Floyd-Warshall algorithm

Floyd-Warshall algorithm
You are encouraged to solve this task according to the task description, using any language you may know.

The Floyd–Warshall algorithm is an algorithm for finding shortest paths in a weighted graph with positive or negative edge weights.

Find the lengths of the shortest paths between all pairs of vertices of the given directed graph. Your code may assume that the input has already been checked for loops, parallel edges and negative cycles.

Print the pair, the distance and (optionally) the path.

Example
```pair     dist    path
1 -> 2    -1     1 -> 3 -> 4 -> 2
1 -> 3    -2     1 -> 3
1 -> 4     0     1 -> 3 -> 4
2 -> 1     4     2 -> 1
2 -> 3     2     2 -> 1 -> 3
2 -> 4     4     2 -> 1 -> 3 -> 4
3 -> 1     5     3 -> 4 -> 2 -> 1
3 -> 2     1     3 -> 4 -> 2
3 -> 4     2     3 -> 4
4 -> 1     3     4 -> 2 -> 1
4 -> 2    -1     4 -> 2
4 -> 3     1     4 -> 2 -> 1 -> 3```

11l

Translation of: Python
```F floyd_warshall(n, edge)
V rn = 0 .< n
V dist = rn.map(i -> [1'000'000] * @n)
V nxt  = rn.map(i -> [0]         * @n)
L(i) rn
dist[i][i] = 0
L(u, v, w) edge
dist[u - 1][v - 1] = w
nxt[u - 1][v - 1] = v - 1
L(k, i, j) cart_product(rn, rn, rn)
V sum_ik_kj = dist[i][k] + dist[k][j]
I dist[i][j] > sum_ik_kj
dist[i][j] = sum_ik_kj
nxt[i][j] = nxt[i][k]
print(‘pair      dist     path’)
L(i, j) cart_product(rn, rn)
I i != j
V path = [i]
L path.last != j
path.append(nxt[path.last][j])
print(‘#. -> #.  #4       #.’.format(i + 1, j + 1, dist[i][j], path.map(p -> String(p + 1)).join(‘ -> ’)))

floyd_warshall(4, [(1, 3, -2), (2, 1, 4), (2, 3, 3), (3, 4, 2), (4, 2, -1)])```
Output:
```pair      dist     path
1 -> 2    -1       1 -> 3 -> 4 -> 2
1 -> 3    -2       1 -> 3
1 -> 4     0       1 -> 3 -> 4
2 -> 1     4       2 -> 1
2 -> 3     2       2 -> 1 -> 3
2 -> 4     4       2 -> 1 -> 3 -> 4
3 -> 1     5       3 -> 4 -> 2 -> 1
3 -> 2     1       3 -> 4 -> 2
3 -> 4     2       3 -> 4
4 -> 1     3       4 -> 2 -> 1
4 -> 2    -1       4 -> 2
4 -> 3     1       4 -> 2 -> 1 -> 3
```

360 Assembly

Translation of: Rexx
```*        Floyd-Warshall algorithm - 06/06/2018
FLOYDWAR CSECT
USING  FLOYDWAR,R13       base register
B      72(R15)            skip savearea
DC     17F'0'             savearea
SAVE   (14,12)            save previous context
MVC    A+8,=F'-2'         a(1,3)=-2
MVC    A+VV*4,=F'4'       a(2,1)= 4
MVC    A+VV*4+8,=F'3'     a(2,3)= 3
MVC    A+VV*8+12,=F'2'    a(3,4)= 2
MVC    A+VV*12+4,=F'-1'   a(4,2)=-1
LA     R8,1               k=1
DO WHILE=(C,R8,LE,V)        do k=1 to v
LA     R10,A                @a
LA     R6,1                 i=1
DO WHILE=(C,R6,LE,V)          do i=1 to v
LA     R7,1                   j=1
DO WHILE=(C,R7,LE,V)            do j=1 to v
LR     R1,R6                    i
BCTR   R1,0
MH     R1,=AL2(VV)
AR     R1,R8                    k
SLA    R1,2
L      R9,A-4(R1)               a(i,k)
LR     R1,R8                    k
BCTR   R1,0
MH     R1,=AL2(VV)
AR     R1,R7                    j
SLA    R1,2
L      R3,A-4(R1)               a(k,j)
AR     R9,R3                    w=a(i,k)+a(k,j)
L      R2,0(R10)                a(i,j)
IF CR,R2,GT,R9 THEN               if a(i,j)>w then
ST     R9,0(R10)                  a(i,j)=w
ENDIF    ,                        endif
LA     R10,4(R10)               next @a
LA     R7,1(R7)                 j++
ENDDO    ,                      enddo j
LA     R6,1(R6)               i++
ENDDO    ,                    enddo i
LA     R8,1(R8)             k++
ENDDO    ,                  enddo k
LA     R10,A              @a
LA     R6,1               f=1
DO WHILE=(C,R6,LE,V)        do f=1 to v
LA     R7,1                 t=1
DO WHILE=(C,R7,LE,V)          do t=1 to v
IF CR,R6,NE,R7 THEN             if f^=t then do
LR     R1,R6                    f
XDECO  R1,XDEC                  edit f
MVC    PG+0(4),XDEC+8           output f
LR     R1,R7                    t
XDECO  R1,XDEC                  edit t
MVC    PG+8(4),XDEC+8           output t
L      R2,0(R10)                a(f,t)
XDECO  R2,XDEC                  edit a(f,t)
MVC    PG+12(4),XDEC+8          output a(f,t)
XPRNT  PG,L'PG                  print
ENDIF    ,                      endif
LA     R10,4(R10)             next @a
LA     R7,1(R7)               t++
ENDDO    ,                    enddo t
LA     R6,1(R6)             f++
ENDDO    ,                  enddo f
L      R13,4(0,R13)       restore previous savearea pointer
RETURN (14,12),RC=0       restore registers from calling sav
VV       EQU    4
V        DC     A(VV)
A        DC     (VV*VV)F'99999999' a(vv,vv)
PG       DC     CL80'   . ->    .   .'
XDEC     DS     CL12
YREGS
END    FLOYDWAR```
Output:
```   1 ->    2  -1
1 ->    3  -2
1 ->    4   0
2 ->    1   4
2 ->    3   2
2 ->    4   4
3 ->    1   5
3 ->    2   1
3 ->    4   2
4 ->    1   3
4 ->    2  -1
4 ->    3   1
```

Translation of: Scheme

```--
-- Floyd-Warshall algorithm.
--
-- See https://en.wikipedia.org/w/index.php?title=Floyd%E2%80%93Warshall_algorithm&oldid=1082310013
--

with Interfaces;  use Interfaces;

is
Floyd_Warshall_Exception : exception;

-- The floating point type we shall use is one that has infinities.
subtype FloatPt is IEEE_Float_32;
.Generic_Elementary_Functions
(FloatPt);
use FloatPt_Elementary_Functions;

-- The following should overflow and give us an IEEE infinity. But I
-- have kept the code so you could use some non-IEEE floating point
-- format and set ENORMOUS_FloatPt to some value that is finite but
-- much larger than actual graph traversal distances.
ENORMOUS_FloatPt : constant FloatPt :=
(FloatPt (1.0) / FloatPt (1.0e-37))**1.0e37;

--
-- Input is a Vector of records representing the edges of a graph.
--
-- Vertices are identified by integers from 1 .. n.
--

type edge is record
u      : Positive;
weight : FloatPt;
v      : Positive;
end record;

(Index_Type => Positive, Element_Type => edge);
use Edge_Vectors;
subtype edge_vector is Edge_Vectors.Vector;

--
-- Floyd-Warshall.
--

type distance_array is
array (Positive range <>, Positive range <>) of FloatPt;

type next_vertex_array is
array (Positive range <>, Positive range <>) of Natural;
Nil_Vertex : constant Natural := 0;

function find_max_vertex      -- Find the maximum vertex number.
(edges : in edge_vector)
return Positive
is
max_vertex : Positive;
begin
if Is_Empty (edges) then
raise Floyd_Warshall_Exception with "no edges";
end if;
max_vertex := 1;
for i in edges.First_Index .. edges.Last_Index loop
max_vertex := Positive'Max (max_vertex, edges.Element (i).u);
max_vertex := Positive'Max (max_vertex, edges.Element (i).v);
end loop;
return max_vertex;
end find_max_vertex;

procedure floyd_warshall      -- Perform Floyd-Warshall.
(edges       : in     edge_vector;
max_vertex  : in     Positive;
distance    :    out distance_array;
next_vertex :    out next_vertex_array)
is
u, v     : Positive;
dist_ikj : FloatPt;
begin

-- Initialize.

for i in 1 .. max_vertex loop
for j in 1 .. max_vertex loop
distance (i, j)    := ENORMOUS_FloatPt;
next_vertex (i, j) := Nil_Vertex;
end loop;
end loop;
for i in edges.First_Index .. edges.Last_Index loop
u                  := edges.Element (i).u;
v                  := edges.Element (i).v;
distance (u, v)    := edges.Element (i).weight;
next_vertex (u, v) := v;
end loop;
for i in 1 .. max_vertex loop
distance (i, i) :=
FloatPt (0.0);           -- Distance from a vertex to itself.
next_vertex (i, i) := i;
end loop;

-- Perform the algorithm.

for k in 1 .. max_vertex loop
for i in 1 .. max_vertex loop
for j in 1 .. max_vertex loop
dist_ikj := distance (i, k) + distance (k, j);
if dist_ikj < distance (i, j) then
distance (i, j)    := dist_ikj;
next_vertex (i, j) := next_vertex (i, k);
end if;
end loop;
end loop;
end loop;

end floyd_warshall;

--
-- Path reconstruction.
--

procedure put_path
(next_vertex : in next_vertex_array;
u, v        : in Positive)
is
i : Positive;
begin
if next_vertex (u, v) /= Nil_Vertex then
i := u;
Put (Positive'Image (i));
while i /= v loop
Put (" ->");
i := next_vertex (i, v);
Put (Positive'Image (i));
end loop;
end if;
end put_path;

example_graph : edge_vector;
max_vertex    : Positive;

begin
Append (example_graph, (u => 1, weight => FloatPt (-2.0), v => 3));
Append (example_graph, (u => 3, weight => FloatPt (+2.0), v => 4));
Append (example_graph, (u => 4, weight => FloatPt (-1.0), v => 2));
Append (example_graph, (u => 2, weight => FloatPt (+4.0), v => 1));
Append (example_graph, (u => 2, weight => FloatPt (+3.0), v => 3));

max_vertex := find_max_vertex (example_graph);

declare

distance    : distance_array (1 .. max_vertex, 1 .. max_vertex);
next_vertex : next_vertex_array
(1 .. max_vertex, 1 .. max_vertex);

begin

floyd_warshall (example_graph, max_vertex, distance, next_vertex);

Put_Line ("  pair       distance        path");
Put_Line ("---------------------------------------------");
for u in 1 .. max_vertex loop
for v in 1 .. max_vertex loop
if u /= v then
Put (Positive'Image (u));
Put (" ->");
Put (Positive'Image (v));
Put ("    ");
Put (FloatPt'Image (distance (u, v)));
Put ("    ");
put_path (next_vertex, u, v);
Put_Line ("");
end if;
end loop;
end loop;

end;
```
Output:
```\$ gnatmake -q floyd_warshall_task.adb && ./floyd_warshall_task
pair       distance        path
---------------------------------------------
1 -> 2    -1.00000E+00     1 -> 3 -> 4 -> 2
1 -> 3    -2.00000E+00     1 -> 3
1 -> 4     0.00000E+00     1 -> 3 -> 4
2 -> 1     4.00000E+00     2 -> 1
2 -> 3     2.00000E+00     2 -> 1 -> 3
2 -> 4     4.00000E+00     2 -> 1 -> 3 -> 4
3 -> 1     5.00000E+00     3 -> 4 -> 2 -> 1
3 -> 2     1.00000E+00     3 -> 4 -> 2
3 -> 4     2.00000E+00     3 -> 4
4 -> 1     3.00000E+00     4 -> 2 -> 1
4 -> 2    -1.00000E+00     4 -> 2
4 -> 3     1.00000E+00     4 -> 2 -> 1 -> 3```

ALGOL 68

Translation of: Lua
```BEGIN # Floyd-Warshall algorithm - translated from the Lua sample #

OP   FMT = ( REAL v )STRING:
BEGIN
STRING result := fixed( ABS v, 0, 15 );
IF result[ LWB result ] = "." THEN "0" +=: result FI;
WHILE result[ UPB result ] = "0" DO result := result[ : UPB result - 1 ] OD;
IF result[ UPB result ] = "." THEN result := result[ : UPB result - 1 ] FI;
IF v < 0 THEN "-" ELSE " " FI + result
END # FMT # ;

PROC print result = ( [,]REAL dist, [,]INT nxt )VOID:
BEGIN
print( ( "pair     dist    path", newline ) );
FOR i FROM 1 LWB nxt TO 1 UPB nxt DO
FOR j FROM 2 LWB nxt TO 2 UPB nxt DO
IF i /= j THEN
INT    u    := i + 1;
INT    v     = j + 1;
print( ( whole( u, 0 ),    " -> ",  whole( v, 0 ), "    "
, FMT dist[ i, j ], "     ", whole( u, 0 )
)
);
WHILE u := nxt[ u - 1, v - 1 ];
print( ( " -> " +whole( u, 0 ) ) );
u /= v
DO SKIP OD;
print( ( newline ) )
FI
OD
OD
END # print result # ;

PROC floyd warshall = ( [,]INT weights, INT num vertices )VOID:
BEGIN

REAL infinity = max real;

[ 0 : num vertices - 1, 0 : num vertices - 1 ]REAL dist;
FOR i FROM LWB dist TO 1 UPB dist DO
FOR j FROM 2 LWB dist TO 2 UPB dist DO
dist[ i, j ] := infinity
OD
OD;

FOR i FROM 1 LWB weights TO 1 UPB weights DO
# the weights array is one based #
[]INT w = weights[ i, : ];
dist[ w[ 1 ] - 1, w[ 2 ] - 1 ] := w[ 3 ]
OD;

[ 0 : num vertices - 1, 0 : num vertices - 1 ]INT nxt;
FOR i FROM LWB nxt TO 1 UPB nxt DO
FOR j FROM 2 LWB nxt TO 2 UPB nxt DO
nxt[ i, j ] := IF i /= j THEN j + 1 ELSE 0 FI
OD
OD;

FOR k FROM 2 LWB dist TO 2 UPB dist DO
FOR i FROM 1 LWB dist TO 1 UPB dist DO
FOR j FROM 2 LWB dist TO 2 UPB dist DO
IF dist[ i, k ] /= infinity AND dist[ k, j ] /= infinity THEN
IF dist[ i, k ] + dist[ k, j ] < dist[ i, j ] THEN
dist[ i, j ] := dist[ i, k ] + dist[ k, j ];
nxt[  i, j ] := nxt[  i, k ]
FI
FI
OD
OD
OD;

print result( dist, nxt )
END # floyd warshall # ;

[,]INT weights = ( ( 1, 3, -2 )
, ( 2, 1,  4 )
, ( 2, 3,  3 )
, ( 3, 4,  2 )
, ( 4, 2, -1 )
);
INT num vertices = 4;
floyd warshall( weights, num vertices )

END```
Output:
```pair     dist    path
1 -> 2    -1     1 -> 3 -> 4 -> 2
1 -> 3    -2     1 -> 3
1 -> 4     0     1 -> 3 -> 4
2 -> 1     4     2 -> 1
2 -> 3     2     2 -> 1 -> 3
2 -> 4     4     2 -> 1 -> 3 -> 4
3 -> 1     5     3 -> 4 -> 2 -> 1
3 -> 2     1     3 -> 4 -> 2
3 -> 4     2     3 -> 4
4 -> 1     3     4 -> 2 -> 1
4 -> 2    -1     4 -> 2
4 -> 3     1     4 -> 2 -> 1 -> 3
```

ATS

A first implementation

Translation of: RATFOR

This implementation uses non-linear types that will leak memory. However, such memory leaks are what Boehm GC is made to deal with. (Also, such leaks are inconsequential in a program like this one.)

Removing one of the runtime assertions (assertloc) might prevent compilation. This is a difference between ATS and most other languages. For the template functions square_array_get_at and square_array_set_at, there is a praxi (an axiom) instead of assertions, and so, by contrast, there is no runtime penalty. A proof of the "axiom" could have been derived from the properties of multiplication, in case I had any doubts (and one may be surprised how often one is wrong about a lemma), but I simply declared it as an axiom.

```(*
Floyd-Warshall algorithm.

See https://en.wikipedia.org/w/index.php?title=Floyd%E2%80%93Warshall_algorithm&oldid=1082310013
*)

#define NIL list_nil ()
#define :: list_cons

typedef Pos = [i : pos] int i

(*------------------------------------------------------------------*)

(* Square arrays with 1-based indexing. *)

extern praxi
lemma_square_array_indices {n    : pos}
{i, j : pos | i <= n; j <= n}
() :<prf>
[0 <= (i - 1) + ((j - 1) * n);
(i - 1) + ((j - 1) * n) < n * n]
void

typedef square_array (t : t@ype+, n : int) =
'{
side_length = int n,
elements = arrayref (t, n * n)
}

fn {t : t@ype}
make_square_array {n    : nat}
(n    : int n,
fill : t) : square_array (t, n) =
let
prval () = mul_gte_gte_gte {n, n} ()
in
'{
side_length = n,
elements = arrayref_make_elt (i2sz (n * n), fill)
}
end

fn {t : t@ype}
square_array_get_at {n    : pos}
{i, j : pos | i <= n; j <= n}
(arr  : square_array (t, n),
i    : int i,
j    : int j) : t =
let
prval () = lemma_square_array_indices {n} {i, j} ()
in
arrayref_get_at (arr.elements,
(i - 1) + ((j - 1) * arr.side_length))
end

fn {t : t@ype}
square_array_set_at {n    : pos}
{i, j : pos | i <= n; j <= n}
(arr  : square_array (t, n),
i    : int i,
j    : int j,
x    : t) : void =
let
prval () = lemma_square_array_indices {n} {i, j} ()
in
arrayref_set_at (arr.elements,
(i - 1) + ((j - 1) * arr.side_length),
x)
end

(*------------------------------------------------------------------*)

typedef floatpt = float
extern castfn i2floatpt : int -<> floatpt
macdef arbitrary_floatpt = i2floatpt (12345)

typedef distance_array (n : int) = square_array (floatpt, n)

typedef vertex = [i : nat] int i
#define NIL_VERTEX 0
typedef next_vertex_array (n : int) = square_array (vertex, n)

typedef edge =
'{      (* The ' means this is allocated by the garbage collector.*)
u = vertex,
weight = floatpt,
v = vertex
}
typedef edge_list (n : int) = list (edge, n)
typedef edge_list = [n : int] edge_list (n)

prfn                           (* edge_list have non-negative size. *)
lemma_edge_list_param {n : int} (edges : edge_list n)
:<prf> [0 <= n] void =
lemma_list_param edges

(*------------------------------------------------------------------*)

fn
find_max_vertex (edges : edge_list) : vertex =
let
fun
loop {n : nat} .<n>.
(p : edge_list n,
u : vertex) : vertex =
case+ p of
| NIL => u

prval () = lemma_edge_list_param edges
in
assertloc (isneqz edges);
loop (edges, 0)
end

fn
floyd_warshall {n           : int}
(edges       : edge_list,
n           : int n,
distance    : distance_array n,
next_vertex : next_vertex_array n) : void =
let
val () = assertloc (1 <= n)
in

(* This implementation does NOT initialize (to any meaningful
value) elements of "distance" that would be set "infinite" in
the Wikipedia pseudocode. Instead you should use the
"next_vertex" array to determine whether there exists a finite
path from one vertex to another.

Thus we avoid any dependence on IEEE floating point or on the
settings of the FPU. *)

(* Initialize. *)

let
var i : Pos
in
for (i := 1; i <= n; i := succ i)
let
var j : Pos
in
for (j := 1; j <= n; j := succ j)
next_vertex[i, j] := NIL_VERTEX
end
end;
let
var p : edge_list
in
for (p := edges; list_is_cons p; p := list_tail p)
let
val () = assertloc (u <> NIL_VERTEX)
val () = assertloc (u <= n)
val () = assertloc (v <> NIL_VERTEX)
val () = assertloc (v <= n)
in
next_vertex[u, v] := v
end
end;
let
var i : Pos
in
for (i := 1; i <= n; i := succ i)
begin
(* Distance from a vertex to itself is zero. *)
distance[i, i] := i2floatpt (0);
next_vertex[i, i] := i
end
end;

(* Perform the algorithm. *)

let
var k : Pos
in
for (k := 1; k <= n; k := succ k)
let
var i : Pos
in
for (i := 1; i <= n; i := succ i)
let
var j : Pos
in
for (j := 1; j <= n; j := succ j)
if next_vertex[i, k] <> NIL_VERTEX
&& next_vertex[k, j] <> NIL_VERTEX then
let
val dist_ikj = distance[i, k] + distance[k, j]
in
if next_vertex[i, j] = NIL_VERTEX
|| dist_ikj < distance[i, j] then
begin
distance[i, j] := dist_ikj;
next_vertex[i, j] := next_vertex[i, k]
end
end
end
end
end

end

fn
print_path {n           : int}
(n           : int n,
next_vertex : next_vertex_array n,
u           : Pos,
v           : Pos) : void =
if 0 < n then
let
val () = assertloc (u <= n)
val () = assertloc (v <= n)
in
if next_vertex[u, v] <> NIL_VERTEX then
let
var i : Int
in
i := u;
print! (i);
while (i <> v)
let
val () = assertloc (1 <= i)
val () = assertloc (i <= n)
in
print! (" -> ");
i := next_vertex[i, v];
print! (i)
end
end
end

implement
main0 () =
let

(* One might notice that (because consing prepends rather than
appends) the order of edges here is *opposite* to that of some
other languages' implementations. But the order of the edges is
immaterial. *)
val example_graph = NIL
val example_graph =
'{u = 1, weight = i2floatpt (~2), v = 3} :: example_graph
val example_graph =
'{u = 3, weight = i2floatpt (2), v = 4} :: example_graph
val example_graph =
'{u = 4, weight = i2floatpt (~1), v = 2} :: example_graph
val example_graph =
'{u = 2, weight = i2floatpt (4), v = 1} :: example_graph
val example_graph =
'{u = 2, weight = i2floatpt (3), v = 3} :: example_graph

val n = find_max_vertex (example_graph)
val distance = make_square_array<floatpt> (n, arbitrary_floatpt)
val next_vertex = make_square_array<vertex> (n, NIL_VERTEX)

in

floyd_warshall (example_graph, n, distance, next_vertex);

println! ("  pair      distance      path");
println! ("------------------------------------------");
let
var u : Pos
in
for (u := 1; u <= n; u := succ u)
let
var v : Pos
in
for (v := 1; v <= n; v := succ v)
if u <> v then
begin
print! (" ", u, " -> ", v, "    ");
if i2floatpt (0) <= distance[u, v] then
print! (" ");
print! (distance[u, v], "     ");
print_path (n, next_vertex, u, v);
println! ()
end
end
end

end```
Output:
```\$ patscc -O3 -DATS_MEMALLOC_GCBDW floyd_warshall_task.dats -lgc && ./a.out
pair      distance      path
------------------------------------------
1 -> 2    -1.000000     1 -> 3 -> 4 -> 2
1 -> 3    -2.000000     1 -> 3
1 -> 4     0.000000     1 -> 3 -> 4
2 -> 1     4.000000     2 -> 1
2 -> 3     2.000000     2 -> 1 -> 3
2 -> 4     4.000000     2 -> 1 -> 3 -> 4
3 -> 1     5.000000     3 -> 4 -> 2 -> 1
3 -> 2     1.000000     3 -> 4 -> 2
3 -> 4     2.000000     3 -> 4
4 -> 1     3.000000     4 -> 2 -> 1
4 -> 2    -1.000000     4 -> 2
4 -> 3     1.000000     4 -> 2 -> 1 -> 3```

A second implementation

Translation of: Standard ML

A second version. An explanation of "Why a second version?" is contained in the program text.

