Floyd-Warshall algorithm

From Rosetta Code
Task
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.

Task

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.

Floyd warshall graph.gif

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



See also



EchoLisp[edit]

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))
 
(define (task)
(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))
(task)

  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[edit]

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

FreeBASIC[edit]

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[edit]

package main
 
import (
"fmt"
"math"
)
 
type arc struct {
to int
wt float64
}
 
func fw(g [][]arc) [][]float64 {
dist := make([][]float64, len(g))
for i := range dist {
di := make([]float64, len(g))
for j := range di {
di[j] = math.Inf(1)
}
di[i] = 0
dist[i] = di
}
for u, arcs := range g {
for _, v := range arcs {
dist[u][v.to] = v.wt
}
}
for k, dk := range dist {
for _, di := range dist {
for j, dij := range di {
if d := di[k] + dk[j]; dij > d {
di[j] = d
}
}
}
}
return dist
}
 
func main() {
g := [][]arc{
1: {{3, -2}},
2: {{1, 4}, {3, 3}},
3: {{4, 2}},
4: {{2, -1}},
}
dist := fw(g)
for _, d := range dist {
fmt.Printf("%4g\n", d)
}
}
Output:
[   0 +Inf +Inf +Inf +Inf]
[+Inf    0   -1   -2    0]
[+Inf    4    0    2    4]
[+Inf    5    1    0    2]
[+Inf    3   -1    1    0]

Haskell[edit]

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"]})

J[edit]

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

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=. ((|i.@,~)#y)*1>.y->./(,y)-._
for_j. i.#y do.
d=. y <. j ({"1 +/ {) y
b=. y~:d
y=. d
n=. (n*-.b)+b * j{"1 n
end.
)
 
task=: verb define
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[edit]

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[edit]

var graph = [];
for (i = 0; i < 10; ++i) {
graph.push([]);
for (j = 0; j < 10; ++j)
graph[i].push(i == j ? 0 : 9999999);
}
 
for (i = 1; i < 10; ++i) {
graph[0][i] = graph[i][0] = parseInt(Math.random() * 9 + 1);
}
 
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]
}
}
}
 
console.log(graph);

Kotlin[edit]

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

Perl 6[edit]

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

Phix[edit]

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[edit]

<?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);
?>

Racket[edit]

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)

REXX[edit]

/*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
if @.i.j>_ then @.i.j=_ /*use a new distance (weight) for edge.*/
end /*j*/
end /*i*/
end /*k*/
w=12 /*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) /*show the distance between 2 vertices.*/
end /*t*/
end /*f*/ /*stick a fork in it, we're all done. */

output   when using the defaults:

  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[edit]

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

SequenceL[edit]

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[edit]

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

Tcl[edit]

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

zkl[edit]

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)