      done;
      assert { inv1 m i edges }
    done;
    assert { inv1 m (cardinal vertices) empty };
    let es = get_edges () in
    while not (S.is_empty es) do
      invariant { subset es.S.to_fset edges /\ inv2 m (diff edges es.S.to_fset) }
      variant { S.cardinal es }
      let (u, v) = S.choose_and_remove es in
      if D.lt (D.add m[u] (D.Finite (weight u v))) m[v] then begin
        assert { match m[u], m[v] with
          | D.Infinite, _ -> false
          | D.Finite _, D.Infinite -> false
            by exists l2. path s l2 u
            so let l = l2 ++ Cons u Nil in path s l v
            so exists l. path s l v /\ length l < cardinal vertices
          | D.Finite wu, D.Finite wv ->
            (exists w. negative_cycle w)
            by
            all_path_gt v wv
            so exists l2. path s l2 u /\ path_weight l2 u = wu
            so let l = l2 ++ Cons u Nil in
              path s l v /\ path_weight l v < wv
          end
        };
        raise NegativeCycle
      end
    done;
    assert { inv2 m edges };
    assert { forall u. not (negative_cycle u) };
    m

end

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Why3 Proof Results for Project "bellman_ford"

Theory "bellman_ford.Graph": fully verified

ObligationsAlt-Ergo 2.0.0Alt-Ergo 2.3.0Eprover 2.0
VC for s---0.00---
vertices_cardinal_pos0.00------
path_in_vertices0.01------
long_path_decomposition---------
split_goal_right
long_path_decomposition.00.17------
long_path_decomposition.10.13------
long_path_decomposition.20.17------
long_path_decomposition.30.16------
long_path_reduction---------
split_goal_right
long_path_reduction.0------0.04
long_path_reduction.10.01------
long_path_reduction.20.01------
VC for simple_path0.87------
VC for key_lemma_1---------
split_goal_right
assertion---------
split_goal_right
VC for key_lemma_10.01------
VC for key_lemma_10.13------
VC for key_lemma_10.33------
VC for key_lemma_10.02------
VC for key_lemma_10.04------
VC for key_lemma_10.09------
VC for key_lemma_10.09------
VC for key_lemma_10.27------
VC for key_lemma_10.02------
variant decrease0.02------
precondition0.01------
precondition0.01------
postcondition0.01------
precondition0.02------
precondition0.02------
postcondition0.01------

Theory "bellman_ford.BellmanFord": fully verified

ObligationsAlt-Ergo 2.0.0Alt-Ergo 2.2.0CVC4 1.6Eprover 2.0
VC for initialize_single_source0.01---------
VC for inv2_path------------
split_goal_right
assertion0.02---------
assertion0.08---------
variant decrease0.03---------
precondition0.01---------
precondition0.22---------
postcondition------------
split_goal_right
postcondition0.05---------
postcondition------------
split_goal_right
postcondition0.09---------
postcondition1.65---------
key_lemma_2------------
split_goal_right
key_lemma_2.00.62---------
key_lemma_2.10.02---------
key_lemma_2.20.02---------
key_lemma_2.30.01---------
VC for relax------------
split_goal_right
assertion------------
split_goal_right
VC for relax0.03---------
VC for relax0.04---------
VC for relax---------1.84
VC for relax0.03---------
VC for relax0.06---------
VC for relax0.07---------
VC for relax0.12---------
VC for relax0.13---------
VC for relax0.02---------
VC for relax0.03---------
VC for relax0.04---------
VC for relax0.04---------
postcondition------------
introduce_premises
postcondition------------
inline_goal
postcondition------------
introduce_premises
postcondition------------
case (v=v1)
true case (postcondition)------------
split_goal_right
VC for relax0.04---------
VC for relax0.15---------
VC for relax1.25---------
VC for relax0.02---------
VC for relax0.03---------
false case (postcondition)0.36---------
assertion------------
split_goal_right
assertion0.05---------
assertion0.09---------
VC for relax0.06---------
postcondition------------
introduce_premises
postcondition------------
inline_goal
postcondition------------
split_goal_right
postcondition0.05---------
postcondition0.09---------
postcondition0.09---------
postcondition0.04---------
postcondition0.69---------
VC for bellman_ford------------
split_goal_right
loop invariant init------------
introduce_premises
loop invariant init------------
inline_goal
loop invariant init------------
split_goal_right
loop invariant init---------0.58
loop invariant init0.02---------
loop invariant init0.07---------
loop invariant init0.02---------
loop invariant init0.06---------
loop invariant init0.06---------
precondition0.01---------
precondition0.06---------
precondition0.02---------
loop variant decrease0.03---------
loop invariant preservation------------
split_goal_right
VC for bellman_ford------0.10---
VC for bellman_ford---0.99------
assertion0.04---------
loop invariant preservation0.34---------
assertion0.02---------
loop invariant init0.03---------
precondition0.02---------
assertion------------
split_goal_right
VC for bellman_ford0.06---------
VC for bellman_ford0.07---------
VC for bellman_ford0.04---------
VC for bellman_ford0.32---------
VC for bellman_ford0.05---------
VC for bellman_ford0.04---------
VC for bellman_ford0.20---------
VC for bellman_ford0.14---------
VC for bellman_ford0.08---------
VC for bellman_ford0.21---------
VC for bellman_ford0.04---------
exceptional postcondition0.01---------
loop variant decrease0.03---------
loop invariant preservation0.27---------
assertion0.06---------
assertion0.04---------
postcondition------------
split_goal_right
postcondition0.05---------
postcondition0.32---------
postcondition0.05---------
VC for bellman_ford0.01---------
out of loop bounds0.01---------