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This means it might contain formatting issues, incorrect code, conceptual problems, or other severe issues.

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{{draft task}} Your latest contract has hit a snag. You came to update the army payroll system, but awoke this morning to the sound of mortars landing not far away and panicked generals banging on you door. The President has loaded his gold on trucks and needs to find the shortest route to safety. You are given the following map. The top left hand corner is (0,0). You and The President are located at HQ in the centre of the country (11,11). Cells marked 0 indicate safety. Numbers other than 0 indicate the number of cells that his party will travel in a day in any direction up, down, left, right, or diagonally.

```
00000
00003130000
000321322221000
00231222432132200
0041433223233211100
0232231612142618530
003152122326114121200
031252235216111132210
022211246332311115210
00113232262121317213200
03152118212313211411110
03231234121132221411410
03513213411311414112320
00427534125412213211400
013322444412122123210
015132331312411123120
003333612214233913300
0219126511415312570
0021321524341325100
00211415413523200
000122111322000
00001120000
00000

```

Part 1 Use Dijkstra's algorithm to find a list of the shortest routes from HQ to safety.

Part 2

Six days later and you are called to another briefing. The good news is The President and his gold are safe, so your invoice may be paid if you can get out of here. To do this a number of troop repositions will be required. It is concluded that you need to know the shortest route from each cell to every other cell. You decide to use Floyd's algorithm. Print the shortest route from (21,11) to (1,11) and from (1,11) to (21,11), and the longest shortest route between any two points.

Extra Credit

# [[Floyd-Warshall algorithm]]

### Part 1

```
let safety=readCSV '\t' "gmooh.dat"|>Seq.choose(fun n->if n.value="0" then Some (n.row,n.col) else None)
let board=readCSV '\t' "gmooh.dat"|>Seq.choose(fun n->match n.value with |"0"|"1"|"2"|"3"|"4"|"5"|"6"|"7"|"8"|"9" as g->Some((n.row,n.col),int g)|_->None)|>Map.ofSeq
let adjacent((n,g),v)=List.choose(fun y->if y=(n,g) then None else match Map.tryFind y board with |None->None|_->Some ((y),1)) [(n+v,g);(n-v,g);(n,g+v);(n,g-v);(n+v,g+v);(n+v,g-v);(n-v,g+v);(n-v,g-v);]
let n=((start),Map.find start board)
let G=nodes |>List.collect(fun n->List.map(fun (n',g)->(((n),(n')),g))(adjacencyList.Item n))|>Map.ofList
let paths=Dijkstra nodes G (11,11)
let _,res=safety|>Seq.choose(fun n->paths n) |> Seq.groupBy(fun n->List.length n)|>Seq.minBy fst
res |> Seq.iter (printfn "%A")