```(*
Floyd-Warshall algorithm.

See https://en.wikipedia.org/w/index.php?title=Floyd%E2%80%93Warshall_algorithm&oldid=1082310013

-------------------------
WHY A SECOND ATS VERSION?
-------------------------

From the first ATS version, I derived a version in OCaml, which
modularized the code. From the OCaml, I produced a Standard ML
implementation that also made the types abstract.

Now I am returning to the ATS, to backport (among other things) the
abstraction of types. In fact I increase the abstraction, in a way
that protects the programmer against accidentally using the
"uninitialized" entries of the "distance" array.

Thus one can follow the chain of improvement, and also compare how
type abstraction is done Standard ML and in ATS. In ATS, type
abstraction can be done using "assume" statements or type casts.

*)

#define NIL list_nil ()
#define :: list_cons

typedef Pos = [i : pos] int i

(*------------------------------------------------------------------*)

(* You can change floatpt from "float" to "double" or another type,
if you wish. *)

typedef floatpt = float

extern castfn int2floatpt : int -<> floatpt

(*------------------------------------------------------------------*)

(* Square arrays with 1-based indexing. *)

local

typedef _square_array (t : t@ype+, n : int) =
(* '{ ... } with a "'" means the type is pointer to a record
allocated by the garbage collector. *)
'{
side_length = int n,
elements = arrayref (t, n * n)
}

in

abstype square_array (t : t@ype+, n : int)

assume square_array (t, n) = _square_array (t, n)

extern praxi
lemma_square_array_indices {n    : pos}
{i, j : pos | i <= n; j <= n}
() :<prf>
[0 <= (i - 1) + ((j - 1) * n);
(i - 1) + ((j - 1) * n) < n * n]
void

fn {t : t@ype}
square_array_make {n    : nat}
(n    : int n,
fill : t) :<!wrt> square_array (t, n) =
let
prval () = mul_gte_gte_gte {n, n} ()
in
'{
side_length = n,
elements = arrayref_make_elt (i2sz (n * n), fill)
}
end

fn {t : t@ype}
square_array_get_at {n    : pos}
{i, j : pos | i <= n; j <= n}
(arr  : square_array (t, n),
i    : int i,
j    : int j) :<!ref> t =
let
prval () = lemma_square_array_indices {n} {i, j} ()
in
arrayref_get_at (arr.elements,
(i - 1) + ((j - 1) * arr.side_length))
end

fn {t : t@ype}
square_array_set_at {n    : pos}
{i, j : pos | i <= n; j <= n}
(arr  : square_array (t, n),
i    : int i,
j    : int j,
x    : t) :<!refwrt> void =
let
prval () = lemma_square_array_indices {n} {i, j} ()
in
arrayref_set_at (arr.elements,
(i - 1) + ((j - 1) * arr.side_length),
x)
end

end (* local *)

(*------------------------------------------------------------------*)

(* A vertex made more abstract than simply identifying it with an
integer. *)

(* The following "abst@ype" tells the compiler that "vertex" is the
same size as "int" (as opposed to the size of a pointer, which
"abstype" assumes). It does *not* identify "vertex" with "int". *)
abst@ype vertex (i : int) = int

typedef vertex = [i : nat] vertex i

(* These casts let us convert between int and the abstract type. *)
extern castfn int2vertex : {i : nat} int i -<> vertex i
extern castfn vertex2int : {i : nat} vertex i -<> int i

macdef nil_vertex = int2vertex 0

fn
vertex_is_nil {u : nat}
(u : vertex u) :<> bool (u == 0) =
vertex2int u = vertex2int nil_vertex

fn
vertex_isnot_nil {u : nat}
(u : vertex u) :<> bool (u != 0) =
~vertex_is_nil u

fn
vertex_eq {u, v : nat}
(u    : vertex u,
v    : vertex v) :<> bool (u == v) =
vertex2int u = vertex2int v

fn
vertex_neq {u, v : nat}
(u    : vertex u,
v    : vertex v) :<> bool (u <> v) =
~vertex_eq (u, v)

fn
vertex_max {u, v : nat}
(u    : vertex u,
v    : vertex v) :<> vertex (max (u, v)) =
int2vertex (max (vertex2int u, vertex2int v))

fn
tostring_vertex (u : vertex) :<> string =
tostring_int (vertex2int u)

fn
tostring_directed_vertex_list (lst : List vertex) :<!wrt> string =
let
fun
loop {n   : nat} .<n>.
(lst : list (vertex, n),
s   : string) :<!wrt> string =
case+ lst of
| NIL => s
| u :: tail =>
let
val s_u = tostring_vertex u
in
if s = "" then
loop (tail, s_u)
else
let
val s1 = strptr2string (string_append (s, " -> ", s_u))
in
loop (tail, s1)
end

end

prval () = lemma_list_param lst
in
loop (lst, "")
end

(*------------------------------------------------------------------*)

(* Graph edges, with weights. *)

local

typedef _edge (u : int, v : int) =
(* The type is pointer to a tuple allocated by the garbage
collector. *)
[1 <= u; 1 <= v] '(vertex u, floatpt, vertex v)

in

abstype edge (u : int, v : int)
typedef edge = [u, v : pos] edge (u, v)

assume edge (u, v) = _edge (u, v)

fn
edge_make {u, v   : pos}
(u      : vertex u,
weight : floatpt,
v      : vertex v) :<> edge (u, v) =
'(u, weight, v)

fn
edge_first {u, v : pos}
(edge : edge (u, v)) :<> vertex u =
edge.0

fn
edge_weight (edge : edge) :<> floatpt =
edge.1

fn
edge_second {u, v : pos}
(edge : edge (u, v)) :<> vertex v =
edge.2

fn
max_vertex_in_edge_list (lst : List edge) :<> vertex =
let
fun
loop {n   : nat} .<n>.
(lst : list (edge, n),
x   : vertex) :<> vertex =
case+ lst of
| NIL => x
| edge :: tail =>
loop (tail,
max (max (edge_first edge, edge_second edge), x))

prval () = lemma_list_param lst
in
loop (lst, nil_vertex)
end

end (* local *)

(*------------------------------------------------------------------*)

(* Floyd-Warshall. *)

local

typedef _floyd_warshall_result (n : int) =
'{
n = int n,
dist = square_array (floatpt, n),
next = square_array (vertex, n)
}

fn {}
_dist_get_at {n    : pos}
{i, j : pos | i <= n; j <= n}
(dist : square_array (floatpt, n),
i    : int i,
j    : int j) :<!ref> floatpt =
square_array_get_at (dist, i, j)

fn
_dist_set_at {n    : pos}
{i, j : pos | i <= n; j <= n}
(dist : square_array (floatpt, n),
i    : int i,
j    : int j,
x    : floatpt) :<!refwrt> void =
square_array_set_at (dist, i, j, x)

fn {}
_next_get_at {n    : pos}
{i, j : pos | i <= n; j <= n}
(next : square_array (vertex, n),
i    : int i,
j    : int j) :<!ref> vertex =
square_array_get_at (next, i, j)

fn
_next_set_at {n    : pos}
{i, j : pos | i <= n; j <= n}
(next : square_array (vertex, n),
i    : int i,
j    : int j,
x    : vertex) :<!refwrt> void =
square_array_set_at (next, i, j, x)

in

abstype floyd_warshall_result (n : int)
typedef floyd_warshall_result = [n : nat] floyd_warshall_result n

assume floyd_warshall_result n = _floyd_warshall_result n

exception FloydWarshallError of (string)

fn
vertex_count {n  : pos}
(fw : floyd_warshall_result n) :<> int n =
fw.n

fn
get_distance {n    : pos}
{i, j : pos | i <= n; j <= n}
(fw   : floyd_warshall_result n,
i    : vertex i,
j    : vertex j) :<!ref> Option floatpt =

(* Notice there is *no way* to return one of the "uninitialized"
values in the "dist" array (which were actually set to a
meaningless value, or could have been set to positive

This kind of behavior is better than returning "positive
infinity", because it does not depend on any particular sort of
floating point. Indeed, in Ada you could use fixed point. *)

let
val i = vertex2int i
val j = vertex2int j
val u = _next_get_at (fw.next, i, j)
in
if iseqz u then
None ()                 (* There is no finite path. *)
else
Some (_dist_get_at (fw.dist, i, j))
end

fn
get_next_vertex {n    : pos}
{i, j : pos | i <= n; j <= n}
(fw   : floyd_warshall_result n,
i    : vertex i,
j    : vertex j) :<!ref> vertex =
_next_get_at (fw.next, vertex2int i, vertex2int j)

fn
floyd_warshall (edges : List edge)
:<1> [n : pos] floyd_warshall_result n =
let
val n = vertex2int (max_vertex_in_edge_list edges)
in
if n = 0 then
\$raise FloydWarshallError ("no vertices")
else
let
macdef arbitrary_floatpt = i2fp (12345)
val dist = square_array_make<floatpt> (n, arbitrary_floatpt)
val next = square_array_make<vertex> (n, nil_vertex)
in

(* Initialize. *)

let
var i : Pos
in
for (i := 1; i <= n; i := succ i)
let
var j : Pos
in
for (j := 1; j <= n; j := succ j)
next[i, j] := nil_vertex
end
end;
let
var p : List edge
in
for (p := edges; list_is_cons p; p := list_tail p)
let
val u = edge_first edge
val () = assertloc (isneqz u)
val () = assertloc (vertex2int u <= n)
val v = edge_second edge
val () = assertloc (isneqz v)
val () = assertloc (vertex2int v <= n)
in
dist[vertex2int u, vertex2int v] := edge_weight edge;
next[vertex2int u, vertex2int v] := v
end
end;
let
var i : Pos
in
for (i := 1; i <= n; i := succ i)
begin
(* Distance from a vertex to itself is zero. *)
dist[i, i] := int2floatpt (0);
next[i, i] := int2vertex i
end
end;

(* Perform the algorithm. *)

let
var k : Pos
in
for (k := 1; k <= n; k := succ k)
let
var i : Pos
in
for (i := 1; i <= n; i := succ i)
let
var j : Pos
in
for (j := 1; j <= n; j := succ j)
if isneqz next[i, k] && isneqz next[k, j] then
let
val dist_ikj = dist[i, k] + dist[k, j]
in
if iseqz next[i, j]
|| dist_ikj < dist[i, j] then
begin
dist[i, j] := dist_ikj;
next[i, j] := next[i, k]
end
end
end
end
end;

(* Return the result. *)

'{ n = n, dist = dist, next = next }

end
end

fn
get_path {n    : int}
{u, v : pos}
(fw   : floyd_warshall_result n,
u    : vertex u,
v    : vertex v) :<!refwrt,!exn> List vertex =
if (fw.n) < vertex2int u then
else if (fw.n) < vertex2int v then
else
if iseqz (get_next_vertex (fw, u, v)) then
NIL
else
let
fun
loop (w   : vertex,
lst : List0 vertex) :<!ntm,!refwrt> List vertex =
if w = v then
list_vt2t (list_reverse lst)
else
let
val () =
val () =
\$effmask_exn assertloc (vertex2int w <= (fw.n))
val w = get_next_vertex (fw, w, v)
in
loop (w, w :: lst)
end
in
\$effmask_ntm loop (u, u :: NIL)
end

end (* local *)

(*------------------------------------------------------------------*)

implement
main0 () =
let
val example_graph =
\$list (edge_make (int2vertex 1, i2fp (~2), int2vertex 3),
edge_make (int2vertex 3, i2fp (2), int2vertex 4),
edge_make (int2vertex 4, i2fp (~1), int2vertex 2),
edge_make (int2vertex 2, i2fp (4), int2vertex 1),
edge_make (int2vertex 2, i2fp (3), int2vertex 3))

val fw = floyd_warshall example_graph
in
println! ("  pair      distance      path");
println! ("------------------------------------------");
let
var i : Pos
in
for (i := 1; i <= (fw.n); i := succ i)
let
var j : Pos
in
for (j := 1; j <= (fw.n); j := succ j)
let
val u = int2vertex i
val v = int2vertex j
in
if u <> v then
let
val s_edge =
tostring_directed_vertex_list (\$list (u, v))
val distance_opt = get_distance (fw, u, v)
in
print! (" ", s_edge, "    ");
begin
case+ distance_opt of
| None () => print! " no path"
| Some distance =>
let
val path = get_path (fw, u, v)
val s_path =
tostring_directed_vertex_list path
in
if int2floatpt (0) <= distance then
print! " ";
print! distance;
print! "     ";
print! s_path
end
end;
println! ()
end
end
end
end
end

(*------------------------------------------------------------------*)```
Output:
```\$ patscc -O3 -DATS_MEMALLOC_GCBDW floyd_warshall_task_2.dats -lgc && ./a.out
pair      distance      path
------------------------------------------
1 -> 2    -1.000000     1 -> 3 -> 4 -> 2
1 -> 3    -2.000000     1 -> 3
1 -> 4     0.000000     1 -> 3 -> 4
2 -> 1     4.000000     2 -> 1
2 -> 3     2.000000     2 -> 1 -> 3
2 -> 4     4.000000     2 -> 1 -> 3 -> 4
3 -> 1     5.000000     3 -> 4 -> 2 -> 1
3 -> 2     1.000000     3 -> 4 -> 2
3 -> 4     2.000000     3 -> 4
4 -> 1     3.000000     4 -> 2 -> 1
4 -> 2    -1.000000     4 -> 2
4 -> 3     1.000000     4 -> 2 -> 1 -> 3```

C

Reads the graph from a file, prints out usage on incorrect invocation.

```#include<limits.h>
#include<stdlib.h>
#include<stdio.h>

typedef struct{
int sourceVertex, destVertex;
int edgeWeight;
}edge;

typedef struct{
int vertices, edges;
edge* edgeMatrix;
}graph;

FILE* fp = fopen(fileName,"r");

graph G;
int i;

fscanf(fp,"%d%d",&G.vertices,&G.edges);

G.edgeMatrix = (edge*)malloc(G.edges*sizeof(edge));

for(i=0;i<G.edges;i++)
fscanf(fp,"%d%d%d",&G.edgeMatrix[i].sourceVertex,&G.edgeMatrix[i].destVertex,&G.edgeMatrix[i].edgeWeight);

fclose(fp);

return G;
}

void floydWarshall(graph g){
int processWeights[g.vertices][g.vertices], processedVertices[g.vertices][g.vertices];
int i,j,k;

for(i=0;i<g.vertices;i++)
for(j=0;j<g.vertices;j++){
processWeights[i][j] = SHRT_MAX;
processedVertices[i][j] = (i!=j)?j+1:0;
}

for(i=0;i<g.edges;i++)
processWeights[g.edgeMatrix[i].sourceVertex-1][g.edgeMatrix[i].destVertex-1] = g.edgeMatrix[i].edgeWeight;

for(i=0;i<g.vertices;i++)
for(j=0;j<g.vertices;j++)
for(k=0;k<g.vertices;k++){
if(processWeights[j][i] + processWeights[i][k] < processWeights[j][k]){
processWeights[j][k] = processWeights[j][i] + processWeights[i][k];
processedVertices[j][k] = processedVertices[j][i];
}
}

printf("pair    dist   path");
for(i=0;i<g.vertices;i++)
for(j=0;j<g.vertices;j++){
if(i!=j){
printf("\n%d -> %d %3d %5d",i+1,j+1,processWeights[i][j],i+1);
k = i+1;
do{
k = processedVertices[k-1][j];
printf("->%d",k);
}while(k!=j+1);
}
}
}

int main(int argC,char* argV[]){
if(argC!=2)
printf("Usage : %s <file containing graph data>");
else
return 0;
}
```

Input file, first row specifies number of vertices and edges.

```4 5
1 3 -2
3 4 2
4 2 -1
2 1 4
2 3 3
```

Invocation and output:

```C:\rosettaCode>fwGraph.exe fwGraph.txt
pair    dist   path
1 -> 2  -1     1->3->4->2
1 -> 3  -2     1->3
1 -> 4   0     1->3->4
2 -> 1   4     2->1
2 -> 3   2     2->1->3
2 -> 4   4     2->1->3->4
3 -> 1   5     3->4->2->1
3 -> 2   1     3->4->2
3 -> 4   2     3->4
4 -> 1   3     4->2->1
4 -> 2  -1     4->2
4 -> 3   1     4->2->1->3
```

VERSION 2. Using Gadget, an a "C" library.

```#include <limits.h>

/* algunos datos globales */
int vertices,edges;

/* algunos prototipos */
F_STAT DatosdeArchivo( const char *cFile);
int * CargaMatriz(int * mat, DS_ARRAY * mat_data, const char * cFile, F_STAT stat );
int * CargaGrafo(int * graph, DS_ARRAY * graph_data, const char *cFile);
void Floyd_Warshall(int * graph, DS_ARRAY graph_data);

/* bloque principal */
Main
if ( Arg_count != 2 ){
Msg_yellow("Modo de uso:\n   ./floyd <archivo_de_vertices>\n");
Stop(1);
}
Get_arg_str (cFile,1);
Set_token_sep(' ');
Cls;
if(Exist_file(cFile)){
New array graph as int;
graph = CargaGrafo( pSDS(graph), cFile);
if(graph){
/* calcula Floyd-Warshall */
Print "Vertices=%d, edges=%d\n",vertices,edges;

Floyd_Warshall( SDS(graph) ); Prnl;

Free array graph;
}

}else{
Msg_redf("No existe el archivo %s",cFile);
}
Free secure cFile;
End

void Floyd_Warshall( RDS(int,graph) ){

Array processedVertices as int(vertices,vertices);
Fill array processWeights as int(vertices,vertices) with SHRT_MAX;

int i,j,k;
Range for processWeights [0:1:vertices, 0:1:vertices ];

Compute_for( processWeights, i,j,
\$processedVertices[i,j] = (i!=j)?j+1:0;
)

#define    VERT_ORIG 0
#define    VERT_DEST 1
#define    WEIGHT    2

Iterator up i [0:1:edges] {
\$2processWeights[ \$graph[i,VERT_ORIG]-1, \$graph[i,VERT_DEST]-1 ] = \$graph[i,WEIGHT];
}

Compute_for (processWeights,i,j,
Iterator up k [0:1:vertices] {
if( \$processWeights[j,i] + \$processWeights[i,k] < \$processWeights[j,k] )
{
\$processWeights[j,k] = \$processWeights[j,i] + \$processWeights[i,k];
\$processedVertices[j,k] = \$processedVertices[j,i];
}
} );

Print "pair    dist   path";

// ya existen rangos definios para "processWeights":
Compute_for(processWeights, i, j,
if(i!=j)
{
Print "\n%d -> %d %3d %5d", i+1, j+1, \$processWeights[i,j], i+1;
int k = i+1;
do{
k = \$processedVertices[k-1,j];
Print " -> %d", k;
}while(k!=j+1);
}
);

Free array processWeights, processedVertices;
}

F_STAT DatosdeArchivo( const char *cFile){
return Stat_file(cFile);
}

int * CargaMatriz( pRDS(int, mat), const char * cFile, F_STAT stat ){
}

int * CargaGrafo( pRDS(int, graph), const char *cFile){

F_STAT dataFile = DatosdeArchivo(cFile);
if(dataFile.is_matrix){

Range ptr graph [0:1:dataFile.total_lines-1, 0:1:dataFile.max_tokens_per_line-1];

graph = CargaMatriz( SDS(graph), cFile, dataFile);

if( graph ){
/* obtengo vertices = 4 y edges = 5 */
edges = dataFile.total_lines;

Block( vertices, Range ptr graph [ 0:1:pRows(graph), 0:1:1 ];
DS_MAXMIN  maxNode = Max_array( SDS(graph) );
Out_int( \$graph[maxNode.local] ) );
}else{
Msg_redf("Archivo \"%s\" no ha podido ser cargado",cFile);
}

}else{
}
return graph;
}
```
Output:

Archivo fuente: floyd_data.txt

```1 3 -2
3 4 2
4 2 -1
2 1 4
2 3 3
```

Salida:

```\$ ./floydWarshall floyd_data.txt
Vertices=4, edges=5
pair    dist   path
1 -> 2  -1     1->3->4->2
1 -> 3  -2     1->3
1 -> 4   0     1->3->4
2 -> 1   4     2->1
2 -> 3   2     2->1->3
2 -> 4   4     2->1->3->4
3 -> 1   5     3->4->2->1
3 -> 2   1     3->4->2
3 -> 4   2     3->4
4 -> 1   3     4->2->1
4 -> 2  -1     4->2
4 -> 3   1     4->2->1->3
```

C#

Translation of: Java
```using System;

namespace FloydWarshallAlgorithm {
class Program {
static void FloydWarshall(int[,] weights, int numVerticies) {
double[,] dist = new double[numVerticies, numVerticies];
for (int i = 0; i < numVerticies; i++) {
for (int j = 0; j < numVerticies; j++) {
dist[i, j] = double.PositiveInfinity;
}
}

for (int i = 0; i < weights.GetLength(0); i++) {
dist[weights[i, 0] - 1, weights[i, 1] - 1] = weights[i, 2];
}

int[,] next = new int[numVerticies, numVerticies];
for (int i = 0; i < numVerticies; i++) {
for (int j = 0; j < numVerticies; j++) {
if (i != j) {
next[i, j] = j + 1;
}
}
}

for (int k = 0; k < numVerticies; k++) {
for (int i = 0; i < numVerticies; i++) {
for (int j = 0; j < numVerticies; j++) {
if (dist[i, k] + dist[k, j] < dist[i, j]) {
dist[i, j] = dist[i, k] + dist[k, j];
next[i, j] = next[i, k];
}
}
}
}

PrintResult(dist, next);
}

static void PrintResult(double[,] dist, int[,] next) {
Console.WriteLine("pair     dist    path");
for (int i = 0; i < next.GetLength(0); i++) {
for (int j = 0; j < next.GetLength(1); j++) {
if (i != j) {
int u = i + 1;
int v = j + 1;
string path = string.Format("{0} -> {1}    {2,2:G}     {3}", u, v, dist[i, j], u);
do {
u = next[u - 1, v - 1];
path += " -> " + u;
} while (u != v);
Console.WriteLine(path);
}
}
}
}

static void Main(string[] args) {
int[,] weights = { { 1, 3, -2 }, { 2, 1, 4 }, { 2, 3, 3 }, { 3, 4, 2 }, { 4, 2, -1 } };
int numVerticies = 4;

FloydWarshall(weights, numVerticies);
}
}
}
```

C++

```#include <iostream>
#include <vector>
#include <sstream>

void print(std::vector<std::vector<double>> dist, std::vector<std::vector<int>> next) {
std::cout << "(pair, dist, path)" << std::endl;
const auto size = std::size(next);
for (auto i = 0; i < size; ++i) {
for (auto j = 0; j < size; ++j) {
if (i != j) {
auto u = i + 1;
auto v = j + 1;
std::cout << "(" << u << " -> " << v << ", " << dist[i][j]
<< ", ";
std::stringstream path;
path << u;
do {
u = next[u - 1][v - 1];
path << " -> " << u;
} while (u != v);
std::cout << path.str() << ")" << std::endl;
}
}
}
}

void solve(std::vector<std::vector<int>> w_s, const int num_vertices) {
std::vector<std::vector<double>> dist(num_vertices);
for (auto& dim : dist) {
for (auto i = 0; i < num_vertices; ++i) {
dim.push_back(INT_MAX);
}
}
for (auto& w : w_s) {
dist[w[0] - 1][w[1] - 1] = w[2];
}
std::vector<std::vector<int>> next(num_vertices);
for (auto i = 0; i < num_vertices; ++i) {
for (auto j = 0; j < num_vertices; ++j) {
next[i].push_back(0);
}
for (auto j = 0; j < num_vertices; ++j) {
if (i != j) {
next[i][j] = j + 1;
}
}
}
for (auto k = 0; k < num_vertices; ++k) {
for (auto i = 0; i < num_vertices; ++i) {
for (auto j = 0; j < num_vertices; ++j) {
if (dist[i][j] > dist[i][k] + dist[k][j]) {
dist[i][j] = dist[i][k] + dist[k][j];
next[i][j] = next[i][k];
}
}
}
}
print(dist, next);
}

int main() {
std::vector<std::vector<int>> w = {
{ 1, 3, -2 },
{ 2, 1, 4 },
{ 2, 3, 3 },
{ 3, 4, 2 },
{ 4, 2, -1 },
};
int num_vertices = 4;
solve(w, num_vertices);
std::cin.ignore();
std::cin.get();
return 0;
}
```
Output:
```(pair, dist, path)
(1 -> 2, -1, 1 -> 3 -> 4 -> 2)
(1 -> 3, -2, 1 -> 3)
(1 -> 4, 0, 1 -> 3 -> 4)
(2 -> 1, 4, 2 -> 1)
(2 -> 3, 2, 2 -> 1 -> 3)
(2 -> 4, 4, 2 -> 1 -> 3 -> 4)
(3 -> 1, 5, 3 -> 4 -> 2 -> 1)
(3 -> 2, 1, 3 -> 4 -> 2)
(3 -> 4, 2, 3 -> 4)
(4 -> 1, 3, 4 -> 2 -> 1)
(4 -> 2, -1, 4 -> 2)
(4 -> 3, 1, 4 -> 2 -> 1 -> 3)```

Common Lisp

Translation of: Scheme

I have wrapped the Common Lisp program in a Roswell script.

