## Dijkstra's shortest path algorithm

Auteurs: Jean-Christophe Filliâtre

Catégories: Graph Algorithms

Outils: Why3

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```(* Dijkstra's shortest path algorithm.

This proof follows Cormen et al's "Algorithms".

Author: Jean-Christophe FilliĆ¢tre (CNRS) *)

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 DijkstraShortestPath

use int.Int
use ref.Ref

```

The graph is introduced as a set v of vertices and a function g_succ returning the successors of a given vertex. The weight of an edge is defined independently, using function weight. The weight is an integer.

```  type vertex

clone set.SetImp with type elt = vertex
clone ImpmapNoDom with type key = vertex

constant v: fset vertex

val ghost function g_succ (_x: vertex) : fset vertex
ensures { subset result v }

val get_succs (x: vertex): set
ensures { result = g_succ x  }

val function weight vertex vertex : int (* edge weight, if there is an edge *)
ensures { result >= 0 }

```

Data structures for the algorithm.

```  (* The set of already visited vertices. *)

val visited: set

(* Map d holds the current distances from the source.
There is no need to introduce infinite distances. *)

val d: t int

(* The priority queue. *)

val q: set

predicate min (m: vertex) (q: set) (d: t int) =
mem m q /\
forall x: vertex. mem x q -> d[m] <= d[x]

val q_extract_min () : vertex writes {q}
requires { not is_empty q }
ensures  { min result (old q) d }
ensures  { q = remove result (old q) }

(* Initialisation of visited, q, and d. *)

val init (src: vertex) : unit writes { visited, q, d }
ensures { is_empty visited }
ensures { q = singleton src }
ensures { d = (old d)[src <- 0] }

(* Relaxation of edge u->v. *)

let relax u v
ensures {
(mem v visited /\ q = old q /\ d = old d)
\/
(mem v q /\ d[u] + weight u v >= d[v] /\ q = old q /\ d = old d)
\/
(mem v q /\ (old d)[u] + weight u v < (old d)[v] /\
q = old q /\ d = (old d)[v <- (old d)[u] + weight u v])
\/
(not mem v visited /\ not mem v (old q) /\
q = add v (old q) /\
d = (old d)[v <- (old d)[u] + weight u v]) }
= if not mem v visited then
let x = d[u] + weight u v in
if mem v q then begin
if x < d[v] then d[v] <- x
end else begin
d[v] <- x
end

(* Paths and shortest paths.

path x y d =
there is a path from x to y of length d

shortest_path x y d =
there is a path from x to y of length d, and no shorter path *)

inductive path vertex vertex int =
| Path_nil :
forall x: vertex. path x x 0
| Path_cons:
forall x y z: vertex. forall d: int.
path x y d -> mem z (g_succ y) -> path x z (d + weight y z)

lemma Length_nonneg: forall x y: vertex. forall d: int. path x y d -> d >= 0

predicate shortest_path (x y: vertex) (d: int) =
path x y d /\ forall d': int. path x y d' -> d <= d'

lemma Path_inversion:
forall src v:vertex. forall d:int. path src v d ->
(v = src /\ d = 0) \/
(exists v':vertex. path src v' (d - weight v' v) /\ mem v (g_succ v'))

lemma Path_shortest_path:
forall src v: vertex. forall d: int. path src v d ->
exists d': int. shortest_path src v d' /\ d' <= d

(* Lemmas (requiring induction). *)

lemma Main_lemma:
forall src v: vertex. forall d: int.
path src v d -> not (shortest_path src v d) ->
v = src /\ d > 0
\/
exists v': vertex. exists d': int.
shortest_path src v' d' /\ mem v (g_succ v') /\ d' + weight v' v < d

lemma Completeness_lemma:
forall s: set.
(* if s is closed under g_succ *)
(forall v: vertex.
mem v s -> forall w: vertex. mem w (g_succ v) -> mem w s) ->
(* and if s contains src *)
forall src: vertex. mem src s ->
(* then any vertex reachable from s is also in s *)
forall dst: vertex. forall d: int.
path src dst d -> mem dst s

(* Definitions used in loop invariants. *)

predicate inv_src (src: vertex) (s q: set) =
mem src s \/ mem src q

predicate inv (src: vertex) (s q: set) (d: t int) =
inv_src src s q /\ d[src] = 0 /\
(* S and Q are contained in V *)
subset s v /\ subset q v /\
(* S and Q are disjoint *)
(forall v: vertex. mem v q -> mem v s -> false) /\
(* we already found the shortest paths for vertices in S *)
(forall v: vertex. mem v s -> shortest_path src v d[v]) /\
(* there are paths for vertices in Q *)
(forall v: vertex. mem v q -> path src v d[v])

predicate inv_succ (_src: vertex) (s q: set) (d: t int) =
(* successors of vertices in S are either in S or in Q *)
forall x: vertex. mem x s ->
forall y: vertex. mem y (g_succ x) ->
(mem y s \/ mem y q) /\ d[y] <= d[x] + weight x y

predicate inv_succ2 (_src: vertex) (s q: set) (d: t int) (u: vertex) (su: set) =
(* successors of vertices in S are either in S or in Q,
unless they are successors of u still in su *)
forall x: vertex. mem x s ->
forall y: vertex. mem y (g_succ x) ->
(x<>u \/ (x=u /\ not (mem y su))) ->
(mem y s \/ mem y q) /\ d[y] <= d[x] + weight x y

lemma inside_or_exit:
forall s, src, v, d. mem src s -> path src v d ->
mem v s
\/
exists y. exists z. exists dy.
mem y s /\ not (mem z s) /\ mem z (g_succ y) /\
path src y dy /\ (dy + weight y z <= d)

(* Algorithm's code. *)

let shortest_path_code (src dst: vertex)
requires { mem src v /\ mem dst v }
ensures  { forall v: vertex. mem v visited ->
shortest_path src v d[v] }
ensures  { forall v: vertex. not mem v visited ->
forall dv: int. not path src v dv }
= init src;
while not is_empty q do
invariant { inv src visited q d }
invariant { inv_succ src visited q d }
invariant { (* vertices at distance < min(Q) are already in S *)
forall m: vertex. min m q d ->
forall x: vertex. forall dx: int. path src x dx ->
dx < d[m] -> mem x visited }
variant   { cardinal v - cardinal visited }
let u = q_extract_min () in
assert { shortest_path src u d[u] };
let su = get_succs u in
while not is_empty su do
invariant { subset su (g_succ u) }
invariant { inv src visited q d  }
invariant { inv_succ2 src visited q d u su }
variant   { cardinal su }
let v = choose_and_remove su in
relax u v;
assert { d[v] <= d[u] + weight u v }
done
done

end
```