Notice how in Common Lisp you have to specially quote the name of a function to call that function as an argument, whereas in Scheme no such thing is necessary. (In fact, a Scheme procedure does not really have a name; you are giving the name of a variable that holds the procedure.)

"Looping" (or tail recursion) is done differently, although it is common for a Common Lisp-like loop macro to be available in Scheme. A Common Lisp-like format also often is available.

```#!/bin/sh
#|-*- mode:lisp -*-|#
#|
exec ros -Q -- \$0 "\$@"
|#
(progn ;;init forms
(ros:ensure-asdf)
)

(defpackage :ros.script.floyd-warshall.3861181636
(:use :cl))
(in-package :ros.script.floyd-warshall.3861181636)

;;;
;;; Floyd-Warshall algorithm.
;;;
;;; See https://en.wikipedia.org/w/index.php?title=Floyd%E2%80%93Warshall_algorithm&oldid=1082310013
;;;
;;; Translated from the Scheme. Small improvements (or what might be
;;; considered improvements), and some type specialization, have been
;;;

;;;-------------------------------------------------------------------
;;;
;;; A square array will be represented by an ordinary Common Lisp
;;; array, but accessed through our own functions (which look similar
;;; to, although not identical to, the corresponding Scheme
;;; functions).
;;;
;;; Square arrays are indexed *starting at one*.
;;;

(defun make-arr (n &key (element-type t) initial-element)
(make-array (list n n) :element-type element-type
:initial-element initial-element))

(defun arr-set (arr i j x)
(setf (aref arr (- i 1) (- j 1)) x))

(defun arr-ref (arr i j)
(aref arr (- i 1) (- j 1)))

;;;-------------------------------------------------------------------
;;;
;;; Floyd-Warshall.
;;;
;;; Input is a list of length-3 lists representing edges; each entry
;;; is:
;;;
;;;    (start-vertex edge-weight end-vertex)
;;;
;;; where vertex identifiers are integers from 1 .. n.
;;;
;;; A difference from the Scheme implementation is that here we do not
;;; assume the floating point supports "infinities". In the Scheme we
;;; did, because in R7RS small there is support for such infinities
;;; (although the standard does not *require* them). Also because
;;; alternatives were not yet apparent to this author. :)
;;;

(defvar *floatpt* 'single-float)
(defconstant nil-vertex 0)

(defun floyd-warshall (edges)
(let* ((n
;; Set n to the maximum vertex number. By design, n also
;; equals the number of vertices.
(max (apply #'max (mapcar #'car edges))

(distance
;; The distances are initialized to a purely arbitrary
;; value. An entry in the "distance" array is meaningful
;; *only* if the corresponding entry in "next-vertex" is
;; not the nil-vertex.
(make-arr n :element-type *floatpt*
:initial-element (coerce 12345 *floatpt*)))

(next-vertex
;; Unless later set otherwise, an entry in "next-vertex"
;; will be the nil-vertex.
(make-arr n :element-type 'fixnum
:initial-element nil-vertex)))

(defun dist (p q) (arr-ref distance p q))
(defun next (p q) (arr-ref next-vertex p q))

(defun set-dist (p q x) (arr-set distance p q x))
(defun set-next (p q x) (arr-set next-vertex p q x))

(defun nilnext (p q) (= (next p q) nil-vertex))

;; Initialize "distance" and "next-vertex".
(loop for edge in edges
do (let ((u (car edge))
(set-dist u v weight)
(set-next u v v)))
(loop for v from 1 to n
do (progn
;; The distance from a vertex to itself = 0.0.
(set-dist v v (coerce 0 *floatpt*))
(set-next v v v)))

;; Perform the algorithm.
(loop
for k from 1 to n
do (loop
for i from 1 to n
do (loop
for j from 1 to n
do (and (not (nilnext i k))
(not (nilnext k j))
(let* ((dist-ikj (+ (dist i k) (dist k j))))
(when (or (nilnext i j)
(< dist-ikj (dist i j)))
(set-dist i j dist-ikj)
(set-next i j (next i k))))))))

;; Return the results.
(values n distance next-vertex)))

;;;-------------------------------------------------------------------
;;;
;;; Path reconstruction from the "next-vertex" array.
;;;
;;; The return value is a list of vertices.
;;;

(defun find-path (next-vertex u v)
(if (= (arr-ref next-vertex u v) nil-vertex)
(list)
(cons u (let ((i u))
(loop while (/= i v)
do (setf i (arr-ref next-vertex i v))
collect i)))))

;;;-------------------------------------------------------------------

(defun directed-vertex-list-to-string (lst)
(if (not lst)
""
(let ((s (write-to-string (car lst))))
(loop for u in (cdr lst)
do (setf s (concatenate 'string s " -> "
(write-to-string u))))
s)))

;;;-------------------------------------------------------------------

(defun main (&rest argv)
(declare (ignorable argv))
(let ((example-graph
(mapcar (lambda (x) (list (coerce (car x) 'fixnum)
'((1 -2 3)
(3 2 4)
(4 -1 2)
(2 4 1)
(2 3 3)))))
(multiple-value-bind (n distance next-vertex)
(floyd-warshall example-graph)
(princ "  pair    distance   path")
(terpri)
(princ "-------------------------------------")
(terpri)
(loop
for u from 1 to n
do (loop
for v from 1 to n
do (unless (= u v)
(format
t " ~A ~7@A     ~A~%"
(directed-vertex-list-to-string (list u v))
(if (= (arr-ref next-vertex u v) nil-vertex)
"   no path"
(write-to-string (arr-ref distance u v)))
(directed-vertex-list-to-string
(find-path next-vertex u v)))))))))

;;;-------------------------------------------------------------------
;;; vim: set ft=lisp lisp:
```
Output:
```\$ ./floyd-warshall.ros
pair    distance   path
-------------------------------------
1 -> 2    -1.0     1 -> 3 -> 4 -> 2
1 -> 3    -2.0     1 -> 3
1 -> 4     0.0     1 -> 3 -> 4
2 -> 1     4.0     2 -> 1
2 -> 3     2.0     2 -> 1 -> 3
2 -> 4     4.0     2 -> 1 -> 3 -> 4
3 -> 1     5.0     3 -> 4 -> 2 -> 1
3 -> 2     1.0     3 -> 4 -> 2
3 -> 4     2.0     3 -> 4
4 -> 1     3.0     4 -> 2 -> 1
4 -> 2    -1.0     4 -> 2
4 -> 3     1.0     4 -> 2 -> 1 -> 3```

D

Translation of: Java
```import std.stdio;

void main() {
int[][] weights = [
[1, 3, -2],
[2, 1, 4],
[2, 3, 3],
[3, 4, 2],
[4, 2, -1]
];
int numVertices = 4;

floydWarshall(weights, numVertices);
}

void floydWarshall(int[][] weights, int numVertices) {
import std.array;

real[][] dist = uninitializedArray!(real[][])(numVertices, numVertices);
foreach(dim; dist) {
dim[] = real.infinity;
}

foreach (w; weights) {
dist[w[0]-1][w[1]-1] = w[2];
}

int[][] next = uninitializedArray!(int[][])(numVertices, numVertices);
for (int i=0; i<next.length; i++) {
for (int j=0; j<next.length; j++) {
if (i != j) {
next[i][j] = j+1;
}
}
}

for (int k=0; k<numVertices; k++) {
for (int i=0; i<numVertices; i++) {
for (int j=0; j<numVertices; j++) {
if (dist[i][j] > dist[i][k] + dist[k][j]) {
dist[i][j] = dist[i][k] + dist[k][j];
next[i][j] = next[i][k];
}
}
}
}

printResult(dist, next);
}

void printResult(real[][] dist, int[][] next) {
import std.conv;
import std.format;

writeln("pair     dist    path");
for (int i=0; i<next.length; i++) {
for (int j=0; j<next.length; j++) {
if (i!=j) {
int u = i+1;
int v = j+1;
string path = format("%d -> %d    %2d     %s", u, v, cast(int) dist[i][j], u);
do {
u = next[u-1][v-1];
path ~= text(" -> ", u);
} while (u != v);
writeln(path);
}
}
}
}
```
Output:
```pair     dist    path
1 -> 2    -1     1 -> 3 -> 4 -> 2
1 -> 3    -2     1 -> 3
1 -> 4     0     1 -> 3 -> 4
2 -> 1     4     2 -> 1
2 -> 3     2     2 -> 1 -> 3
2 -> 4     4     2 -> 1 -> 3 -> 4
3 -> 1     5     3 -> 4 -> 2 -> 1
3 -> 2     1     3 -> 4 -> 2
3 -> 4     2     3 -> 4
4 -> 1     3     4 -> 2 -> 1
4 -> 2    -1     4 -> 2
4 -> 3     1     4 -> 2 -> 1 -> 3```

EchoLisp

Transcription of the Floyd-Warshall algorithm, with best path computation.

```(lib 'matrix)

;; in : initialized dist and next matrices
;; out : dist and next matrices
;; O(n^3)

(define (floyd-with-path n dist next (d 0))
(for* ((k n) (i n) (j n))
#:break (< (array-ref dist j j) 0) => 'negative-cycle
(set! d (+ (array-ref dist i k) (array-ref dist k j)))
(when (< d (array-ref dist i j))
(array-set! dist i j d)
(array-set! next i j (array-ref next i k)))))

;; utilities

;; init random edges costs, matrix 66% filled
(define (init-edges n dist next)
(for* ((i n) (j n))
(array-set! dist i i 0)
(array-set! next i j null)
#:continue (= j i)
(array-set! dist i j Infinity)
#:continue (< (random) 0.3)
(array-set! dist i j (1+ (random 100)))
(array-set! next i j j)))

;; show path from u to v
(define (path u v)
(cond
((= u v) (list u))
((null? (array-ref next u v)) null)
(else (cons u (path (array-ref next u v) v)))))

(define( mdist u v) ;; show computed distance
(array-ref dist u v))

(init-edges n dist next)
(array-print dist) ;; show init distances
(floyd-with-path n dist next))
```
Output:
```(define n 8)
(define next (make-array n n))
(define dist (make-array n n))

0    Infinity   Infinity   13         98         Infinity   35         47
8    0          Infinity   Infinity   83         77         16         3
73   3          0          3          76         84         91         Infinity
30   49         Infinity   0          41         Infinity   4          4
22   83         92         Infinity   0          30         27         98
6    Infinity   Infinity   24         59         0          Infinity   Infinity
60   Infinity   45         Infinity   67         100        0          Infinity
72   15         95         21         Infinity   Infinity   27         0

(array-print dist) ;; computed distances

0    32   62   13   54   84   17   17
8    0    61   21   62   77   16   3
11   3    0    3    44   74   7    6
27   19   49   0    41   71   4    4
22   54   72   35   0    30   27   39
6    38   68   19   59   0    23   23
56   48   45   48   67   97   0    51
23   15   70   21   62   92   25   0

(path 1 3)  → (1 0 3)
(mdist 1 0) → 8
(mdist 0 3) → 13
(mdist 1 3) → 21 ;; = 8 + 13
(path 7 6) → (7 3 6)
(path 6 7) → (6 2 1 7)

```

Elixir

```defmodule Floyd_Warshall do
def main(n, edge) do
{dist, next} = setup(n, edge)
{dist, next} = shortest_path(n, dist, next)
print(n, dist, next)
end

defp setup(n, edge) do
big = 1.0e300
dist = for i <- 1..n, j <- 1..n, into: %{}, do: {{i,j},(if i==j, do: 0, else: big)}
next = for i <- 1..n, j <- 1..n, into: %{}, do: {{i,j}, nil}
Enum.reduce(edge, {dist,next}, fn {u,v,w},{dst,nxt} ->
{ Map.put(dst, {u,v}, w), Map.put(nxt, {u,v}, v) }
end)
end

defp shortest_path(n, dist, next) do
(for k <- 1..n, i <- 1..n, j <- 1..n, do: {k,i,j})
|> Enum.reduce({dist,next}, fn {k,i,j},{dst,nxt} ->
if dst[{i,j}] > dst[{i,k}] + dst[{k,j}] do
{Map.put(dst, {i,j}, dst[{i,k}] + dst[{k,j}]), Map.put(nxt, {i,j}, nxt[{i,k}])}
else
{dst, nxt}
end
end)
end

defp print(n, dist, next) do
IO.puts "pair     dist    path"
for i <- 1..n, j <- 1..n, i != j,
do: :io.format "~w -> ~w  ~4w     ~s~n", [i, j, dist[{i,j}], path(next, i, j)]
end

defp path(next, i, j), do: path(next, i, j, [i]) |> Enum.join(" -> ")

defp path(_next, i, i, list), do: Enum.reverse(list)
defp path(next, i, j, list) do
u = next[{i,j}]
path(next, u, j, [u | list])
end
end

edge = [{1, 3, -2}, {2, 1, 4}, {2, 3, 3}, {3, 4, 2}, {4, 2, -1}]
Floyd_Warshall.main(4, edge)
```
Output:
```pair     dist    path
1 -> 2    -1     1 -> 3 -> 4 -> 2
1 -> 3    -2     1 -> 3
1 -> 4     0     1 -> 3 -> 4
2 -> 1     4     2 -> 1
2 -> 3     2     2 -> 1 -> 3
2 -> 4     4     2 -> 1 -> 3 -> 4
3 -> 1     5     3 -> 4 -> 2 -> 1
3 -> 2     1     3 -> 4 -> 2
3 -> 4     2     3 -> 4
4 -> 1     3     4 -> 2 -> 1
4 -> 2    -1     4 -> 2
4 -> 3     1     4 -> 2 -> 1 -> 3
```

F#

Floyd's algorithm

```//Floyd's algorithm: Nigel Galloway August 5th 2018
let Floyd (n:'a[]) (g:Map<('a*'a),int>)= //nodes graph(Map of adjacency list)
let ix n g=Seq.init (pown g n) (fun x->List.unfold(fun (a,b)->if a=0 then None else Some(b%g,(a-1,b/g)))(n,x))
let fN w (i,j,k)=match Map.tryFind(i,j) w,Map.tryFind(i,k) w,Map.tryFind(k,j) w with
|(None  ,Some j,Some k)->Some(j+k)
|(Some i,Some j,Some k)->if (j+k) < i then Some(j+k) else None
|_                     ->None
let n,z=ix 3 (Array.length n)|>Seq.choose(fun (i::j::k::_)->if i<>j&&i<>k&&j<>k then Some(n.[i],n.[j],n.[k]) else None)
|>Seq.fold(fun (n,n') ((i,j,k) as g)->match fN n g with |Some g->(Map.add (i,j) g n,Map.add (i,j) k n')|_->(n,n')) (g,Map.empty)
(n,(fun x y->seq{
let rec fN n g=seq{
match Map.tryFind (n,g) z with
|Some r->yield! fN n r; yield Some r;yield! fN r g
|_->yield None}
yield! fN x y |> Seq.choose id; yield y}))
```

```let fW=Map[((1,3),-2);((3,4),2);((4,2),-1);((2,1),4);((2,3),3)]
let N,G=Floyd [|1..4|] fW
List.allPairs [1..4] [1..4]|>List.filter(fun (n,g)->n<>g)|>List.iter(fun (n,g)->printfn "%d->%d %d %A" n g N.[(n,g)] (n::(List.ofSeq (G n g))))
```
Output:
```1->2 -1 [1; 3; 4; 2]
1->3 -2 [1; 3]
1->4 0 [1; 3; 4]
2->1 4 [2; 1]
2->3 2 [2; 1; 3]
2->4 4 [2; 1; 3; 4]
3->1 5 [3; 4; 2; 1]
3->2 1 [3; 4; 2]
3->4 2 [3; 4]
4->1 3 [4; 2; 1]
4->2 -1 [4; 2]
4->3 1 [4; 2; 1; 3]
```

Fortran

Works with: gfortran version 11.3.0

```module floyd_warshall_algorithm

use, intrinsic :: ieee_arithmetic

implicit none

integer, parameter :: floating_point_kind = &
& ieee_selected_real_kind (6, 37)
integer, parameter :: fpk = floating_point_kind

integer, parameter :: nil_vertex = 0

type :: edge
integer :: u
real(kind = fpk) :: weight
integer :: v
end type edge

type :: edge_list
type(edge), allocatable :: element(:)
end type edge_list

contains

subroutine make_example_graph (edges)
type(edge_list), intent(out) :: edges

allocate (edges%element(1:5))
edges%element(1) = edge (1, -2.0, 3)
edges%element(2) = edge (3, +2.0, 4)
edges%element(3) = edge (4, -1.0, 2)
edges%element(4) = edge (2, +4.0, 1)
edges%element(5) = edge (2, +3.0, 3)
end subroutine make_example_graph

function find_max_vertex (edges) result (n)
type(edge_list), intent(in) :: edges
integer n

integer i

n = 1
do i = lbound (edges%element, 1), ubound (edges%element, 1)
n = max (n, edges%element(i)%u)
n = max (n, edges%element(i)%v)
end do
end function find_max_vertex

subroutine floyd_warshall (edges, max_vertex, distance, next_vertex)

type(edge_list), intent(in) :: edges
integer, intent(out) :: max_vertex
real(kind = fpk), allocatable, intent(out) :: distance(:,:)
integer, allocatable, intent(out) :: next_vertex(:,:)

integer :: n
integer :: i, j, k
integer :: u, v
real(kind = fpk) :: dist_ikj
real(kind = fpk) :: infinity

n = find_max_vertex (edges)
max_vertex = n

allocate (distance(1:n, 1:n))
allocate (next_vertex(1:n, 1:n))

infinity = ieee_value (1.0_fpk,  ieee_positive_inf)

! Initialize.

do i = 1, n
do j = 1, n
distance(i, j) = infinity
next_vertex (i, j) = nil_vertex
end do
end do
do i = lbound (edges%element, 1), ubound (edges%element, 1)
u = edges%element(i)%u
v = edges%element(i)%v
distance(u, v) = edges%element(i)%weight
next_vertex(u, v) = v
end do
do i = 1, n
distance(i, i) = 0.0_fpk ! Distance from a vertex to itself.
next_vertex(i, i) = i
end do

! Perform the algorithm.

do k = 1, n
do i = 1, n
do j = 1, n
dist_ikj = distance(i, k) + distance(k, j)
if (dist_ikj < distance(i, j)) then
distance(i, j) = dist_ikj
next_vertex(i, j) = next_vertex(i, k)
end if
end do
end do
end do

end subroutine floyd_warshall

subroutine print_path (next_vertex, u, v)
integer, intent(in) :: next_vertex(:,:)
integer, intent(in) :: u, v

integer i

if (next_vertex(u, v) /= nil_vertex) then
i = u
write (*, '(I0)', advance = 'no') i
do while (i /= v)
i = next_vertex(i, v)
write (*, '('' -> '', I0)', advance = 'no') i
end do
end if
end subroutine print_path

end module floyd_warshall_algorithm

use, non_intrinsic :: floyd_warshall_algorithm

implicit none

type(edge_list) :: example_graph
integer :: max_vertex
real(kind = fpk), allocatable :: distance(:,:)
integer, allocatable :: next_vertex(:,:)
integer :: u, v

call make_example_graph (example_graph)
call floyd_warshall (example_graph, max_vertex, distance, &
&               next_vertex)

1000 format (1X, I0, ' -> ', I0, 5X, F4.1, 6X)

write (*, '(''  pair     distance    path'')')
write (*, '(''---------------------------------------'')')
do u = 1, max_vertex
do v = 1, max_vertex
if (u /= v) then
write (*, 1000, advance = 'no') u, v, distance(u, v)
call print_path (next_vertex, u, v)
write (*, '()', advance = 'yes')
end if
end do
end do

```
Output:
```\$ gfortran -g -std=f2018 -fcheck=all -fno-unsafe-math-optimizations -frounding-math -fsignaling-nans floyd_warshall_task.f90 && ./a.out
pair     distance    path
---------------------------------------
1 -> 2     -1.0      1 -> 3 -> 4 -> 2
1 -> 3     -2.0      1 -> 3
1 -> 4      0.0      1 -> 3 -> 4
2 -> 1      4.0      2 -> 1
2 -> 3      2.0      2 -> 1 -> 3
2 -> 4      4.0      2 -> 1 -> 3 -> 4
3 -> 1      5.0      3 -> 4 -> 2 -> 1
3 -> 2      1.0      3 -> 4 -> 2
3 -> 4      2.0      3 -> 4
4 -> 1      3.0      4 -> 2 -> 1
4 -> 2     -1.0      4 -> 2
4 -> 3      1.0      4 -> 2 -> 1 -> 3```

FreeBASIC

Translation of: Java
```' FB 1.05.0 Win64

Const POSITIVE_INFINITY As Double = 1.0/0.0

Sub printResult(dist(any, any) As Double, nxt(any, any) As Integer)
Dim As Integer u, v
Print("pair     dist    path")
For i As Integer = 0 To UBound(nxt, 1)
For j As Integer = 0 To UBound(nxt, 1)
If i <> j Then
u = i + 1
v = j + 1
Print Str(u); " -> "; Str(v); "    "; dist(i, j); "     "; Str(u);
Do
u = nxt(u - 1, v - 1)
Print " -> "; Str(u);
Loop While u <> v
Print
End If
Next j
Next i
End Sub

Sub floydWarshall(weights(Any, Any) As Integer, numVertices As Integer)
Dim dist(0 To numVertices - 1, 0 To numVertices - 1) As Double
For i As Integer = 0 To numVertices - 1
For j As Integer = 0 To numVertices - 1
dist(i, j) = POSITIVE_INFINITY
Next j
Next i

For x As Integer = 0 To UBound(weights, 1)
dist(weights(x, 0) - 1, weights(x, 1) - 1) = weights(x, 2)
Next x

Dim nxt(0 To numVertices - 1, 0 To numVertices - 1) As Integer
For i As Integer = 0 To numVertices - 1
For j As Integer = 0 To numVertices - 1
If i <> j Then nxt(i, j) = j + 1
Next j
Next i

For k As Integer = 0 To numVertices - 1
For i As Integer = 0 To numVertices - 1
For j As Integer = 0 To numVertices - 1
If (dist(i, k) + dist(k, j)) < dist(i, j) Then
dist(i, j) = dist(i, k) + dist(k, j)
nxt(i, j) = nxt(i, k)
End If
Next j
Next i
Next k

printResult(dist(), nxt())
End Sub

Dim weights(4, 2) As Integer = {{1, 3, -2}, {2, 1, 4}, {2, 3, 3}, {3, 4, 2}, {4, 2, -1}}
Dim numVertices As Integer = 4
floydWarshall(weights(), numVertices)
Print
Print "Press any key to quit"
Sleep
```
Output:
```pair     dist    path
1 -> 2    -1     1 -> 3 -> 4 -> 2
1 -> 3    -2     1 -> 3
1 -> 4     0     1 -> 3 -> 4
2 -> 1     4     2 -> 1
2 -> 3     2     2 -> 1 -> 3
2 -> 4     4     2 -> 1 -> 3 -> 4
3 -> 1     5     3 -> 4 -> 2 -> 1
3 -> 2     1     3 -> 4 -> 2
3 -> 4     2     3 -> 4
4 -> 1     3     4 -> 2 -> 1
4 -> 2    -1     4 -> 2
4 -> 3     1     4 -> 2 -> 1 -> 3
```

Go

```package main

import (
"fmt"
"strconv"
)

// A Graph is the interface implemented by graphs that
// this algorithm can run on.
type Graph interface {
Vertices() []Vertex
Neighbors(v Vertex) []Vertex
Weight(u, v Vertex) int
}

// Nonnegative integer ID of vertex
type Vertex int

// ig is a graph of integers that satisfies the Graph interface.
type ig struct {
vert  []Vertex
edges map[Vertex]map[Vertex]int
}

func (g ig) edge(u, v Vertex, w int) {
if _, ok := g.edges[u]; !ok {
g.edges[u] = make(map[Vertex]int)
}
g.edges[u][v] = w
}
func (g ig) Vertices() []Vertex { return g.vert }
func (g ig) Neighbors(v Vertex) (vs []Vertex) {
for k := range g.edges[v] {
vs = append(vs, k)
}
return vs
}
func (g ig) Weight(u, v Vertex) int { return g.edges[u][v] }
func (g ig) path(vv []Vertex) (s string) {
if len(vv) == 0 {
return ""
}
s = strconv.Itoa(int(vv[0]))
for _, v := range vv[1:] {
s += " -> " + strconv.Itoa(int(v))
}
return s
}

const Infinity = int(^uint(0) >> 1)

func FloydWarshall(g Graph) (dist map[Vertex]map[Vertex]int, next map[Vertex]map[Vertex]*Vertex) {
vert := g.Vertices()
dist = make(map[Vertex]map[Vertex]int)
next = make(map[Vertex]map[Vertex]*Vertex)
for _, u := range vert {
dist[u] = make(map[Vertex]int)
next[u] = make(map[Vertex]*Vertex)
for _, v := range vert {
dist[u][v] = Infinity
}
dist[u][u] = 0
for _, v := range g.Neighbors(u) {
v := v
dist[u][v] = g.Weight(u, v)
next[u][v] = &v
}
}
for _, k := range vert {
for _, i := range vert {
for _, j := range vert {
if dist[i][k] < Infinity && dist[k][j] < Infinity {
if dist[i][j] > dist[i][k]+dist[k][j] {
dist[i][j] = dist[i][k] + dist[k][j]
next[i][j] = next[i][k]
}
}
}
}
}
return dist, next
}

func Path(u, v Vertex, next map[Vertex]map[Vertex]*Vertex) (path []Vertex) {
if next[u][v] == nil {
return
}
path = []Vertex{u}
for u != v {
u = *next[u][v]
path = append(path, u)
}
return path
}

func main() {
g := ig{[]Vertex{1, 2, 3, 4}, make(map[Vertex]map[Vertex]int)}
g.edge(1, 3, -2)
g.edge(3, 4, 2)
g.edge(4, 2, -1)
g.edge(2, 1, 4)
g.edge(2, 3, 3)

dist, next := FloydWarshall(g)
fmt.Println("pair\tdist\tpath")
for u, m := range dist {
for v, d := range m {
if u != v {
fmt.Printf("%d -> %d\t%3d\t%s\n", u, v, d, g.path(Path(u, v, next)))
}
}
}
}
```
Output:
```pair    dist    path
1 -> 2   -1 1 -> 3 -> 4 -> 2
1 -> 3   -2 1 -> 3
1 -> 4    0 1 -> 3 -> 4
2 -> 1    4 2 -> 1
2 -> 3    2 2 -> 1 -> 3
2 -> 4    4 2 -> 1 -> 3 -> 4
3 -> 1    5 3 -> 4 -> 2 -> 1
3 -> 2    1 3 -> 4 -> 2
3 -> 4    2 3 -> 4
4 -> 1    3 4 -> 2 -> 1
4 -> 2   -1 4 -> 2
4 -> 3    1 4 -> 2 -> 1 -> 3
```

Groovy

Translation of: Java
```class FloydWarshall {
static void main(String[] args) {
int[][] weights = [[1, 3, -2], [2, 1, 4], [2, 3, 3], [3, 4, 2], [4, 2, -1]]
int numVertices = 4

floydWarshall(weights, numVertices)
}

static void floydWarshall(int[][] weights, int numVertices) {
double[][] dist = new double[numVertices][numVertices]
for (double[] row : dist) {
Arrays.fill(row, Double.POSITIVE_INFINITY)
}

for (int[] w : weights) {
dist[w[0] - 1][w[1] - 1] = w[2]
}

int[][] next = new int[numVertices][numVertices]
for (int i = 0; i < next.length; i++) {
for (int j = 0; j < next.length; j++) {
if (i != j) {
next[i][j] = j + 1
}
}
}

for (int k = 0; k < numVertices; k++) {
for (int i = 0; i < numVertices; i++) {
for (int j = 0; j < numVertices; j++) {
if (dist[i][k] + dist[k][j] < dist[i][j]) {
dist[i][j] = dist[i][k] + dist[k][j]
next[i][j] = next[i][k]
}
}
}
}

printResult(dist, next)
}

static void printResult(double[][] dist, int[][] next) {
println("pair     dist    path")
for (int i = 0; i < next.length; i++) {
for (int j = 0; j < next.length; j++) {
if (i != j) {
int u = i + 1
int v = j + 1
String path = String.format("%d -> %d    %2d     %s", u, v, (int) dist[i][j], u)
boolean loop = true
while (loop) {
u = next[u - 1][v - 1]
path += " -> " + u
loop = u != v
}
println(path)
}
}
}
}
}
```
Output:
```pair     dist    path
1 -> 2    -1     1 -> 3 -> 4 -> 2
1 -> 3    -2     1 -> 3
1 -> 4     0     1 -> 3 -> 4
2 -> 1     4     2 -> 1
2 -> 3     2     2 -> 1 -> 3
2 -> 4     4     2 -> 1 -> 3 -> 4
3 -> 1     5     3 -> 4 -> 2 -> 1
3 -> 2     1     3 -> 4 -> 2
3 -> 4     2     3 -> 4
4 -> 1     3     4 -> 2 -> 1
4 -> 2    -1     4 -> 2
4 -> 3     1     4 -> 2 -> 1 -> 3```

Necessary imports

```import Control.Monad (join)
import Data.List (union)
import Data.Map hiding (foldr, union)
import Data.Maybe (fromJust, isJust)
import Data.Semigroup
import Prelude hiding (lookup, filter)
```

First we define a general datatype to represent the shortest path. Type `a` represents a distance. It could be a number, in case of weighted graph or boolean value for just a directed graph. Type `b` goes for vertice labels (integers, chars, strings...)

```data Shortest b a = Shortest { distance :: a, path :: [b] }
deriving Show
```

Next we note that shortest paths form a semigroup with following "addition" rule:

```instance (Ord a, Eq b) => Semigroup (Shortest b a) where
a <> b = case distance a `compare` distance b of
GT -> b
LT -> a
EQ -> a { path = path a `union` path b }
```

It finds minimal path by `distance`, and in case of equal distances joins both paths. We will lift this semigroup to monoid using `Maybe` wrapper.

Graph is represented as a `Map`, containing pairs of vertices and corresponding weigts. The distance table is a `Map`, containing pairs of joint vertices and corresponding shortest paths.

Now we are ready to define the main part of the Floyd-Warshall algorithm, which processes properly prepared distance table `dist` for given list of vertices `v`:

```floydWarshall v dist = foldr innerCycle (Just <\$> dist) v
where
innerCycle k dist = (newDist <\$> v <*> v) `setTo` dist
where
newDist i j =
((i,j), do a <- join \$ lookup (i, k) dist
b <- join \$ lookup (k, j) dist
return \$ Shortest (distance a <> distance b) (path a))

setTo = unionWith (<>) . fromList
```

The `floydWarshall` produces only first steps of shortest paths. Whole paths are build by following function:

```buildPaths d = mapWithKey (\pair s -> s { path = buildPath pair}) d
where
buildPath (i,j)
| i == j    = [[j]]
| otherwise = do k <- path \$ fromJust \$ lookup (i,j) d
p <- buildPath (k,j)
[i : p]
```

All pre- and postprocessing is done by the main function `findMinDistances`:

```findMinDistances v g =
let weights = mapWithKey (\(_,j) w -> Shortest w [j]) g
trivial = fromList [ ((i,i), Shortest mempty []) | i <- v ]
clean d = fromJust <\$> filter isJust (d \\ trivial)
in buildPaths \$ clean \$ floydWarshall v (weights <> trivial)
```

Examples:

The sample graph:

```g = fromList [((2,1), 4)
,((2,3), 3)
,((1,3), -2)
,((3,4), 2)
,((4,2), -1)]
```

the helper function

```showShortestPaths v g = mapM_ print \$ toList \$ findMinDistances v g
```
Output:

Weights as distances:

```λ> showShortestPaths [1..4] (Sum <\$> g)
((1,2),Shortest {distance = Sum {getSum = -1}, path = [[1,3,4,2]]})
((1,3),Shortest {distance = Sum {getSum = -2}, path = [[1,3]]})
((1,4),Shortest {distance = Sum {getSum = 0}, path = [[1,3,4]]})
((2,1),Shortest {distance = Sum {getSum = 4}, path = [[2,1]]})
((2,3),Shortest {distance = Sum {getSum = 2}, path = [[2,1,3]]})
((2,4),Shortest {distance = Sum {getSum = 4}, path = [[2,1,3,4]]})
((3,1),Shortest {distance = Sum {getSum = 5}, path = [[3,4,2,1]]})
((3,2),Shortest {distance = Sum {getSum = 1}, path = [[3,4,2]]})
((3,4),Shortest {distance = Sum {getSum = 2}, path = [[3,4]]})
((4,1),Shortest {distance = Sum {getSum = 3}, path = [[4,2,1]]})
((4,2),Shortest {distance = Sum {getSum = -1}, path = [[4,2]]})
((4,3),Shortest {distance = Sum {getSum = 1}, path = [[4,2,1,3]]})```

Unweighted directed graph

```λ> showShortestPaths [1..4] (Any . (/= 0) <\$> g)
((1,2),Shortest {distance = Any {getAny = True}, path = [[1,3,4,2]]})
((1,3),Shortest {distance = Any {getAny = True}, path = [[1,3]]})
((1,4),Shortest {distance = Any {getAny = True}, path = [[1,3,4]]})
((2,1),Shortest {distance = Any {getAny = True}, path = [[2,1]]})
((2,3),Shortest {distance = Any {getAny = True}, path = [[2,1,3],[2,3]]})
((2,4),Shortest {distance = Any {getAny = True}, path = [[2,1,3,4],[2,3,4]]})
((3,1),Shortest {distance = Any {getAny = True}, path = [[3,4,2,1]]})
((3,2),Shortest {distance = Any {getAny = True}, path = [[3,4,2]]})
((3,4),Shortest {distance = Any {getAny = True}, path = [[3,4]]})
((4,1),Shortest {distance = Any {getAny = True}, path = [[4,2,1]]})
((4,2),Shortest {distance = Any {getAny = True}, path = [[4,2]]})
((4,3),Shortest {distance = Any {getAny = True}, path = [[4,2,1,3],[4,2,3]]})```

For some pairs several possible paths are found.

Uniformly weighted graph:

```λ> showShortestPaths [1..4] (const (Sum 1) <\$> g)
((1,2),Shortest {distance = Sum {getSum = 3}, path = [[1,3,4,2]]})
((1,3),Shortest {distance = Sum {getSum = 1}, path = [[1,3]]})
((1,4),Shortest {distance = Sum {getSum = 2}, path = [[1,3,4]]})
((2,1),Shortest {distance = Sum {getSum = 1}, path = [[2,1]]})
((2,3),Shortest {distance = Sum {getSum = 1}, path = [[2,3]]})
((2,4),Shortest {distance = Sum {getSum = 2}, path = [[2,3,4]]})
((3,1),Shortest {distance = Sum {getSum = 3}, path = [[3,4,2,1]]})
((3,2),Shortest {distance = Sum {getSum = 2}, path = [[3,4,2]]})
((3,4),Shortest {distance = Sum {getSum = 1}, path = [[3,4]]})
((4,1),Shortest {distance = Sum {getSum = 2}, path = [[4,2,1]]})
((4,2),Shortest {distance = Sum {getSum = 1}, path = [[4,2]]})
((4,3),Shortest {distance = Sum {getSum = 2}, path = [[4,2,3]]})```

Graph labeled by chars:

```g2 = fromList [(('A','S'), 1)
,(('A','D'), -1)
,(('S','E'), 2)
,(('D','E'), 4)]
```
```λ> showShortestPaths "ASDE" (Sum <\$> g2)
(('A','D'),Shortest {distance = Sum {getSum = -1}, path = ["AD"]})
(('A','E'),Shortest {distance = Sum {getSum = 3}, path = ["ASE","ADE"]})
(('A','S'),Shortest {distance = Sum {getSum = 1}, path = ["AS"]})
(('D','E'),Shortest {distance = Sum {getSum = 4}, path = ["DE"]})
(('S','E'),Shortest {distance = Sum {getSum = 2}, path = ["SE"]})```

Icon

Translation of: Scheme
Works with: Icon version 9.5.20i

```#
# Floyd-Warshall algorithm.
#
# See https://en.wikipedia.org/w/index.php?title=Floyd%E2%80%93Warshall_algorithm&oldid=1082310013
#

record fw_results (n, distance, next_vertex)

procedure main ()
local example_graph
local fw
local u, v

example_graph := [[1, -2.0, 3],
[3, +2.0, 4],
[4, -1.0, 2],
[2, +4.0, 1],
[2, +3.0, 3]]

fw := floyd_warshall (example_graph)

printf ("  pair    distance   path\n")
printf ("-------------------------------------\n")
every u := 1 to fw.n do {
every v := 1 to fw.n do {
if u ~= v then {
printf (" %d -> %d    %4s     %s\n", u, v,
string (ref_array (fw.distance, u, v)),
path_to_string (find_path (fw.next_vertex, u, v)))
}
}
}
end

procedure floyd_warshall (edges)
local n, distance, next_vertex
local e
local i, j, k
local dist_ij, dist_ik, dist_kj, dist_ikj

n := max_vertex (edges)
distance := create_array ([1, 1], [n, n], &null)
next_vertex := create_array ([1, 1], [n, n], &null)

# Initialization.
every e := !edges do {
ref_array (distance, e[1], e[3]) := e[2]
ref_array (next_vertex, e[1], e[3]) := e[3]
}
every i := 1 to n do {
ref_array (distance, i, i) := 0.0 # Distance to self = 0.
ref_array (next_vertex, i, i) := i
}

# Perform the algorithm. Here &null will play the role of
# "infinity": "\" means a value is finite, "/" that it is infinite.
every k := 1 to n do {
every i := 1 to n do {
every j := 1 to n do {
dist_ij := ref_array (distance, i, j)
dist_ik := ref_array (distance, i, k)
dist_kj := ref_array (distance, k, j)
if \dist_ik & \dist_kj then {
dist_ikj := dist_ik + dist_kj
if /dist_ij | dist_ikj < dist_ij then {
ref_array (distance, i, j) := dist_ikj
ref_array (next_vertex, i, j) :=
ref_array (next_vertex, i, k)
}
}
}
}
}

return fw_results (n, distance, next_vertex)
end

procedure find_path (next_vertex, u, v)
local path

if / (ref_array (next_vertex, u, v)) then {
path := []
} else {
path := [u]
while u ~= v do {
u := ref_array (next_vertex, u, v)
put (path, u)
}
}
return path
end

procedure path_to_string (path)
local s

if *path = 0 then {
s := ""
} else {
s := string (path[1])
every s ||:= (" -> " || !path[2 : 0])
}
return s
end

procedure max_vertex (edges)
local e
local m

*edges = 0 & stop ("no edges")
m := 1
every e := !edges do m := max (m, e[1], e[3])
return m
end
```
Output:
```\$ icon floyd-warshall-in-Icon.icn
pair    distance   path
-------------------------------------
1 -> 2    -1.0     1 -> 3 -> 4 -> 2
1 -> 3    -2.0     1 -> 3
1 -> 4     0.0     1 -> 3 -> 4
2 -> 1     4.0     2 -> 1
2 -> 3     2.0     2 -> 1 -> 3
2 -> 4     4.0     2 -> 1 -> 3 -> 4
3 -> 1     5.0     3 -> 4 -> 2 -> 1
3 -> 2     1.0     3 -> 4 -> 2
3 -> 4     2.0     3 -> 4
4 -> 1     3.0     4 -> 2 -> 1
4 -> 2    -1.0     4 -> 2
4 -> 3     1.0     4 -> 2 -> 1 -> 3```

J

```floyd=: verb define
for_j. i.#y do.
y=. y <. j ({"1 +/ {) y
end.
)
```

Alternate implementation (same behavior):

```floyd=: ]F..(]<.{"1+/{) i.@#
```

Example use:

```graph=: ".;._2]0 :0
0  _ _2 _  NB. 1->3 costs _2
4  0  3 _  NB. 2->1 costs 4; 2->3 costs 3
_  _  0 2  NB. 3->4 costs 2
_ _1  _ 0  NB. 4->2 costs _1
)

floyd graph
0 _1 _2 0
4  0  2 4
5  1  0 2
3 _1  1 0
```

The graph matrix holds the costs of each directed node. Row index corresponds to starting node. Column index corresponds to ending node. Unconnected nodes have infinite cost.

This approach turns out to be faster than the more concise <./ .+~^:_ for many relatively small graphs (though `floyd` happens to be slightly slower for the task example).

Path Reconstruction

This draft task currently asks for path reconstruction, which is a different (related) algorithm:

```floydrecon=: verb define
n=. (\$y)\$_(I._=,y)},(\$\$i.@#)y
for_j. i.#y do.
d=. y <. j ({"1 +/ {) y
b=. y~:d
y=. d
n=. (n*-.b)+b * j{"1 n
end.
)

dist=. floyd y
next=. floydrecon y
echo 'pair  dist   path'
for_i. i.#y do.
for_k. i.#y do.
ndx=. <i,k
if. (i~:k)*_>ndx{next do.
txt=. (":1+i),'->',(":1+k)
txt=. txt,_5{.":ndx{dist
txt=. txt,'    ',":1+i
j=. i
while. j~:k do.
assert. j~:(<j,k){next
j=. (<j,k){next
txt=. txt,'->',":1+j
end.
echo txt
end.
end.
end.
i.0 0
)
```

Draft output:

```   task graph
pair  dist   path
1->2   _1    1->3->4->2
1->3   _2    1->3
1->4    0    1->3->4
2->1    4    2->1
2->3    2    2->1->3
2->4    4    2->1->3->4
3->1    5    3->4->2->1
3->2    1    3->4->2
3->4    2    3->4
4->1    3    4->2->1
4->2   _1    4->2
4->3    1    4->2->1->3
```

Java

```import static java.lang.String.format;
import java.util.Arrays;

public class FloydWarshall {

public static void main(String[] args) {
int[][] weights = {{1, 3, -2}, {2, 1, 4}, {2, 3, 3}, {3, 4, 2}, {4, 2, -1}};
int numVertices = 4;

floydWarshall(weights, numVertices);
}

static void floydWarshall(int[][] weights, int numVertices) {

double[][] dist = new double[numVertices][numVertices];
for (double[] row : dist)
Arrays.fill(row, Double.POSITIVE_INFINITY);

for (int[] w : weights)
dist[w[0] - 1][w[1] - 1] = w[2];

int[][] next = new int[numVertices][numVertices];
for (int i = 0; i < next.length; i++) {
for (int j = 0; j < next.length; j++)
if (i != j)
next[i][j] = j + 1;
}

for (int k = 0; k < numVertices; k++)
for (int i = 0; i < numVertices; i++)
for (int j = 0; j < numVertices; j++)
if (dist[i][k] + dist[k][j] < dist[i][j]) {
dist[i][j] = dist[i][k] + dist[k][j];
next[i][j] = next[i][k];
}

printResult(dist, next);
}

static void printResult(double[][] dist, int[][] next) {
System.out.println("pair     dist    path");
for (int i = 0; i < next.length; i++) {
for (int j = 0; j < next.length; j++) {
if (i != j) {
int u = i + 1;
int v = j + 1;
String path = format("%d -> %d    %2d     %s", u, v,
(int) dist[i][j], u);
do {
u = next[u - 1][v - 1];
path += " -> " + u;
} while (u != v);
System.out.println(path);
}
}
}
}
}
```
```pair     dist    path
1 -> 2    -1     1 -> 3 -> 4 -> 2
1 -> 3    -2     1 -> 3
1 -> 4     0     1 -> 3 -> 4
2 -> 1     4     2 -> 1
2 -> 3     2     2 -> 1 -> 3
2 -> 4     4     2 -> 1 -> 3 -> 4
3 -> 1     5     3 -> 4 -> 2 -> 1
3 -> 2     1     3 -> 4 -> 2
3 -> 4     2     3 -> 4
4 -> 1     3     4 -> 2 -> 1
4 -> 2    -1     4 -> 2
4 -> 3     1     4 -> 2 -> 1 -> 3```

JavaScript

Using output code translated from the Lua sample.

```'use strict'
let numVertices = 4;
let weights = [ [ 1, 3, -2 ], [ 2, 1, 4 ], [ 2, 3, 3 ], [ 3, 4, 2 ], [ 4, 2, -1 ] ];

let graph = [];
for (let i = 0; i < numVertices; ++i) {
graph.push([]);
for (let j = 0; j < numVertices; ++j)
graph[i].push(i == j ? 0 : 9999999);
}

for (let i = 0; i < weights.length; ++i) {
let w = weights[i];
graph[w[0] - 1][w[1] - 1] = w[2];
}

let nxt = [];
for (let i = 0; i < numVertices; ++i) {
nxt.push([]);
for (let j = 0; j < numVertices; ++j)
nxt[i].push(i == j ? 0 : j + 1);
}

for (let k = 0; k < numVertices; ++k) {
for (let i = 0; i < numVertices; ++i) {
for (let j = 0; j < numVertices; ++j) {
if (graph[i][j] > graph[i][k] + graph[k][j]) {
graph[i][j] = graph[i][k] + graph[k][j];
nxt[i][j] = nxt[i][k];
}
}
}
}

console.log("pair     dist    path");
for (let i = 0; i < numVertices; ++i) {
for (let j = 0; j < numVertices; ++j) {
if (i != j) {
let u = i + 1;
let v = j + 1;
let path = u + " -> " + v + "    " + graph[i][j].toString().padStart(2) + "     " + u;
do {
u = nxt[u - 1][v - 1];
path = path + " -> " + u;
} while (u != v);
console.log(path)
}
}
}
```
Output:
```pair     dist    path
1 -> 2    -1     1 -> 3 -> 4 -> 2
1 -> 3    -2     1 -> 3
1 -> 4     0     1 -> 3 -> 4
2 -> 1     4     2 -> 1
2 -> 3     2     2 -> 1 -> 3
2 -> 4     4     2 -> 1 -> 3 -> 4
3 -> 1     5     3 -> 4 -> 2 -> 1
3 -> 2     1     3 -> 4 -> 2
3 -> 4     2     3 -> 4
4 -> 1     3     4 -> 2 -> 1
4 -> 2    -1     4 -> 2
4 -> 3     1     4 -> 2 -> 1 -> 3
```

jq

Works with: jq version 1.5

In this section, we represent the graph by a JSON object giving the weights: if u and v are the (string) labels of two nodes connected with an arrow from u to v, then .[u][v] is the associated weight:

```def weights: {
"1": {"3": -2},
"2": {"1" : 4, "3": 3},
"3": {"4": 2},
"4": {"2": -1}
};```

The algorithm given here is a direct implementation of the definitional algorithm:

```def fwi:
. as \$weights
| keys_unsorted as \$nodes
# construct the dist matrix
| reduce \$nodes[] as \$u ({};
reduce \$nodes[] as \$v (.;
.[\$u][\$v] = infinite))
| reduce \$nodes[] as \$u (.; .[\$u][\$u] = 0 )
| reduce \$nodes[] as \$u (.;
reduce (\$weights[\$u]|keys_unsorted[]) as \$v (.;
.[\$u][\$v] = \$weights[\$u][\$v] ))
| reduce \$nodes[] as \$w (.;
reduce \$nodes[] as \$u (.;
reduce \$nodes[] as \$v (.;
(.[\$u][\$w] + .[\$w][\$v]) as \$x
| if .[\$u][\$v] > \$x then .[\$u][\$v] = \$x
else . end )))
;

weights | fwi```
Output:
```{
"1": {
"1": 0,
"2": -1,
"3": -2,
"4": 0
},
"2": {
"1": 4,
"2": 0,
"3": 2,
"4": 4
},
"3": {
"1": 5,
"2": 1,
"3": 0,
"4": 2
},
"4": {
"1": 3,
"2": -1,
"3": 1,
"4": 0
}
}```

Julia

Translation of: Java
```# Floyd-Warshall algorithm: https://rosettacode.org/wiki/Floyd-Warshall_algorithm
# v0.6

function floydwarshall(weights::Matrix, nvert::Int)
dist = fill(Inf, nvert, nvert)
for i in 1:size(weights, 1)
dist[weights[i, 1], weights[i, 2]] = weights[i, 3]
end
# return dist
next = collect(j != i ? j : 0 for i in 1:nvert, j in 1:nvert)

for k in 1:nvert, i in 1:nvert, j in 1:nvert
if dist[i, k] + dist[k, j] < dist[i, j]
dist[i, j] = dist[i, k] + dist[k, j]
next[i, j] = next[i, k]
end
end

# return next
function printresult(dist, next)
println("pair     dist    path")
for i in 1:size(next, 1), j in 1:size(next, 2)
if i != j
u = i
path = @sprintf "%d -> %d    %2d     %s" i j dist[i, j] i
while true
u = next[u, j]
path *= " -> \$u"
if u == j break end
end
println(path)
end
end
end
printresult(dist, next)
end

floydwarshall([1 3 -2; 2 1 4; 2 3 3; 3 4 2; 4 2 -1], 4)
```

Kotlin

Translation of: Java
```// version 1.1

object FloydWarshall {
fun doCalcs(weights: Array<IntArray>, nVertices: Int) {
val dist = Array(nVertices) { DoubleArray(nVertices) { Double.POSITIVE_INFINITY } }
for (w in weights) dist[w[0] - 1][w[1] - 1] = w[2].toDouble()
val next = Array(nVertices) { IntArray(nVertices) }
for (i in 0 until next.size) {
for (j in 0 until next.size) {
if (i != j) next[i][j] = j + 1
}
}
for (k in 0 until nVertices) {
for (i in 0 until nVertices) {
for (j in 0 until nVertices) {
if (dist[i][k] + dist[k][j] < dist[i][j]) {
dist[i][j] = dist[i][k] + dist[k][j]
next[i][j] = next[i][k]
}
}
}
}
printResult(dist, next)
}

private fun printResult(dist: Array<DoubleArray>, next: Array<IntArray>) {
var u: Int
var v: Int
var path: String
println("pair     dist    path")
for (i in 0 until next.size) {
for (j in 0 until next.size) {
if (i != j) {
u = i + 1
v = j + 1
path = ("%d -> %d    %2d     %s").format(u, v, dist[i][j].toInt(), u)
do {
u = next[u - 1][v - 1]
path += " -> " + u
} while (u != v)
println(path)
}
}
}
}
}

fun main(args: Array<String>) {
val weights = arrayOf(
intArrayOf(1, 3, -2),
intArrayOf(2, 1, 4),
intArrayOf(2, 3, 3),
intArrayOf(3, 4, 2),
intArrayOf(4, 2, -1)
)
val nVertices = 4
FloydWarshall.doCalcs(weights, nVertices)
}
```
Output:
```pair     dist    path
1 -> 2    -1     1 -> 3 -> 4 -> 2
1 -> 3    -2     1 -> 3
1 -> 4     0     1 -> 3 -> 4
2 -> 1     4     2 -> 1
2 -> 3     2     2 -> 1 -> 3
2 -> 4     4     2 -> 1 -> 3 -> 4
3 -> 1     5     3 -> 4 -> 2 -> 1
3 -> 2     1     3 -> 4 -> 2
3 -> 4     2     3 -> 4
4 -> 1     3     4 -> 2 -> 1
4 -> 2    -1     4 -> 2
4 -> 3     1     4 -> 2 -> 1 -> 3
```

Lua

Translation of: D
```function printResult(dist, nxt)
print("pair     dist    path")
for i=0, #nxt do
for j=0, #nxt do
if i ~= j then
u = i + 1
v = j + 1
path = string.format("%d -> %d    %2d     %s", u, v, dist[i][j], u)
repeat
u = nxt[u-1][v-1]
path = path .. " -> " .. u
until (u == v)
print(path)
end
end
end
end

function floydWarshall(weights, numVertices)
dist = {}
for i=0, numVertices-1 do
dist[i] = {}
for j=0, numVertices-1 do
dist[i][j] = math.huge
end
end

for _,w in pairs(weights) do
-- the weights array is one based
dist[w[1]-1][w[2]-1] = w[3]
end

nxt = {}
for i=0, numVertices-1 do
nxt[i] = {}
for j=0, numVertices-1 do
if i ~= j then
nxt[i][j] = j+1
end
end
end

for k=0, numVertices-1 do
for i=0, numVertices-1 do
for j=0, numVertices-1 do
if dist[i][k] + dist[k][j] < dist[i][j] then
dist[i][j] = dist[i][k] + dist[k][j]
nxt[i][j] = nxt[i][k]
end
end
end
end

printResult(dist, nxt)
end

weights = {
{1, 3, -2},
{2, 1, 4},
{2, 3, 3},
{3, 4, 2},
{4, 2, -1}
}
numVertices = 4
floydWarshall(weights, numVertices)
```
Output:
```pair     dist    path
1 -> 2    -1     1 -> 3 -> 4 -> 2
1 -> 3    -2     1 -> 3
1 -> 4     0     1 -> 3 -> 4
2 -> 1     4     2 -> 1
2 -> 3     2     2 -> 1 -> 3
2 -> 4     4     2 -> 1 -> 3 -> 4
3 -> 1     5     3 -> 4 -> 2 -> 1
3 -> 2     1     3 -> 4 -> 2
3 -> 4     2     3 -> 4
4 -> 1     3     4 -> 2 -> 1
4 -> 2    -1     4 -> 2
4 -> 3     1     4 -> 2 -> 1 -> 3```

Mathematica / Wolfram Language

```g = Graph[{1 \[DirectedEdge] 3, 3 \[DirectedEdge] 4,
4 \[DirectedEdge] 2, 2 \[DirectedEdge] 1, 2 \[DirectedEdge] 3},
EdgeWeight -> {(1 \[DirectedEdge] 3) -> -2, (3 \[DirectedEdge] 4) ->
2, (4 \[DirectedEdge] 2) -> -1, (2 \[DirectedEdge] 1) ->
4, (2 \[DirectedEdge] 3) -> 3}]
vl = VertexList[g];
dm = GraphDistanceMatrix[g];
Grid[LexicographicSort[
DeleteCases[
Catenate[
Table[{vl[[i]], vl[[j]], dm[[i, j]]}, {i, Length[vl]}, {j,
Length[vl]}]], {x_, x_, _}]]]
```
Output:
```1	2	-1.
1	3	-2.
1	4	0.
2	1	4.
2	3	2.
2	4	4.
3	1	5.
3	2	1.
3	4	2.
4	1	3.
4	2	-1.
4	3	1.```

Mercury

Translation of: Scheme
Works with: Mercury version 20.06.1

```:- module floyd_warshall_task.

:- interface.
:- import_module io.
:- pred main(io, io).
:- mode main(di, uo) is det.

:- implementation.
:- import_module float.
:- import_module int.
:- import_module list.
:- import_module string.
:- import_module version_array2d.

%%%-------------------------------------------------------------------

%% Square arrays with 1-based indexing.

:- func arr_init(int, T) = version_array2d(T).
arr_init(N, Fill) = version_array2d.init(N, N, Fill).

:- func arr_get(version_array2d(T), int, int) = T.
arr_get(Arr, I, J) = Elem :-
I1 = I - 1,
J1 = J - 1,
Elem = Arr^elem(I1, J1).

:- func arr_set(version_array2d(T), int, int, T) = version_array2d(T).
arr_set(Arr0, I, J, Elem) = Arr :-
I1 = I - 1,
J1 = J - 1,
Arr = (Arr0^elem(I1, J1) := Elem).

%%%-------------------------------------------------------------------

:- func find_max_vertex(list({int, float, int})) = int.
find_max_vertex(Edges) = find_max_vertex_(Edges, 0).

:- func find_max_vertex_(list({int, float, int}), int) = int.
find_max_vertex_([], MaxVertex0) = MaxVertex0.
find_max_vertex_([{U, _, V} | Tail], MaxVertex0) = MaxVertex :-
MaxVertex = find_max_vertex_(Tail, max(max(MaxVertex0, U), V)).

%%%-------------------------------------------------------------------

:- func arbitrary_float = float.
arbitrary_float = (12345.0).

:- func nil_vertex = int.
nil_vertex = 0.

:- func floyd_warshall(list({int, float, int})) =
{int, version_array2d(float), version_array2d(int)}.
floyd_warshall(Edges) = {N, Dist, Next} :-
N = find_max_vertex(Edges),
Dist0 = arr_init(N, arbitrary_float),
Next0 = arr_init(N, nil_vertex),
(if (N = 0) then (Dist = Dist0,
Next = Next0)
else ({Dist1, Next1} = floyd_warshall_initialize(Edges, N,
Dist0, Next0),
{Dist, Next} = floyd_warshall_loop_k(N, 1, Dist1, Next1))).

:- func floyd_warshall_initialize(list({int, float, int}),
int,
version_array2d(float),
version_array2d(int)) =
{version_array2d(float), version_array2d(int)}.
floyd_warshall_initialize(Edges, N, Dist0, Next0) = {Dist1, Next1} :-
floyd_warshall_read_edges(Edges, Dist0, Next0) = {D1, X1},
floyd_warshall_diagonals(N, 1, D1, X1) = {Dist1, Next1}.

version_array2d(float),
version_array2d(int)) =
{version_array2d(float), version_array2d(int)}.
floyd_warshall_read_edges([], Dist0, Next0) = {Dist0, Next0}.
Dist0, Next0) = {Dist1, Next1} :-
D1 = arr_set(Dist0, U, V, Weight),
X1 = arr_set(Next0, U, V, V),
floyd_warshall_read_edges(Tail, D1, X1) = {Dist1, Next1}.

:- func floyd_warshall_diagonals(int, int,
version_array2d(float),
version_array2d(int)) =
{version_array2d(float), version_array2d(int)}.
floyd_warshall_diagonals(N, I, Dist0, Next0) = {Dist1, Next1} :-
N1 = N + 1,
(if (I = N1) then (Dist1 = Dist0,
Next1 = Next0)
else (
%% The distance from a vertex to itself = 0.0.
D1 = arr_set(Dist0, I, I, 0.0),
X1 = arr_set(Next0, I, I, I),
I1 = I + 1,
floyd_warshall_diagonals(N, I1, D1, X1) = {Dist1, Next1})).

:- func floyd_warshall_loop_k(int, int,
version_array2d(float),
version_array2d(int)) =
{version_array2d(float), version_array2d(int)}.
floyd_warshall_loop_k(N, K, Dist0, Next0) = {Dist1, Next1} :-
N1 = N + 1,
(if (K = N1) then (Dist1 = Dist0,
Next1 = Next0)
else ({D1, X1} = floyd_warshall_loop_i(N, K, 1, Dist0, Next0),
K1 = K + 1,
{Dist1, Next1} = floyd_warshall_loop_k(N, K1, D1, X1))).

:- func floyd_warshall_loop_i(int, int, int,
version_array2d(float),
version_array2d(int)) =
{version_array2d(float), version_array2d(int)}.
floyd_warshall_loop_i(N, K, I, Dist0, Next0) = {Dist1, Next1} :-
N1 = N + 1,
(if (I = N1) then (Dist1 = Dist0,
Next1 = Next0)
else ({D1, X1} = floyd_warshall_loop_j(N, K, I, 1, Dist0, Next0),
I1 = I + 1,
{Dist1, Next1} = floyd_warshall_loop_i(N, K, I1, D1, X1))).

:- func floyd_warshall_loop_j(int, int, int, int,
version_array2d(float),
version_array2d(int)) =
{version_array2d(float), version_array2d(int)}.
floyd_warshall_loop_j(N, K, I, J, Dist0, Next0) = {Dist1, Next1} :-
J1 = J + 1,
N1 = N + 1,
(if (J = N1) then (Dist1 = Dist0,
Next1 = Next0)
else (if ((arr_get(Next0, I, K) = nil_vertex);
(arr_get(Next0, K, J) = nil_vertex))
then ({Dist1, Next1} =
floyd_warshall_loop_j(N, K, I, J1, Dist0, Next0))
else (Dist_ikj = arr_get(Dist0, I, K) + arr_get(Dist0, K, J),
(if (arr_get(Next0, I, J) = nil_vertex;
Dist_ikj < arr_get(Dist0, I, J))
then (D1 = arr_set(Dist0, I, J, Dist_ikj),
X1 = arr_set(Next0, I, J, arr_get(Next0, I, K)),
{Dist1, Next1} =
floyd_warshall_loop_j(N, K, I, J1, D1, X1))
else ({Dist1, Next1} =
floyd_warshall_loop_j(N, K, I, J1,
Dist0, Next0)))))).

%%%-------------------------------------------------------------------

:- func path_string(version_array2d(int), int, int) = string.
path_string(Next, U, V) = S :-
if (arr_get(Next, U, V) = nil_vertex) then S = ""
else S = path_string_(Next, U, V, int_to_string(U)).

:- func path_string_(version_array2d(int), int, int, string) = string.
path_string_(Next, U, V, S0) = S :-
(if (U = V) then (S = S0)
else (U1 = arr_get(Next, U, V),
S1 = append(append(S0, " -> "), int_to_string(U1)),
path_string_(Next, U1, V, S1) = S)).

%%%-------------------------------------------------------------------

main(!IO) :-
Example_graph = [{1, -2.0, 3},
{3, 2.0, 4},
{4, -1.0, 2},
{2, 4.0, 1},
{2, 3.0, 3}],
{N, Dist, Next} = floyd_warshall(Example_graph),
format("  pair    distance   path\n", [], !IO),
format("-------------------------------------\n", [], !IO),
main_loop_u(N, 1, Dist, Next, !IO).

:- pred main_loop_u(int, int,
version_array2d(float),
version_array2d(int),
io, io).
:- mode main_loop_u(in, in, in, in, di, uo) is det.
main_loop_u(N, U, Dist, Next, !IO) :-
N1 = N + 1,
(if (U = N1) then true
else (main_loop_v(N, U, 1, Dist, Next, !IO),
U1 = U + 1,
main_loop_u(N, U1, Dist, Next, !IO))).

:- pred main_loop_v(int, int, int,
version_array2d(float),
version_array2d(int),
io, io).
:- mode main_loop_v(in, in, in, in, in, di, uo) is det.
main_loop_v(N, U, V, Dist, Next, !IO) :-
V1 = V + 1,
N1 = N + 1,
(if (V = N1) then true
else if (U = V) then main_loop_v(N, U, V1, Dist, Next, !IO)
else (format(" %d -> %d    %4.1f     %s\n",
[i(U), i(V), f(arr_get(Dist, U, V)),
s(path_string(Next, U, V))],
!IO),
main_loop_v(N, U, V1, Dist, Next, !IO))).

%%%-------------------------------------------------------------------
%%% local variables:
%%% mode: mercury
%%% prolog-indent-width: 2
%%% end:```
Output:
```\$ mmc floyd_warshall_task.m && ./floyd_warshall_task
pair    distance   path
-------------------------------------
1 -> 2    -1.0     1 -> 3 -> 4 -> 2
1 -> 3    -2.0     1 -> 3
1 -> 4     0.0     1 -> 3 -> 4
2 -> 1     4.0     2 -> 1
2 -> 3     2.0     2 -> 1 -> 3
2 -> 4     4.0     2 -> 1 -> 3 -> 4
3 -> 1     5.0     3 -> 4 -> 2 -> 1
3 -> 2     1.0     3 -> 4 -> 2
3 -> 4     2.0     3 -> 4
4 -> 1     3.0     4 -> 2 -> 1
4 -> 2    -1.0     4 -> 2
4 -> 3     1.0     4 -> 2 -> 1 -> 3```

Modula-2

```MODULE FloydWarshall;
FROM FormatString IMPORT FormatString;
FROM SpecialReals IMPORT Infinity;

CONST NUM_VERTICIES = 4;
TYPE
IntArray = ARRAY[0..NUM_VERTICIES-1],[0..NUM_VERTICIES-1] OF INTEGER;
RealArray = ARRAY[0..NUM_VERTICIES-1],[0..NUM_VERTICIES-1] OF REAL;

PROCEDURE FloydWarshall(weights : ARRAY OF ARRAY OF INTEGER);
VAR
dist : RealArray;
next : IntArray;
i,j,k : INTEGER;
BEGIN
FOR i:=0 TO NUM_VERTICIES-1 DO
FOR j:=0 TO NUM_VERTICIES-1 DO
dist[i,j] := Infinity;
END
END;
k := HIGH(weights);
FOR i:=0 TO k DO
dist[weights[i,0]-1,weights[i,1]-1] := FLOAT(weights[i,2]);
END;
FOR i:=0 TO NUM_VERTICIES-1 DO
FOR j:=0 TO NUM_VERTICIES-1 DO
IF i#j THEN
next[i,j] := j+1;
END
END
END;
FOR k:=0 TO NUM_VERTICIES-1 DO
FOR i:=0 TO NUM_VERTICIES-1 DO
FOR j:=0 TO NUM_VERTICIES-1 DO
IF dist[i,j] > dist[i,k] + dist[k,j] THEN
dist[i,j] := dist[i,k] + dist[k,j];
next[i,j] := next[i,k];
END
END
END
END;
PrintResult(dist, next);
END FloydWarshall;

PROCEDURE PrintResult(dist : RealArray; next : IntArray);
VAR
i,j,u,v : INTEGER;
buf : ARRAY[0..63] OF CHAR;
BEGIN
WriteString("pair     dist    path");
WriteLn;
FOR i:=0 TO NUM_VERTICIES-1 DO
FOR j:=0 TO NUM_VERTICIES-1 DO
IF i#j THEN
u := i + 1;
v := j + 1;
FormatString("%i -> %i    %2i     %i", buf, u, v, TRUNC(dist[i,j]), u);
WriteString(buf);
REPEAT
u := next[u-1,v-1];
FormatString(" -> %i", buf, u);
WriteString(buf);
UNTIL u=v;
WriteLn
END
END
END
END PrintResult;

TYPE WeightArray = ARRAY[0..4],[0..2] OF INTEGER;
VAR weights : WeightArray;
BEGIN
weights := WeightArray{
{1,  3, -2},
{2,  1,  4},
{2,  3,  3},
{3,  4,  2},
{4,  2, -1}
};

FloydWarshall(weights);

END FloydWarshall.
```

Nim

Translation of: D
```import sequtils, strformat

type
Weight = tuple[src, dest, value: int]
Weights = seq[Weight]

#---------------------------------------------------------------------------------------------------

proc printResult(dist: seq[seq[float]]; next: seq[seq[int]]) =

echo "pair     dist    path"
for i in 0..next.high:
for j in 0..next.high:
if i != j:
var u = i + 1
let v = j + 1
var path = fmt"{u} -> {v}    {dist[i][j].toInt:2d}     {u}"
while true:
u = next[u-1][v-1]
path &= fmt" -> {u}"
if u == v: break
echo path

#---------------------------------------------------------------------------------------------------

proc floydWarshall(weights: Weights; numVertices: Positive) =

var dist = repeat(repeat(Inf, numVertices), numVertices)
for w in weights:
dist[w.src - 1][w.dest - 1] = w.value.toFloat

var next = repeat(newSeq[int](numVertices), numVertices)
for i in 0..<numVertices:
for j in 0..<numVertices:
if i != j:
next[i][j] = j + 1

for k in 0..<numVertices:
for i in 0..<numVertices:
for j in 0..<numVertices:
if dist[i][j] > dist[i][k] + dist[k][j]:
dist[i][j] = dist[i][k] + dist[k][j]
next[i][j] = next[i][k]

printResult(dist, next)

#———————————————————————————————————————————————————————————————————————————————————————————————————

let weights: Weights = @[(1, 3, -2), (2, 1, 4), (2, 3, 3), (3, 4, 2), (4, 2, -1)]
let numVertices = 4

floydWarshall(weights, numVertices)
```
Output:
```pair     dist    path
1 -> 2    -1     1 -> 3 -> 4 -> 2
1 -> 3    -2     1 -> 3
1 -> 4     0     1 -> 3 -> 4
2 -> 1     4     2 -> 1
2 -> 3     2     2 -> 1 -> 3
2 -> 4     4     2 -> 1 -> 3 -> 4
3 -> 1     5     3 -> 4 -> 2 -> 1
3 -> 2     1     3 -> 4 -> 2
3 -> 4     2     3 -> 4
4 -> 1     3     4 -> 2 -> 1
4 -> 2    -1     4 -> 2
4 -> 3     1     4 -> 2 -> 1 -> 3```

ObjectIcon

Translation of: Icon

The only changes needed from the classical Icon were in library linkage and code order. (The record definition had to come after the library linkages.)

Certainly there are better ways to write an Object Icon implementation (for example, using a class instead of record), but this helps show that most of the classical dialect is still there.

```#
# Floyd-Warshall algorithm.
#
# See https://en.wikipedia.org/w/index.php?title=Floyd%E2%80%93Warshall_algorithm&oldid=1082310013
#

import io
import ipl.array
import ipl.printf

record fw_results (n, distance, next_vertex)

procedure main ()
local example_graph
local fw
local u, v

example_graph := [[1, -2.0, 3],
[3, +2.0, 4],
[4, -1.0, 2],
[2, +4.0, 1],
[2, +3.0, 3]]

fw := floyd_warshall (example_graph)

printf ("  pair    distance   path\n")
printf ("-------------------------------------\n")
every u := 1 to fw.n do {
every v := 1 to fw.n do {
if u ~= v then {
printf (" %d -> %d    %4s     %s\n", u, v,
string (ref_array (fw.distance, u, v)),
path_to_string (find_path (fw.next_vertex, u, v)))
}
}
}
end

procedure floyd_warshall (edges)
local n, distance, next_vertex
local e
local i, j, k
local dist_ij, dist_ik, dist_kj, dist_ikj

n := max_vertex (edges)
distance := create_array ([1, 1], [n, n], &null)
next_vertex := create_array ([1, 1], [n, n], &null)

# Initialization.
every e := !edges do {
ref_array (distance, e[1], e[3]) := e[2]
ref_array (next_vertex, e[1], e[3]) := e[3]
}
every i := 1 to n do {
ref_array (distance, i, i) := 0.0 # Distance to self = 0.
ref_array (next_vertex, i, i) := i
}

# Perform the algorithm. Here &null will play the role of
# "infinity": "\" means a value is finite, "/" that it is infinite.
every k := 1 to n do {
every i := 1 to n do {
every j := 1 to n do {
dist_ij := ref_array (distance, i, j)
dist_ik := ref_array (distance, i, k)
dist_kj := ref_array (distance, k, j)
if \dist_ik & \dist_kj then {
dist_ikj := dist_ik + dist_kj
if /dist_ij | dist_ikj < dist_ij then {
ref_array (distance, i, j) := dist_ikj
ref_array (next_vertex, i, j) :=
ref_array (next_vertex, i, k)
}
}
}
}
}

return fw_results (n, distance, next_vertex)
end

procedure find_path (next_vertex, u, v)
local path

if / (ref_array (next_vertex, u, v)) then {
path := []
} else {
path := [u]
while u ~= v do {
u := ref_array (next_vertex, u, v)
put (path, u)
}
}
return path
end

procedure path_to_string (path)
local s

if *path = 0 then {
s := ""
} else {
s := string (path[1])
every s ||:= (" -> " || !path[2 : 0])
}
return s
end

procedure max_vertex (edges)
local e
local m

*edges = 0 & stop ("no edges")
m := 1
every e := !edges do m := max (m, e[1], e[3])
return m
end```
Output:
```\$ oiscript floyd-warshall-in-OI.icn
pair    distance   path
-------------------------------------
1 -> 2    -1.0     1 -> 3 -> 4 -> 2
1 -> 3    -2.0     1 -> 3
1 -> 4     0.0     1 -> 3 -> 4
2 -> 1     4.0     2 -> 1
2 -> 3     2.0     2 -> 1 -> 3
2 -> 4     4.0     2 -> 1 -> 3 -> 4
3 -> 1     5.0     3 -> 4 -> 2 -> 1
3 -> 2     1.0     3 -> 4 -> 2
3 -> 4     2.0     3 -> 4
4 -> 1     3.0     4 -> 2 -> 1
4 -> 2    -1.0     4 -> 2
4 -> 3     1.0     4 -> 2 -> 1 -> 3```

OCaml

Translation of: ATS

This implementation was written by referring frequently to the ATS, but differs from it considerably. For example, it assumes IEEE floating point, whereas the ATS purposely avoided that assumption. However, the "square array" and "edge" types are very similar to the ATS equivalents.

```(*
Floyd-Warshall algorithm.

See https://en.wikipedia.org/w/index.php?title=Floyd%E2%80%93Warshall_algorithm&oldid=1082310013
*)

module Square_array =

(* Square arrays with 1-based indexing. *)

struct
type 'a t =
{
n : int;
r : 'a Array.t
}

let make n fill =
let r = Array.make (n * n) fill in
{ n = n; r = r }

let get arr (i, j) =
Array.get arr.r ((i - 1) + (arr.n * (j - 1)))

let set arr (i, j) x =
Array.set arr.r ((i - 1) + (arr.n * (j - 1))) x
end

module Vertex =

(* A vertex is a positive integer, or 0 for the nil object. *)

struct
type t = int

let nil = 0

let print_vertex u =
print_int u

let rec print_directed_list lst =
match lst with
| [] -> ()
| [u] -> print_vertex u
| u :: tail ->
begin
print_vertex u;
print_string " -> ";
print_directed_list tail
end
end

module Edge =

(* A graph edge. *)

struct
type t =
{
u : Vertex.t;
weight : Float.t;
v : Vertex.t
}

let make u weight v =
{ u = u; weight = weight; v = v }
end

module Paths =

(* The "next vertex" array and its operations. *)

struct
type t = Vertex.t Square_array.t

let make n =
Square_array.make n Vertex.nil

let get = Square_array.get
let set = Square_array.set

let path paths u v =
(* Path reconstruction. In the finest tradition of the standard
List module, this implementation is *not* tail recursive. *)
if Square_array.get paths (u, v) = Vertex.nil then
[]
else
let rec build_path paths u v =
if u = v then
[v]
else
let i = Square_array.get paths (u, v) in
u :: build_path paths i v
in
build_path paths u v

let print_path paths u v =
Vertex.print_directed_list (path paths u v)
end

module Distances =

(* The "distance" array and its operations. *)

struct
type t = Float.t Square_array.t

let make n =
Square_array.make n Float.infinity

let get = Square_array.get
let set = Square_array.set
end

let find_max_vertex edges =
(* This implementation is *not* tail recursive. *)
let rec find_max =
function
| [] -> Vertex.nil
| edge :: tail -> max (max Edge.(edge.u) Edge.(edge.v))
(find_max tail)
in
find_max edges

let floyd_warshall edges =
(* This implementation assumes IEEE floating point. The OCaml Float
module explicitly specifies 64-bit IEEE floating point. *)
let _ = assert (edges <> []) in
let n = find_max_vertex edges in
let dist = Distances.make n in
let next = Paths.make n in
function
| [] -> ()
| edge :: tail ->
let u = Edge.(edge.u) in
let v = Edge.(edge.v) in
let weight = Edge.(edge.weight) in
begin
Distances.set dist (u, v) weight;
Paths.set next (u, v) v;
end
in
begin

(* Initialization. *)

for i = 1 to n do
(* Distance from a vertex to itself = 0.0 *)
Distances.set dist (i, i) 0.0;
Paths.set next (i, i) i
done;

(* Perform the algorithm. *)

for k = 1 to n do
for i = 1 to n do
for j = 1 to n do
let dist_ij = Distances.get dist (i, j) in
let dist_ik = Distances.get dist (i, k) in
let dist_kj = Distances.get dist (k, j) in
let dist_ikj = dist_ik +. dist_kj in
if dist_ikj < dist_ij then
begin
Distances.set dist (i, j) dist_ikj;
Paths.set next (i, j) (Paths.get next (i, k))
end
done
done
done;

(* Return the results, as a 3-tuple. *)

(n, dist, next)

end

let example_graph =
[Edge.make 1 (-2.0) 3;
Edge.make 3 (+2.0) 4;
Edge.make 4 (-1.0) 2;
Edge.make 2 (+4.0) 1;
Edge.make 2 (+3.0) 3]
;;

let (n, dist, next) =
floyd_warshall example_graph
;;

print_string "  pair     distance    path";
print_newline ();
print_string "---------------------------------------";
print_newline ();
for u = 1 to n do
for v = 1 to n do
if u <> v then
begin
print_string " ";
Vertex.print_directed_list [u; v];
print_string "     ";
Printf.printf "%4.1f" (Distances.get dist (u, v));
print_string "      ";
Paths.print_path next u v;
print_newline ()
end
done
done
;;
```
Output:
```\$ ocamlopt floyd_warshall_task.ml && ./a.out
pair     distance    path
---------------------------------------
1 -> 2     -1.0      1 -> 3 -> 4 -> 2
1 -> 3     -2.0      1 -> 3
1 -> 4      0.0      1 -> 3 -> 4
2 -> 1      4.0      2 -> 1
2 -> 3      2.0      2 -> 1 -> 3
2 -> 4      4.0      2 -> 1 -> 3 -> 4
3 -> 1      5.0      3 -> 4 -> 2 -> 1
3 -> 2      1.0      3 -> 4 -> 2
3 -> 4      2.0      3 -> 4
4 -> 1      3.0      4 -> 2 -> 1
4 -> 2     -1.0      4 -> 2
4 -> 3      1.0      4 -> 2 -> 1 -> 3```

Perl

```sub FloydWarshall{
my \$edges = shift;
my (@dist, @seq);
my \$num_vert = 0;
# insert given dists into dist matrix
map {
\$dist[\$_->[0] - 1][\$_->[1] - 1] = \$_->[2];
\$num_vert = \$_->[0] if \$num_vert < \$_->[0];
\$num_vert = \$_->[1] if \$num_vert < \$_->[1];
} @\$edges;
my @vertices = 0..(\$num_vert - 1);
# init sequence/"next" table
for my \$i(@vertices){
for my \$j(@vertices){
\$seq[\$i][\$j] = \$j if \$i != \$j;
}
}
# diagonal of dists matrix
#map {\$dist[\$_][\$_] = 0} @vertices;
for my \$k(@vertices){
for my \$i(@vertices){
next unless defined \$dist[\$i][\$k];
for my \$j(@vertices){
next unless defined \$dist[\$k][\$j];
if(\$i != \$j && (!defined(\$dist[\$i][\$j])
|| \$dist[\$i][\$j] > \$dist[\$i][\$k] + \$dist[\$k][\$j])){
\$dist[\$i][\$j] = \$dist[\$i][\$k] + \$dist[\$k][\$j];
\$seq[\$i][\$j] = \$seq[\$i][\$k];
}
}
}
}
# print table
print "pair     dist    path\n";
for my \$i(@vertices){
for my \$j(@vertices){
next if \$i == \$j;
my @path = (\$i + 1);
while(\$seq[\$path[-1] - 1][\$j] != \$j){
push @path, \$seq[\$path[-1] - 1][\$j] + 1;
}
push @path, \$j + 1;
printf "%d -> %d  %4d     %s\n",
\$path[0], \$path[-1], \$dist[\$i][\$j], join(' -> ', @path);
}
}
}

my \$graph = [[1, 3, -2], [2, 1, 4], [2, 3, 3], [3, 4, 2], [4, 2, -1]];
FloydWarshall(\$graph);
```
Output:
```pair     dist    path
1 -> 2    -1     1 -> 3 -> 4 -> 2
1 -> 3    -2     1 -> 3
1 -> 4     0     1 -> 3 -> 4
2 -> 1     4     2 -> 1
2 -> 3     2     2 -> 1 -> 3
2 -> 4     4     2 -> 1 -> 3 -> 4
3 -> 1     5     3 -> 4 -> 2 -> 1
3 -> 2     1     3 -> 4 -> 2
3 -> 4     2     3 -> 4
4 -> 1     3     4 -> 2 -> 1
4 -> 2    -1     4 -> 2
4 -> 3     1     4 -> 2 -> 1 -> 3```

Phix

Direct translation of the wikipedia pseudocode

```constant inf = 1e300*1e300

function Path(integer u, integer v, sequence next)
if next[u,v]=null then
return ""
end if
sequence path = {sprintf("%d",u)}
while u!=v do
u = next[u,v]
path = append(path,sprintf("%d",u))
end while
return join(path,"->")
end function

procedure FloydWarshall(integer V, sequence weights)
sequence dist = repeat(repeat(inf,V),V)
sequence next = repeat(repeat(null,V),V)
for k=1 to length(weights) do
integer {u,v,w} = weights[k]
dist[u,v] := w  -- the weight of the edge (u,v)
next[u,v] := v
end for
-- standard Floyd-Warshall implementation
for k=1 to V do
for i=1 to V do
for j=1 to V do
atom d = dist[i,k] + dist[k,j]
if dist[i,j] > d then
dist[i,j] := d
next[i,j] := next[i,k]
end if
end for
end for
end for
printf(1,"pair  dist  path\n")
for u=1 to V do
for v=1 to V do
if u!=v then
printf(1,"%d->%d   %2d   %s\n",{u,v,dist[u,v],Path(u,v,next)})
end if
end for
end for
end procedure

constant V = 4
constant weights = {{1, 3, -2}, {2, 1, 4}, {2, 3, 3}, {3, 4, 2}, {4, 2, -1}}
FloydWarshall(V,weights)
```
Output:
```pair  dist  path
1->2   -1   1->3->4->2
1->3   -2   1->3
1->4    0   1->3->4
2->1    4   2->1
2->3    2   2->1->3
2->4    4   2->1->3->4
3->1    5   3->4->2->1
3->2    1   3->4->2
3->4    2   3->4
4->1    3   4->2->1
4->2   -1   4->2
4->3    1   4->2->1->3
```

PHP

```<?php
\$graph = array();
for (\$i = 0; \$i < 10; ++\$i) {
\$graph[] = array();
for (\$j = 0; \$j < 10; ++\$j)
\$graph[\$i][] = \$i == \$j ? 0 : 9999999;
}

for (\$i = 1; \$i < 10; ++\$i) {
\$graph[0][\$i] = \$graph[\$i][0] = rand(1, 9);
}

for (\$k = 0; \$k < 10; ++\$k) {
for (\$i = 0; \$i < 10; ++\$i) {
for (\$j = 0; \$j < 10; ++\$j) {
if (\$graph[\$i][\$j] > \$graph[\$i][\$k] + \$graph[\$k][\$j])
\$graph[\$i][\$j] = \$graph[\$i][\$k] + \$graph[\$k][\$j];
}
}
}

print_r(\$graph);
?>
```

Prolog

Works with SWI-Prolog as of Jan 2019

```:- use_module(library(clpfd)).

path(List, To, From, [From], W) :-
select([To,From,W],List,_).
path(List, To, From, [Link|R], W) :-
W #= W1 + W2,

find_path(Din, From, To, [From|Pout], Wout) :-
between(1, 4, From),
between(1, 4, To),
dif(From, To),
findall([W,P], (
path(Din, From, To, P, W),
all_distinct(P)
), Paths),
sort(Paths, [[Wout,Pout]|_]).

print_all_paths :-
D = [[1, 3, -2], [2, 3, 3], [2, 1, 4], [3, 4, 2], [4, 2, -1]],
format('Pair\t  Dist\tPath~n'),
forall(
find_path(D, From, To, Path, Weight),(
atomic_list_concat(Path, ' -> ', PPath),
format('~p -> ~p\t  ~p\t~w~n', [From, To, Weight, PPath]))).
```
Output:
```?- print_all_paths.
Pair      Dist  Path
1 -> 2    -1    1 -> 3 -> 4 -> 2
1 -> 3    -2    1 -> 3
1 -> 4    0     1 -> 3 -> 4
2 -> 1    4     2 -> 1
2 -> 3    2     2 -> 1 -> 3
2 -> 4    4     2 -> 1 -> 3 -> 4
3 -> 1    5     3 -> 4 -> 2 -> 1
3 -> 2    1     3 -> 4 -> 2
3 -> 4    2     3 -> 4
4 -> 1    3     4 -> 2 -> 1
4 -> 2    -1    4 -> 2
4 -> 3    1     4 -> 2 -> 1 -> 3
true.

?- ```

Python

Translation of: Ruby
```from math import inf
from itertools import product

def floyd_warshall(n, edge):
rn = range(n)
dist = [[inf] * n for i in rn]
nxt  = [[0]   * n for i in rn]
for i in rn:
dist[i][i] = 0
for u, v, w in edge:
dist[u-1][v-1] = w
nxt[u-1][v-1] = v-1
for k, i, j in product(rn, repeat=3):
sum_ik_kj = dist[i][k] + dist[k][j]
if dist[i][j] > sum_ik_kj:
dist[i][j] = sum_ik_kj
nxt[i][j]  = nxt[i][k]
print("pair     dist    path")
for i, j in product(rn, repeat=2):
if i != j:
path = [i]
while path[-1] != j:
path.append(nxt[path[-1]][j])
print("%d → %d  %4d       %s"
% (i + 1, j + 1, dist[i][j],
' → '.join(str(p + 1) for p in path)))

if __name__ == '__main__':
floyd_warshall(4, [[1, 3, -2], [2, 1, 4], [2, 3, 3], [3, 4, 2], [4, 2, -1]])
```
Output:
```pair     dist    path
1 → 2    -1       1 → 3 → 4 → 2
1 → 3    -2       1 → 3
1 → 4     0       1 → 3 → 4
2 → 1     4       2 → 1
2 → 3     2       2 → 1 → 3
2 → 4     4       2 → 1 → 3 → 4
3 → 1     5       3 → 4 → 2 → 1
3 → 2     1       3 → 4 → 2
3 → 4     2       3 → 4
4 → 1     3       4 → 2 → 1
4 → 2    -1       4 → 2
4 → 3     1       4 → 2 → 1 → 3```

Racket

Translation of: EchoLisp
```#lang typed/racket
(require math/array)

;; in : initialized dist and next matrices
;; out : dist and next matrices
;; O(n^3)
(define-type Next-T (Option Index))
(define-type Dist-T Real)
(define-type Dists (Array Dist-T))
(define-type Nexts (Array Next-T))
(define-type Settable-Dists (Settable-Array Dist-T))
(define-type Settable-Nexts (Settable-Array Next-T))

(: floyd-with-path (-> Index Dists Nexts (Values Dists Nexts)))
(: init-edges (-> Index (Values Settable-Dists Settable-Nexts)))

(define (floyd-with-path n dist-in next-in)
(define dist : Settable-Dists (array->mutable-array dist-in))
(define next : Settable-Nexts (array->mutable-array next-in))
(for* ((k n) (i n) (j n))
(when (negative? (array-ref dist (vector j j)))
(raise 'negative-cycle))
(define i.k (vector i k))
(define i.j (vector i j))
(define d (+ (array-ref dist i.k) (array-ref dist (vector k j))))
(when (< d (array-ref dist i.j))
(array-set! dist i.j d)
(array-set! next i.j (array-ref next i.k))))
(values dist next))

;; utilities

;; init random edges costs, matrix 66% filled
(define (init-edges n)
(define dist : Settable-Dists (array->mutable-array (make-array (vector n n) 0)))
(define next : Settable-Nexts (array->mutable-array (make-array (vector n n) #f)))
(for* ((i n) (j n) #:unless (= i j))
(define i.j (vector i j))
(array-set! dist i.j +Inf.0)
(unless (< (random) 0.3)
(array-set! dist i.j (add1 (random 100)))
(array-set! next i.j j)))
(values dist next))

;; show path from u to v
(: path (-> Nexts Index Index (Listof Index)))
(define (path next u v)
(let loop : (Listof Index) ((u : Index u) (rv : (Listof Index) null))
(if (= u v)
(reverse (cons u rv))
(let ((nxt (array-ref next (vector u v))))
(if nxt (loop nxt (cons u rv)) null)))))

;; show computed distance
(: mdist (-> Dists Index Index Dist-T))
(define (mdist dist u v)
(array-ref dist (vector u v)))

(module+ main
(define n 8)
(define-values (dist next) (init-edges n))
(define-values (dist+ next+) (floyd-with-path n dist next))
(displayln "original dist")
dist
(displayln "new dist and next")
dist+
next+
;; note, these path and dist calls are not as carefully crafted as
;; the echolisp ones (in fact they're verbatim copied)
(displayln "paths and distances")
(path  next+ 1 3)
(mdist dist+ 1 0)
(mdist dist+ 0 3)
(mdist dist+ 1 3)
(path next+ 7 6)
(path next+ 6 7))
```
Output:
```original dist
(mutable-array
#[#[0 51 +inf.0 11 44 13 +inf.0 86]
#[48 0 70 +inf.0 65 78 77 54]
#[29 +inf.0 0 +inf.0 78 14 +inf.0 24]
#[40 79 52 0 +inf.0 99 37 88]
#[71 62 +inf.0 7 0 +inf.0 +inf.0 +inf.0]
#[89 65 83 +inf.0 91 0 41 70]
#[69 34 +inf.0 49 +inf.0 89 0 20]
#[2 56 +inf.0 60 +inf.0 75 +inf.0 0]])
new dist and next
(mutable-array
#[#[0 51 63 11 44 13 48 68]
#[48 0 70 59 65 61 77 54]
#[26 77 0 37 70 14 55 24]
#[40 71 52 0 84 53 37 57]
#[47 62 59 7 0 60 44 64]
#[63 65 83 74 91 0 41 61]
#[22 34 85 33 66 35 0 20]
#[2 53 65 13 46 15 50 0]])
(mutable-array
#[#[#f 1 3 3 4 5 3 3]
#[0 #f 2 0 4 0 6 7]
#[7 7 #f 7 7 5 5 7]
#[0 6 2 #f 0 0 6 6]
#[3 1 3 3 #f 3 3 3]
#[6 1 2 6 4 #f 6 6]
#[7 1 7 7 7 7 #f 7]
#[0 0 0 0 0 0 0 #f]])
paths and distances
'(1 0 3)
48
11
59
'(7 0 3 6)
'(6 7)```

Raku

(formerly Perl 6)

Works with: Rakudo version 2016.12
Translation of: Ruby
```sub Floyd-Warshall (Int \$n, @edge) {
my @dist = [0, |(Inf xx \$n-1)], *.Array.rotate(-1) … !*[*-1];
my @next = [0 xx \$n] xx \$n;

for @edge -> (\$u, \$v, \$w) {
@dist[\$u-1;\$v-1] = \$w;
@next[\$u-1;\$v-1] = \$v-1;
}

for [X] ^\$n xx 3 -> (\$k, \$i, \$j) {
if @dist[\$i;\$j] > my \$sum = @dist[\$i;\$k] + @dist[\$k;\$j] {
@dist[\$i;\$j] = \$sum;
@next[\$i;\$j] = @next[\$i;\$k];
}
}

say ' Pair  Distance     Path';
for [X] ^\$n xx 2 -> (\$i, \$j){
next if \$i == \$j;
my @path = \$i;
@path.push: @next[@path[*-1];\$j] until @path[*-1] == \$j;
printf("%d → %d  %4d       %s\n", \$i+1, \$j+1, @dist[\$i;\$j],
@path.map( *+1 ).join(' → '));
}
}

Floyd-Warshall(4, [[1, 3, -2], [2, 1, 4], [2, 3, 3], [3, 4, 2], [4, 2, -1]]);
```
Output:
``` Pair  Distance     Path
1 → 2    -1       1 → 3 → 4 → 2
1 → 3    -2       1 → 3
1 → 4     0       1 → 3 → 4
2 → 1     4       2 → 1
2 → 3     2       2 → 1 → 3
2 → 4     4       2 → 1 → 3 → 4
3 → 1     5       3 → 4 → 2 → 1
3 → 2     1       3 → 4 → 2
3 → 4     2       3 → 4
4 → 1     3       4 → 2 → 1
4 → 2    -1       4 → 2
4 → 3     1       4 → 2 → 1 → 3
```

RATFOR

Translation of: Fortran
Works with: ratfor77 version public domain 1.0
Works with: gfortran version 11.3.0
Works with: f2c version 20100827

```#
# Floyd-Warshall algorithm.
#
# See https://en.wikipedia.org/w/index.php?title=Floyd%E2%80%93Warshall_algorithm&oldid=1082310013
#

#
# A C programmer might take note that the most rapid stride in an
# array is on the *leftmost* index, rather than the *rightmost* as in
# C.
#
# (In other words, Fortran has "column-major order", whereas C has
# "row-major order". I prefer to think of it in terms of strides. For
# one thing, in my opinion, which index is for a "column" and which
# for a "row" should be considered arbitrary unless dictated by
# context.)
#

# VLIMIT = the maximum number of vertices the program can handle.
define(VLIMIT, 100)

# NILVTX = the nil vertex.
define(NILVTX, 0)

# STRSZ = a buffer size used in some character-handling routines.
define(STRSZ, 300)

# BUFSZ = a buffer size used in some character-handling routines.
define(BUFSZ, 20)

function maxvtx (numedg, edges)

# Find the maximum vertex number.

implicit none

integer numedg
real edges(1:3, 1:numedg)     # Notice Fortran's column-major order!
integer maxvtx

integer n, i

n = 1
for (i = 1; i <= numedg; i = i + 1)
{
n = max (n, int (edges(1, i)))
n = max (n, int (edges(3, i)))
}
maxvtx = n
end

subroutine floyd (numedg, edges, n, dist, nxtvtx)

# Floyd-Warshall.

implicit none

integer numedg
real edges(1:3, 1:numedg)     # Notice Fortran's column-major order!
integer n
real dist(1:VLIMIT, 1:VLIMIT)
integer nxtvtx(1:VLIMIT, 1:VLIMIT)

#
# This implementation does NOT initialize elements of "dist" that
# would be set "infinite" in the original Fortran 90. Such elements
# are left uninitialized. Instead you should use the "nxtvtx" array
# to determine whether there exists a finite path from one vertex to
# another.
#
# See also the Icon and Object Icon implementations that use "&null"
# as a stand-in for "infinity". This implementation is similar to
# those. In this Ratfor, the nil entry in "nxtvtx" is used instead
# of one in "dist".
#

integer i, j, k
integer u, v
real dstikj

# Initialization.

for (i = 1; i <= n; i = i + 1)
for (j = 1; j <= n; j = j + 1)
nxtvtx(i, j) = NILVTX
for (i = 1; i <= numedg; i = i + 1)
{
u = int (edges(1, i))
v = int (edges(3, i))
dist(u, v) = edges(2, i)
nxtvtx(u, v) = v
}
for (i = 1; i <= n; i = i + 1)
{
dist(i, i) = 0.0          # Distance from a vertex to itself.
nxtvtx(i, i) = i
}

# Perform the algorithm.

for (k = 1; k <= n; k = k + 1)
for (i = 1; i <= n; i = i + 1)
for (j = 1; j <= n; j = j + 1)
if (nxtvtx(i, k) != NILVTX && nxtvtx(k, j) != NILVTX)
{
dstikj = dist(i, k) + dist(k, j)
if (nxtvtx(i, j) == NILVTX)
{
dist(i, j) = dstikj
nxtvtx(i, j) = nxtvtx(i, k)
}
else if (dstikj < dist(i, j))
{
dist(i, j) = dstikj
nxtvtx(i, j) = nxtvtx(i, k)
}
}
end

subroutine cpy (chr, str, j)

# A helper subroutine for pthstr.

implicit none

character*BUFSZ chr
character str*STRSZ
integer j

integer i

i = 1
while (chr(i:i) == ' ')
{
if (i == BUFSZ)
{
write (*, *) "character* boundary exceeded in cpy"
stop
}
i = i + 1
}
while (i <= BUFSZ)
{
if (STRSZ < j)
{
write (*, *) "character* boundary exceeded in cpy"
stop
}
str(j:j) = chr(i:i)
j = j + 1
i = i + 1
}
end

subroutine pthstr (nxtvtx, u, v, str, k)

# Construct a string for a path from u to v. Start at str(k).

implicit none

integer nxtvtx(1:VLIMIT, 1:VLIMIT)
integer u, v
character str*STRSZ
integer k

integer i, j
character*BUFSZ chr
character*25 fmt10
character*25 fmt20

write (fmt10, '(''(I'', I15, '')'')') BUFSZ - 1
write (fmt20, '(''(A'', I15, '')'')') BUFSZ

if (nxtvtx(u, v) != NILVTX)
{
j = k
i = u
chr = ' '
write (chr, fmt10) i
call cpy (chr, str, j)
while (i != v)
{
write (chr, fmt20) "-> "
call cpy (chr, str, j)
i = nxtvtx(i, v)
write (chr, fmt10) i
call cpy (chr, str, j)
}
}
end

function trimr (str)

# Find the length of a character*, if one ignores trailing spaces.

implicit none

character str*STRSZ
integer trimr

logical done

trimr = STRSZ
done = .false.
while (!done)
{
if (trimr == 0)
done = .true.
else if (str(trimr:trimr) != ' ')
done = .true.
else
trimr = trimr - 1
}
end

program demo
implicit none

integer maxvtx
integer trimr

integer exmpsz
real exampl(1:3, 1:5)
integer n
real dist(1:VLIMIT, 1:VLIMIT)
integer nxtvtx(1:VLIMIT, 1:VLIMIT)
character str*STRSZ
integer u, v
integer j

exmpsz = 5
data exampl / 1, -2.0, 3,   _
3, +2.0, 4,   _
4, -1.0, 2,   _
2, +4.0, 1,   _
2, +3.0, 3 /

n = maxvtx (exmpsz, exampl)
call floyd (exmpsz, exampl, n, dist, nxtvtx)

1000 format (I2, ' ->', I2, 5X, F4.1, 6X)

write (*, '(''  pair     distance    path'')')
write (*, '(''---------------------------------------'')')
for (u = 1; u <= n; u = u + 1)
for (v = 1; v <= n; v = v + 1)
if (u != v)
{
str = ' '
write (str, 1000) u, v, dist(u, v)
call pthstr (nxtvtx, u, v, str, 23)
write (* , '(1000A1)') (str(j:j), j = 1, trimr (str))
}
end```
Output:

I get slightly different output, depending on whether I use gfortran or f2c to compile the generated FORTRAN code. The two outputs differ in how 0.0 is printed.

First gfortran:

```\$ ratfor77 -6x floyd_warshall_task.r > floyd_warshall_task.f && gfortran -std=legacy floyd_warshall_task.f && ./a.out
pair     distance    path
---------------------------------------
1 -> 2     -1.0      1 -> 3 -> 4 -> 2
1 -> 3     -2.0      1 -> 3
1 -> 4      0.0      1 -> 3 -> 4
2 -> 1      4.0      2 -> 1
2 -> 3      2.0      2 -> 1 -> 3
2 -> 4      4.0      2 -> 1 -> 3 -> 4
3 -> 1      5.0      3 -> 4 -> 2 -> 1
3 -> 2      1.0      3 -> 4 -> 2
3 -> 4      2.0      3 -> 4
4 -> 1      3.0      4 -> 2 -> 1
4 -> 2     -1.0      4 -> 2
4 -> 3      1.0      4 -> 2 -> 1 -> 3```

Now f2c:

```\$ ratfor77 -6x floyd_warshall_task.r > floyd_warshall_task.f && f2c floyd_warshall_task.f && cc floyd_warshall_task.c -lf2c && ./a.out
maxvtx:
floyd:
cpy:
pthstr:
trimr:
MAIN demo:
pair     distance    path
---------------------------------------
1 -> 2     -1.0      1 -> 3 -> 4 -> 2
1 -> 3     -2.0      1 -> 3
1 -> 4       .0      1 -> 3 -> 4
2 -> 1      4.0      2 -> 1
2 -> 3      2.0      2 -> 1 -> 3
2 -> 4      4.0      2 -> 1 -> 3 -> 4
3 -> 1      5.0      3 -> 4 -> 2 -> 1
3 -> 2      1.0      3 -> 4 -> 2
3 -> 4      2.0      3 -> 4
4 -> 1      3.0      4 -> 2 -> 1
4 -> 2     -1.0      4 -> 2
4 -> 3      1.0      4 -> 2 -> 1 -> 3```

REXX

```/*REXX program uses Floyd─Warshall algorithm to find shortest distance between vertices.*/
v= 4             /*███       {1}       ███*/     /*number of vertices in weighted graph.*/
@.= 99999999     /*███    4 /   \ -2   ███*/     /*the default distance  (edge weight). */
@.1.3= -2        /*███     /  3  \     ███*/     /*the distance (weight) for an edge.   */
@.2.1=  4        /*███  {2} ────► {3}  ███*/     /* "     "         "     "   "   "     */
@.2.3=  3        /*███     \     /     ███*/     /* "     "         "     "   "   "     */
@.3.4=  2        /*███   -1 \   / 2    ███*/     /* "     "         "     "   "   "     */
@.4.2= -1        /*███       {4}       ███*/     /* "     "         "     "   "   "     */

do     k=1  for v
do   i=1  for v
do j=1  for v;  _= @.i.k + @.k.j /*add two nodes together.              */
if @.i.j>_  then @.i.j= _        /*use a new distance (weight) for edge.*/
end   /*j*/
end     /*i*/
end       /*k*/
w= 12;                     \$= left('', 20)       /*width of the columns for the output. */
say \$ center('vertices',w) center('distance', w) /*display the  1st  line of the title. */
say \$ center('pair'    ,w) center('(weight)', w) /*   "     "   2nd    "   "  "    "    */
say \$ copies('═'       ,w) copies('═'       , w) /*   "     "   3rd    "   "  "    "    */
/* [↓]  display edge distances (weight)*/
do   f=1  for v                               /*process each of the "from" vertices. */
do t=1  for v;    if f==t  then iterate     /*   "      "   "  "   "to"      "     */
say  \$      center(f '───►' t, w)        right(@.f.t, w % 2)
end   /*t*/                                 /* [↑]  the distance between 2 vertices*/
end     /*f*/                                 /*stick a fork in it,  we're all done. */
```
output   when using the default inputs:
```                       vertices     distance
pair       (weight)
════════════ ════════════
1 ───► 2       -1
1 ───► 3       -2
1 ───► 4        0
2 ───► 1        4
2 ───► 3        2
2 ───► 4        4
3 ───► 1        5
3 ───► 2        1
3 ───► 4        2
4 ───► 1        3
4 ───► 2       -1
4 ───► 3        1
```

Ruby

```def floyd_warshall(n, edge)
dist = Array.new(n){|i| Array.new(n){|j| i==j ? 0 : Float::INFINITY}}
nxt = Array.new(n){Array.new(n)}
edge.each do |u,v,w|
dist[u-1][v-1] = w
nxt[u-1][v-1] = v-1
end

n.times do |k|
n.times do |i|
n.times do |j|
if dist[i][j] > dist[i][k] + dist[k][j]
dist[i][j] = dist[i][k] + dist[k][j]
nxt[i][j] = nxt[i][k]
end
end
end
end

puts "pair     dist    path"
n.times do |i|
n.times do |j|
next  if i==j
u = i
path = [u]
path << (u = nxt[u][j])  while u != j
path = path.map{|u| u+1}.join(" -> ")
puts "%d -> %d  %4d     %s" % [i+1, j+1, dist[i][j], path]
end
end
end

n = 4
edge = [[1, 3, -2], [2, 1, 4], [2, 3, 3], [3, 4, 2], [4, 2, -1]]
floyd_warshall(n, edge)
```
Output:
```pair     dist    path
1 -> 2    -1     1 -> 3 -> 4 -> 2
1 -> 3    -2     1 -> 3
1 -> 4     0     1 -> 3 -> 4
2 -> 1     4     2 -> 1
2 -> 3     2     2 -> 1 -> 3
2 -> 4     4     2 -> 1 -> 3 -> 4
3 -> 1     5     3 -> 4 -> 2 -> 1
3 -> 2     1     3 -> 4 -> 2
3 -> 4     2     3 -> 4
4 -> 1     3     4 -> 2 -> 1
4 -> 2    -1     4 -> 2
4 -> 3     1     4 -> 2 -> 1 -> 3
```

Rust

The lack of built-in support for multi-dimensional arrays makes the task in Rust a bit lengthy (without additional crates). The used graph representation leverages Rust's generics, so that it works with any type that defines addition and ordering and it requires no special value for infinity.

```pub type Edge = (usize, usize);

#[derive(Clone, Debug, PartialEq, Eq, Hash)]
pub struct Graph<T> {
size: usize,
edges: Vec<Option<T>>,
}

impl<T> Graph<T> {
pub fn new(size: usize) -> Self {
Self {
size,
edges: std::iter::repeat_with(|| None).take(size * size).collect(),
}
}

pub fn new_with(size: usize, f: impl FnMut(Edge) -> Option<T>) -> Self {
let edges = (0..size)
.flat_map(|i| (0..size).map(move |j| (i, j)))
.map(f)
.collect();

Self { size, edges }
}

pub fn with_diagonal(mut self, mut f: impl FnMut(usize) -> Option<T>) -> Self {
self.edges
.iter_mut()
.step_by(self.size + 1)
.enumerate()
.for_each(move |(vertex, edge)| *edge = f(vertex));

self
}

pub fn size(&self) -> usize {
self.size
}

pub fn edge(&self, edge: Edge) -> &Option<T> {
let index = self.edge_index(edge);
&self.edges[index]
}

pub fn edge_mut(&mut self, edge: Edge) -> &mut Option<T> {
let index = self.edge_index(edge);
&mut self.edges[index]
}

fn edge_index(&self, (row, col): Edge) -> usize {
assert!(row < self.size && col < self.size);
row * self.size() + col
}
}

impl<T> std::ops::Index<Edge> for Graph<T> {
type Output = Option<T>;

fn index(&self, index: Edge) -> &Self::Output {
self.edge(index)
}
}

impl<T> std::ops::IndexMut<Edge> for Graph<T> {
fn index_mut(&mut self, index: Edge) -> &mut Self::Output {
self.edge_mut(index)
}
}

#[derive(Clone, Debug, PartialEq, Eq)]
pub struct Paths(Graph<usize>);

impl Paths {
pub fn new<T>(graph: &Graph<T>) -> Self {
Self(Graph::new_with(graph.size(), |(i, j)| {
graph[(i, j)].as_ref().map(|_| j)
}))
}

pub fn vertices(&self, from: usize, to: usize) -> Path<'_> {
assert!(from < self.0.size() && to < self.0.size());

Path {
graph: &self.0,
from: Some(from),
to,
}
}

fn update(&mut self, from: usize, to: usize, via: usize) {
self.0[(from, to)] = self.0[(from, via)];
}
}

#[derive(Clone, Copy, Debug, PartialEq, Eq)]
pub struct Path<'a> {
graph: &'a Graph<usize>,
from: Option<usize>,
to: usize,
}

impl<'a> Iterator for Path<'a> {
type Item = usize;

fn next(&mut self) -> Option<Self::Item> {
self.from.map(|from| {
let result = from;

self.from = if result != self.to {
self.graph[(result, self.to)]
} else {
None
};

result
})
}
}

pub fn floyd_warshall<W>(mut result: Graph<W>) -> (Graph<W>, Option<Paths>)
where
W: Copy + std::ops::Add<W, Output = W> + std::cmp::Ord + Default,
{
let mut without_negative_cycles = true;
let mut paths = Paths::new(&result);
let n = result.size();

for k in 0..n {
for i in 0..n {
for j in 0..n {
// Negative cycle detection with T::default as the negative boundary
if i == j && result[(i, j)].filter(|&it| it < W::default()).is_some() {
without_negative_cycles = false;
continue;
}

if let (Some(ik_weight), Some(kj_weight)) = (result[(i, k)], result[(k, j)]) {
let ij_edge = result.edge_mut((i, j));
let ij_weight = ik_weight + kj_weight;

if ij_edge.is_none() {
*ij_edge = Some(ij_weight);
paths.update(i, j, k);
} else {
ij_edge
.as_mut()
.filter(|it| ij_weight < **it)
.map_or((), |it| {
*it = ij_weight;
paths.update(i, j, k);
});
}
}
}
}
}

(result, Some(paths).filter(|_| without_negative_cycles)) // No paths for negative cycles
}

fn format_path<T: ToString>(path: impl Iterator<Item = T>) -> String {
path.fold(String::new(), |mut acc, x| {
if !acc.is_empty() {
acc.push_str(" -> ");
}

acc.push_str(&x.to_string());
acc
})
}

fn print_results<W, V>(weights: &Graph<W>, paths: Option<&Paths>, vertex: impl Fn(usize) -> V)
where
W: std::fmt::Display + Default + Eq,
V: std::fmt::Display,
{
let n = weights.size();

for from in 0..n {
for to in 0..n {
if let Some(weight) = &weights[(from, to)] {
// Skip trivial information (i.e., default weight on the diagonal)
if from == to && *weight == W::default() {
continue;
}

println!(
"{} -> {}: {} \t{}",
vertex(from),
vertex(to),
weight,
format_path(paths.iter().flat_map(|p| p.vertices(from, to)).map(&vertex))
);
}
}
}
}

fn main() {
let graph = {
let mut g = Graph::new(4).with_diagonal(|_| Some(0));
g[(0, 2)] = Some(-2);
g[(1, 0)] = Some(4);
g[(1, 2)] = Some(3);
g[(2, 3)] = Some(2);
g[(3, 1)] = Some(-1);
g
};

let (weights, paths) = floyd_warshall(graph);
// Fixup the vertex name (as we use zero-based indices)
print_results(&weights, paths.as_ref(), |index| index + 1);
}
```
Output:
```1 -> 2: -1      1 -> 3 -> 4 -> 2
1 -> 3: -2      1 -> 3
1 -> 4: 0       1 -> 3 -> 4
2 -> 1: 4       2 -> 1
2 -> 3: 2       2 -> 1 -> 3
2 -> 4: 4       2 -> 1 -> 3 -> 4
3 -> 1: 5       3 -> 4 -> 2 -> 1
3 -> 2: 1       3 -> 4 -> 2
3 -> 4: 2       3 -> 4
4 -> 1: 3       4 -> 2 -> 1
4 -> 2: -1      4 -> 2
4 -> 3: 1       4 -> 2 -> 1 -> 3
```

Scala

Translation of: Java
```import java.lang.String.format;

object FloydWarshall extends App {

val weights = Array(Array(1, 3, -2), Array(2, 1, 4), Array(2, 3, 3), Array(3, 4, 2), Array(4, 2, -1))
val numVertices = 4

floydWarshall(weights, numVertices)

def floydWarshall(weights: Array[Array[Int]], numVertices: Int): Unit = {

val dist = Array.fill(numVertices, numVertices)(Double.PositiveInfinity)
for (w <- weights)
dist(w(0) - 1)(w(1) - 1) = w(2)

val next = Array.ofDim[Int](numVertices, numVertices)
for (i <- 0 until numVertices; j <- 0 until numVertices if i != j)
next(i)(j) = j + 1

for {
k <- 0 until numVertices
i <- 0 until numVertices
j <- 0 until numVertices
if dist(i)(k) + dist(k)(j) < dist(i)(j)
} {
dist(i)(j) = dist(i)(k) + dist(k)(j)
next(i)(j) = next(i)(k)
}

printResult(dist, next)
}

def printResult(dist: Array[Array[Double]], next: Array[Array[Int]]): Unit = {
println("pair     dist    path")
for {
i <- 0 until next.length
j <- 0 until next.length if i != j
} {
var u = i + 1
val v = j + 1
var path = format("%d -> %d    %2d     %s", u, v,
(dist(i)(j)).toInt, u);
while (u != v) {
u = next(u - 1)(v - 1)
path += s" -> \$u"
}
println(path)
}
}
}
```
Output:
```pair     dist    path
1 -> 2    -1     1 -> 3 -> 4 -> 2
1 -> 3    -2     1 -> 3
1 -> 4     0     1 -> 3 -> 4
2 -> 1     4     2 -> 1
2 -> 3     2     2 -> 1 -> 3
2 -> 4     4     2 -> 1 -> 3 -> 4
3 -> 1     5     3 -> 4 -> 2 -> 1
3 -> 2     1     3 -> 4 -> 2
3 -> 4     2     3 -> 4
4 -> 1     3     4 -> 2 -> 1
4 -> 2    -1     4 -> 2
4 -> 3     1     4 -> 2 -> 1 -> 3

```

Scheme

Works with: Scheme version R7RS small

I have run this program successfully in Chibi, Gauche, and CHICKEN 5 Schemes. (One may need an extension to run R7RS code in CHICKEN.)

```;;; Floyd-Warshall algorithm.
;;;
;;; See https://en.wikipedia.org/w/index.php?title=Floyd%E2%80%93Warshall_algorithm&oldid=1082310013
;;;

(import (scheme base))
(import (scheme cxr))
(import (scheme write))

;;;
;;; A square array will be represented by a cons-pair:
;;;
;;;    (vector-of-length n-squared . n)
;;;
;;; Arrays are indexed *starting at one*.
;;;

(define (make-arr n fill)
(cons (make-vector (* n n) fill) n))

(define (arr-set! arr i j x)
(let ((vec (car arr))
(n (cdr arr)))
(vector-set! vec (+ (- i 1) (* n (- j 1))) x)))

(define (arr-ref arr i j)
(let ((vec (car arr))
(n (cdr arr)))
(vector-ref vec (+ (- i 1) (* n (- j 1))))))

;;;
;;; Floyd-Warshall.
;;;
;;; Input is a list of length-3 lists representing edges; each entry
;;; is:
;;;
;;;    (start-vertex edge-weight end-vertex)
;;;
;;; where vertex identifiers are (to help keep this example brief)
;;; integers from 1 .. n.
;;;

(define (floyd-warshall edges)

(define n
;; Set n to the maximum vertex number. By design, n also equals
;; the number of vertices.
(max (apply max (map car edges))

(define distance (make-arr n +inf.0))
(define next-vertex (make-arr n #f))

;; Initialize "distance" and "next-vertex".
(for-each (lambda (edge)
(let ((u (car edge))
(arr-set! distance u v weight)
(arr-set! next-vertex u v v)))
edges)
(do ((v 1 (+ v 1)))
((< n v))
(arr-set! distance v v 0)
(arr-set! next-vertex v v v))

;; Perform the algorithm.
(do ((k 1 (+ k 1)))
((< n k))
(do ((i 1 (+ i 1)))
((< n i))
(do ((j 1 (+ j 1)))
((< n j))
(let ((dist-ij (arr-ref distance i j))
(dist-ik (arr-ref distance i k))
(dist-kj (arr-ref distance k j)))
(let ((dist-ik+dist-kj (+ dist-ik dist-kj)))
(when (< dist-ik+dist-kj dist-ij)
(arr-set! distance i j dist-ik+dist-kj)
(arr-set! next-vertex i j
(arr-ref next-vertex i k))))))))

;; Return the results.
(values n distance next-vertex))

;;;
;;; Path reconstruction from the "next-vertex" array.
;;;
;;; The return value is a list of vertices.
;;;

(define (find-path next-vertex u v)
(if (not (arr-ref next-vertex u v))
(list)
(let loop ((u u)
(path (list u)))
(if (= u v)
(reverse path)
(let ((u^ (arr-ref next-vertex u v)))
(loop u^ (cons u^ path)))))))

(define (display-path path)
(let loop ((p path))
(cond ((null? p))
((null? (cdr p)) (display (car p)))
(else (display (car p))
(display " -> ")
(loop (cdr p))))))

(define example-graph
'((1 -2 3)
(3 2 4)
(4 -1 2)
(2 4 1)
(2 3 3)))

(let-values (((n distance next-vertex)
(floyd-warshall example-graph)))
(display " pair   distance    path")
(newline)
(display "------------------------------------")
(newline)
(do ((u 1 (+ u 1)))
((< n u))
(do ((v 1 (+ v 1)))
((< n v))
(unless (= u v)
(display u)
(display " -> ")
(display v)
(let* ((s (number->string (arr-ref distance u v)))
(slen (string-length s))
(display s))
(display "      ")
(display-path (find-path next-vertex u v))
(newline)))))
```
Output:
```\$ gosh floyd-warshall.scm
pair   distance    path
------------------------------------
1 -> 2     -1      1 -> 3 -> 4 -> 2
1 -> 3     -2      1 -> 3
1 -> 4      0      1 -> 3 -> 4
2 -> 1      4      2 -> 1
2 -> 3      2      2 -> 1 -> 3
2 -> 4      4      2 -> 1 -> 3 -> 4
3 -> 1      5      3 -> 4 -> 2 -> 1
3 -> 2      1      3 -> 4 -> 2
3 -> 4      2      3 -> 4
4 -> 1      3      4 -> 2 -> 1
4 -> 2     -1      4 -> 2
4 -> 3      1      4 -> 2 -> 1 -> 3```

SequenceL

Translation of: Go
```import <Utilities/Sequence.sl>;
import <Utilities/Math.sl>;

ARC ::= (To: int, Weight: float);
arc(t,w) := (To: t, Weight: w);
VERTEX ::= (Label: int, Arcs: ARC(1));
vertex(l,arcs(1)) := (Label: l, Arcs: arcs);

getArcsFrom(vertex, graph(1)) :=
let
index := firstIndexOf(graph.Label, vertex);
in
[] when index = 0
else
graph[index].Arcs;

getWeightTo(vertex, arcs(1)) :=
let
index := firstIndexOf(arcs.To, vertex);
in
0 when index = 0
else
arcs[index].Weight;

throughK(k, dist(2)) :=
let
newDist[i, j] := min(dist[i][k] + dist[k][j], dist[i][j]);
in
dist when k > size(dist)
else
throughK(k + 1, newDist);

floydWarshall(graph(1)) :=
let
initialResult[i,j] := 1.79769e308 when i /= j else 0
foreach i within 1 ... size(graph),
j within 1 ... size(graph);

singleResult[i,j] := getWeightTo(j, getArcsFrom(i, graph))
foreach i within 1 ... size(graph),
j within 1 ... size(graph);

start[i,j] :=
initialResult[i,j] when singleResult[i,j] = 0
else
singleResult[i,j];
in
throughK(1, start);

main() :=
let
graph := [vertex(1, [arc(3,-2)]),
vertex(2, [arc(1,4), arc(3,3)]),
vertex(3, [arc(4,2)]),
vertex(4, [arc(2,-1)])];
in
floydWarshall(graph);```
Output:
```[[0,-1,-2,0],[4,0,2,4],[5,1,0,2],[3,-1,1,0]]
```

Sidef

Translation of: Ruby
```func floyd_warshall(n, edge) {
var dist = n.of {|i| n.of { |j| i == j ? 0 : Inf }}
var nxt  = n.of { n.of(nil) }
for u,v,w in edge {
dist[u-1][v-1] = w
nxt[u-1][v-1] = v-1
}

[^n] * 3 -> cartesian { |k, i, j|
if (dist[i][j] > dist[i][k]+dist[k][j]) {
dist[i][j] = dist[i][k]+dist[k][j]
nxt[i][j] = nxt[i][k]
}
}

var summary = "pair     dist    path\n"
for i,j (^n ~X ^n) {
i==j && next
var u = i
var path = [u]
while (u != j) {
path << (u = nxt[u][j])
}
path.map!{|u| u+1 }.join!(" -> ")
summary += ("%d -> %d  %4d     %s\n" % (i+1, j+1, dist[i][j], path))
}

return summary
}

var n = 4
var edge = [[1, 3, -2], [2, 1, 4], [2, 3, 3], [3, 4, 2], [4, 2, -1]]
print floyd_warshall(n, edge)
```
Output:
```pair     dist    path
1 -> 2    -1     1 -> 3 -> 4 -> 2
1 -> 3    -2     1 -> 3
1 -> 4     0     1 -> 3 -> 4
2 -> 1     4     2 -> 1
2 -> 3     2     2 -> 1 -> 3
2 -> 4     4     2 -> 1 -> 3 -> 4
3 -> 1     5     3 -> 4 -> 2 -> 1
3 -> 2     1     3 -> 4 -> 2
3 -> 4     2     3 -> 4
4 -> 1     3     4 -> 2 -> 1
4 -> 2    -1     4 -> 2
4 -> 3     1     4 -> 2 -> 1 -> 3
```

Standard ML

Translation of: OCaml
Works with: MLton version 20210117
Works with: Poly/ML version 5.9

You have to comment out the call to main () if you are using Poly/ML. The code as is works with MLton.

(Poly/ML is a separate compiler that, by itself, looks for a main function to start the program at.)

```(*
Floyd-Warshall algorithm.

See https://en.wikipedia.org/w/index.php?title=Floyd%E2%80%93Warshall_algorithm&oldid=1082310013
*)

(*------------------------------------------------------------------(*

In this program, I introduce more "abstraction" than there was in
earlier versions, which were written in the SML-like languages
OCaml and ATS. This is an example of proceeding from where one has
gotten so far, to turn a program into a better one. The
improvements made here could be backported to the other languages.

In most respects, though, this program is very similar to the
OCaml.

Standard ML seems to specify its REAL signature is for IEEE
floating point, so this program assumes there is a positive
"infinity". (The difference is tiny between an algorithm with
"infinity" and one without.)

*)------------------------------------------------------------------*)

(* Square arrays with 1-based indexing. *)

signature SQUARE_ARRAY =
sig
type 'a squareArray
val make : int * 'a -> 'a squareArray
val get : 'a squareArray -> int * int -> 'a
val set : 'a squareArray -> int * int -> 'a -> unit
end

structure SquareArray : SQUARE_ARRAY =
struct

type 'a squareArray = int * 'a array

fun make (n, fill) =
(n, Array.array (n * n, fill))

fun get (n, r) (i, j) =
Array.sub (r, (i - 1) + (n * (j - 1)))

fun set (n, r) (i, j) x =
Array.update (r, (i - 1) + (n * (j - 1)), x)

end

(*------------------------------------------------------------------*)

(* A vertex is, internally, a positive integer, or 0 for the nil
object. *)

signature VERTEX =
sig
exception VertexError
eqtype vertex
val nilVertex : vertex
val isNil : vertex -> bool
val max : vertex * vertex -> vertex
val toInt : vertex -> int
val fromInt : int -> vertex
val toString : vertex -> string
val directedListToString : vertex list -> string
end

structure Vertex : VERTEX =
struct

exception VertexError

type vertex = int

val nilVertex = 0

fun isNil u = u = nilVertex
fun max (u, v) = Int.max (u, v)
fun toInt u = u

fun fromInt i =
if i < nilVertex then
raise VertexError
else
i

fun toString u = Int.toString u

fun directedListToString [] = ""
| directedListToString [u] = toString u
| directedListToString (u :: tail) =
(* This implementation is *not* tail recursive. *)
(toString u) ^ " -> " ^ (directedListToString tail)

end

(*------------------------------------------------------------------*)

(* Graph edges, with weights. *)

signature EDGE =
sig
type edge
val make : Vertex.vertex * real * Vertex.vertex -> edge
val first : edge -> Vertex.vertex
val weight : edge -> real
val second : edge -> Vertex.vertex
end

structure Edge : EDGE =
struct

type edge = Vertex.vertex * real * Vertex.vertex

fun make edge = edge
fun first (u, _, _) = u
fun weight (_, w, _) = w
fun second (_, _, v) = v

end

(*------------------------------------------------------------------*)

(* The "dist" array and its operations. *)

signature DISTANCES =
sig
type distances
val make : int -> distances
val get : distances -> int * int -> real
val set : distances -> int * int -> real -> unit
end

structure Distances : DISTANCES =
struct

type distances = real SquareArray.squareArray

fun make n = SquareArray.make (n, Real.posInf)
val get = SquareArray.get
val set = SquareArray.set

end

(*------------------------------------------------------------------*)

(* The "next" array and its operations. It lets you look up optimum
paths. *)

signature PATHS =
sig
type paths
val make : int -> paths
val get : paths -> int * int -> Vertex.vertex
val set : paths -> int * int -> Vertex.vertex -> unit
val path : (paths * int * int) -> Vertex.vertex list
val pathString : (paths * int * int) -> string
end

structure Paths : PATHS =
struct

type paths = Vertex.vertex SquareArray.squareArray

fun make n = SquareArray.make (n, Vertex.nilVertex)
val get = SquareArray.get
val set = SquareArray.set

fun path (p, u, v) =
if Vertex.isNil (get p (u, v)) then
[]
else
let
fun
build_path (p, u, v) =
if u = v then
[v]
else
let
val i = get p (u, v)
in
u :: build_path (p, i, v)
end
in
build_path (p, u, v)
end

fun pathString (p, u, v) =
Vertex.directedListToString (path (p, u, v))

end

(*------------------------------------------------------------------*)

(* Floyd-Warshall. *)

exception FloydWarshallError

fun find_max_vertex [] = Vertex.nilVertex
| find_max_vertex (edge :: tail) =
(* This implementation is *not* tail recursive. *)
Vertex.max (Vertex.max (Edge.first edge, Edge.second edge),
find_max_vertex tail)

fun floyd_warshall [] = raise FloydWarshallError
| floyd_warshall edges =
let
val n = find_max_vertex edges
val dist = Distances.make n
val next = Paths.make n

| read_edges (edge :: tail) =
let
val u = Edge.first edge
val v = Edge.second edge
val weight = Edge.weight edge
in
(Distances.set dist (u, v) weight;
Paths.set next (u, v) v;
end

val indices =
(* Indices in order from 1 .. n. *)
List.tabulate (n, fn i => i + 1)
in

(* Initialization. *)

List.app (fn i => (Distances.set dist (i, i) 0.0;
Paths.set next (i, i) i))
indices;

(* Perform the algorithm. *)

List.app
(fn k =>
List.app
(fn i =>
List.app
(fn j =>
let
val dist_ij = Distances.get dist (i, j)
val dist_ik = Distances.get dist (i, k)
val dist_kj = Distances.get dist (k, j)
val dist_ikj = dist_ik + dist_kj
in
if dist_ikj < dist_ij then
let
val new_dist = dist_ikj
val new_next = Paths.get next (i, k)
in
Distances.set dist (i, j) new_dist;
Paths.set next (i, j) new_next
end
else
()
end)
indices)
indices)
indices;

(* Return the results, as a 3-tuple. *)

(n, dist, next)

end

(*------------------------------------------------------------------*)

fun tilde_to_minus s =
String.translate (fn c => if c = #"~" then "-" else str c) s

fun main () =
let
val example_graph =
[Edge.make (Vertex.fromInt 1, ~2.0, Vertex.fromInt 3),
Edge.make (Vertex.fromInt 3, 2.0, Vertex.fromInt 4),
Edge.make (Vertex.fromInt 4, ~1.0, Vertex.fromInt 2),
Edge.make (Vertex.fromInt 2, 4.0, Vertex.fromInt 1),
Edge.make (Vertex.fromInt 2, 3.0, Vertex.fromInt 3)]

val (n, dist, next) = floyd_warshall example_graph

val indices =
(* Indices in order from 1 .. n. *)
List.tabulate (n, fn i => i + 1)
in
print "  pair     distance    path\n";
print "---------------------------------------\n";
List.app
(fn u =>
List.app
(fn v =>
if u <> v then
(print " ";
print (Vertex.directedListToString [u, v]);
print "     ";
if 0.0 <= Distances.get dist (u, v) then
print " "
else
();
print (tilde_to_minus
(Real.fmt (StringCvt.FIX (SOME 1))
(Distances.get dist (u, v))));
print "      ";
print (Paths.pathString (next, u, v));
print "\n")
else
())
indices)
indices
end;

(* Comment out the following line, if you are using Poly/ML. *)
main ();

(*------------------------------------------------------------------*)
(* local variables: *)
(* mode: sml *)
(* sml-indent-level: 2 *)
(* sml-indent-args: 2 *)
(* end: *)
```
Output:
```\$ mlton floyd_warshall_task.sml && ./floyd_warshall_task
pair     distance    path
---------------------------------------
1 -> 2     -1.0      1 -> 3 -> 4 -> 2
1 -> 3     -2.0      1 -> 3
1 -> 4      0.0      1 -> 3 -> 4
2 -> 1      4.0      2 -> 1
2 -> 3      2.0      2 -> 1 -> 3
2 -> 4      4.0      2 -> 1 -> 3 -> 4
3 -> 1      5.0      3 -> 4 -> 2 -> 1
3 -> 2      1.0      3 -> 4 -> 2
3 -> 4      2.0      3 -> 4
4 -> 1      3.0      4 -> 2 -> 1
4 -> 2     -1.0      4 -> 2
4 -> 3      1.0      4 -> 2 -> 1 -> 3```

Tcl

Library: Tcllib (Package: struct::graph::op)

The implementation of Floyd-Warshall in tcllib is quite readable; this example merely initialises a graph from an adjacency list then calls the tcllib code:

```package require Tcl 8.5     ;# for {*} and [dict]
package require struct::graph
package require struct::graph::op

struct::graph g

set arclist {
a b
a p
b m
b c
c d
d e
e f
f q
f g
}

g node insert {*}\$arclist

foreach {from to} \$arclist {
set a [g arc insert \$from \$to]
g arc setweight \$a 1.0
}

set paths [::struct::graph::op::FloydWarshall g]

set paths [dict filter \$paths key {a *}]        ;# filter for paths starting at "a"
set paths [dict filter \$paths value {[0-9]*}]   ;# whose cost is not "Inf"
set paths [lsort -stride 2 -index 1 -real -decreasing \$paths]   ;# and print the longest first
puts \$paths
```
Output:
`{a q} 6.0 {a g} 6.0 {a f} 5.0 {a e} 4.0 {a d} 3.0 {a m} 2.0 {a c} 2.0 {a p} 1.0 {a b} 1.0 {a a} 0`

Visual Basic .NET

Translation of: C#
```Module Module1

Sub PrintResult(dist As Double(,), nxt As Integer(,))
Console.WriteLine("pair     dist    path")
For i = 1 To nxt.GetLength(0)
For j = 1 To nxt.GetLength(1)
If i <> j Then
Dim u = i
Dim v = j
Dim path = String.Format("{0} -> {1}    {2,2:G}     {3}", u, v, dist(i - 1, j - 1), u)
Do
u = nxt(u - 1, v - 1)
path += String.Format(" -> {0}", u)
Loop While u <> v
Console.WriteLine(path)
End If
Next
Next
End Sub

Sub FloydWarshall(weights As Integer(,), numVerticies As Integer)
Dim dist(numVerticies - 1, numVerticies - 1) As Double
For i = 1 To numVerticies
For j = 1 To numVerticies
dist(i - 1, j - 1) = Double.PositiveInfinity
Next
Next

For i = 1 To weights.GetLength(0)
dist(weights(i - 1, 0) - 1, weights(i - 1, 1) - 1) = weights(i - 1, 2)
Next

Dim nxt(numVerticies - 1, numVerticies - 1) As Integer
For i = 1 To numVerticies
For j = 1 To numVerticies
If i <> j Then
nxt(i - 1, j - 1) = j
End If
Next
Next

For k = 1 To numVerticies
For i = 1 To numVerticies
For j = 1 To numVerticies
If dist(i - 1, k - 1) + dist(k - 1, j - 1) < dist(i - 1, j - 1) Then
dist(i - 1, j - 1) = dist(i - 1, k - 1) + dist(k - 1, j - 1)
nxt(i - 1, j - 1) = nxt(i - 1, k - 1)
End If
Next
Next
Next

PrintResult(dist, nxt)
End Sub

Sub Main()
Dim weights = {{1, 3, -2}, {2, 1, 4}, {2, 3, 3}, {3, 4, 2}, {4, 2, -1}}
Dim numVeritices = 4

FloydWarshall(weights, numVeritices)
End Sub

End Module
```
Output:
```pair     dist    path
1 -> 2    -1     1 -> 3 -> 4 -> 2
1 -> 3    -2     1 -> 3
1 -> 4     0     1 -> 3 -> 4
2 -> 1     4     2 -> 1
2 -> 3     2     2 -> 1 -> 3
2 -> 4     4     2 -> 1 -> 3 -> 4
3 -> 1     5     3 -> 4 -> 2 -> 1
3 -> 2     1     3 -> 4 -> 2
3 -> 4     2     3 -> 4
4 -> 1     3     4 -> 2 -> 1
4 -> 2    -1     4 -> 2
4 -> 3     1     4 -> 2 -> 1 -> 3```

Wren

Translation of: Kotlin
Library: Wren-fmt
```import "./fmt" for Fmt

class FloydWarshall {
static doCalcs(weights, nVertices) {
var dist = List.filled(nVertices, null)
for (i in 0...nVertices) dist[i] = List.filled(nVertices, 1/0)
for (w in weights) dist[w[0] - 1][w[1] - 1] = w[2]
var next = List.filled(nVertices, null)
for (i in 0...nVertices) next[i] = List.filled(nVertices, 0)
for (i in 0...next.count) {
for (j in 0...next.count) {
if (i != j) next[i][j] = j + 1
}
}
for (k in 0...nVertices) {
for (i in 0...nVertices) {
for (j in 0...nVertices) {
if (dist[i][k] + dist[k][j] < dist[i][j]) {
dist[i][j] = dist[i][k] + dist[k][j]
next[i][j] = next[i][k]
}
}
}
}
printResult_(dist, next)
}

static printResult_(dist,  next) {
System.print("pair     dist    path")
for (i in 0...next.count) {
for (j in 0...next.count) {
if (i != j) {
var u = i + 1
var v = j + 1
var path = Fmt.swrite("\$d -> \$d    \$2d     \$s", u, v, dist[i][j].truncate, u)
while (true) {
u = next[u - 1][v - 1]
path = path +  " -> " + u.toString
if (u == v) break
}
System.print(path)
}
}
}
}
}

var weights = [ [1, 3, -2], [2, 1, 4], [2, 3, 3], [3, 4, 2], [4, 2, -1] ]
var nVertices = 4
FloydWarshall.doCalcs(weights, nVertices)
```
Output:
```pair     dist    path
1 -> 2    -1     1 -> 3 -> 4 -> 2
1 -> 3    -2     1 -> 3
1 -> 4     0     1 -> 3 -> 4
2 -> 1     4     2 -> 1
2 -> 3     2     2 -> 1 -> 3
2 -> 4     4     2 -> 1 -> 3 -> 4
3 -> 1     5     3 -> 4 -> 2 -> 1
3 -> 2     1     3 -> 4 -> 2
3 -> 4     2     3 -> 4
4 -> 1     3     4 -> 2 -> 1
4 -> 2    -1     4 -> 2
4 -> 3     1     4 -> 2 -> 1 -> 3
```

zkl

```fcn FloydWarshallWithPathReconstruction(dist){ // dist is munged
V:=dist[0].len();
next:=V.pump(List,V.pump(List,Void.copy).copy);  // VxV matrix of Void
foreach u,v in (V,V){ if(dist[u][v]!=Void and u!=v) next[u][v] = v }
foreach k,i,j in (V,V,V){
a,b,c:=dist[i][j],dist[i][k],dist[k][j];
if( (a!=Void and b!=Void and c!=Void and a>b+c) or  // Inf math
(a==Void and b!=Void and c!=Void) ){
dist[i][j] = b+c;
next[i][j] = next[i][k];
}
}
return(dist,next)
}
fcn path(next,u,v){
if(Void==next[u][v]) return(T);
path:=List(u);
while(u!=v){ path.append(u = next[u][v]) }
path
}
fcn printM(m){ m.pump(Console.println,rowFmt) }
fcn rowFmt(row){ ("%5s "*row.len()).fmt(row.xplode()) }```
```const V=4;
dist:=V.pump(List,V.pump(List,Void.copy).copy);  // VxV matrix of Void
foreach i in (V){ dist[i][i] = 0 }     // zero vertexes

/* Graph from the Wikipedia:
1  2  3  4
d ----------
1| 0  X -2  X
2| 4  0  3  X
3| X  X  0  2
4| X -1  X  0
*/
dist[0][2]=-2; dist[1][0]=4; dist[1][2]=3; dist[2][3]=2; dist[3][1]=-1;

dist,next:=FloydWarshallWithPathReconstruction(dist);
println("Shortest distance array:"); printM(dist);
println("\nPath array:");        printM(next);
println("\nAll paths:");
foreach u,v in (V,V){
if(p:=path(next,u,v)) p.println();
}```
Output:
```Shortest distance array:
0    -1    -2     0
4     0     2     4
5     1     0     2
3    -1     1     0

Path array:
Void     2     2     2
0  Void     0     0
3     3  Void     3
1     1     1  Void

All paths:
L(0,2,3,1)
L(0,2)
L(0,2,3)
L(1,0)
L(1,0,2)
L(1,0,2,3)
L(2,3,1,0)
L(2,3,1)
L(2,3)
L(3,1,0)
L(3,1)
L(3,1,0,2)
```