```

{{out}}

```
[(11, 11); (10, 11); (7, 11); (6, 12); (0, 12)]
[(11, 11); (10, 11); (7, 11); (7, 12); (1, 6)]
[(11, 11); (10, 10); (8, 8); (4, 8); (1, 8)]
[(11, 11); (11, 12); (8, 9); (2, 9); (1, 9)]
[(11, 11); (10, 10); (8, 10); (5, 13); (1, 13)]
[(11, 11); (10, 11); (7, 8); (4, 11); (1, 14)]
[(11, 11); (11, 12); (8, 9); (2, 15); (1, 15)]
[(11, 11); (11, 12); (8, 9); (2, 15); (1, 16)]
[(11, 11); (10, 10); (8, 10); (5, 7); (2, 4)]
[(11, 11); (10, 11); (7, 8); (7, 5); (2, 5)]
[(11, 11); (11, 12); (8, 15); (9, 16); (2, 16)]
[(11, 11); (12, 10); (11, 9); (9, 9); (3, 3)]
[(11, 11); (10, 11); (7, 8); (4, 5); (3, 4)]
[(11, 11); (12, 11); (12, 14); (8, 18); (3, 18)]
[(11, 11); (12, 11); (9, 14); (6, 17); (4, 19)]
[(11, 11); (11, 12); (8, 9); (8, 3); (6, 1)]
[(11, 11); (12, 11); (12, 8); (8, 4); (6, 2)]
[(11, 11); (11, 12); (11, 15); (11, 17); (7, 21)]
[(11, 11); (11, 12); (8, 9); (8, 3); (8, 1)]
[(11, 11); (12, 11); (12, 8); (12, 4); (9, 1)]
[(11, 11); (11, 12); (8, 9); (14, 3); (11, 0)]
[(11, 11); (10, 11); (7, 8); (7, 5); (12, 0)]
[(11, 11); (12, 10); (13, 10); (13, 5); (13, 0)]
[(11, 11); (12, 11); (12, 8); (16, 4); (13, 1)]
[(11, 11); (12, 11); (12, 14); (16, 18); (13, 21)]
[(11, 11); (12, 11); (12, 8); (12, 4); (15, 1)]
[(11, 11); (11, 12); (11, 15); (11, 17); (15, 21)]
[(11, 11); (12, 11); (12, 8); (16, 4); (16, 1)]
[(11, 11); (10, 11); (10, 14); (12, 16); (16, 20)]
[(11, 11); (12, 11); (12, 14); (16, 18); (16, 21)]
[(11, 11); (12, 11); (15, 8); (15, 5); (18, 2)]
[(11, 11); (10, 11); (13, 8); (14, 7); (18, 3)]
[(11, 11); (12, 11); (15, 8); (18, 5); (19, 4)]
[(11, 11); (11, 12); (14, 15); (16, 15); (19, 18)]
[(11, 11); (12, 11); (15, 11); (16, 12); (20, 16)]
[(11, 11); (10, 11); (13, 11); (17, 15); (20, 18)]
[(11, 11); (12, 10); (13, 10); (18, 15); (21, 15)]
[(11, 11); (11, 12); (14, 9); (18, 13); (22, 9)]
[(11, 11); (12, 11); (15, 8); (18, 11); (22, 11)]
[(11, 11); (11, 12); (14, 9); (18, 13); (22, 13)]

```

### Part 2

```
let board=readCSV '\t' "gmooh.dat"|>Seq.choose(fun n->match n.value with |"0"|"1"|"2"|"3"|"4"|"5"|"6"|"7"|"8"|"9" as g->Some((n.row,n.col),int g)|_->None)|>Map.ofSeq
let nodes=board|>Seq.map(fun n->n.Key)|>Set.ofSeq
let adjacent (n,g) v=List.choose(fun y->if y=(n,g) then None else match Set.contains y nodes with |true->Some ((y),1)|_->None) [(n+v,g);(n-v,g);(n,g+v);(n,g-v);(n+v,g+v);(n+v,g-v);(n-v,g+v);(n-v,g-v);]
paths (21,11) (1,11) |>Seq.iteri(fun n g->if n>0 then printf "->"; printf "%A" g else printf "%A" g) ; printfn ""
paths (1,11) (21,11) |>Seq.iteri(fun n g->if n>0 then printf "->"; printf "%A" g else printf "%A" g) ; printfn ""

```

{{out}}

```
(20, 10)->(19, 9)->(18, 9)->(13, 4)->(6, 11)->(4, 11)->(1, 11)
(2, 10)->(5, 13)->(9, 9)->(15, 3)->(20, 8)->(20, 10)->(21, 11)

```

## Go

{{trans|Phix}}

A more or less faithful translation though adjusted to Go's 0-based indices and the cell coordinates are therefore 1 less than the Phix results.

Initially, using a simple breadth-first search. Parts 1 and 2 and extra credit.

```package main

import (
"fmt"
"strings"
)

var gmooh = strings.Split(
`.........00000.........
......00003130000......
....000321322221000....
...00231222432132200...
..0041433223233211100..
..0232231612142618530..
.003152122326114121200.
.031252235216111132210.
.022211246332311115210.
00113232262121317213200
03152118212313211411110
03231234121132221411410
03513213411311414112320
00427534125412213211400
.013322444412122123210.
.015132331312411123120.
.003333612214233913300.
..0219126511415312570..
..0021321524341325100..
...00211415413523200...
....000122111322000....
......00001120000......
.........00000.........`, "\n")

var width, height = len(gmooh), len(gmooh)

type pyx int // {y, x}

var d = []pyx{{-1, -1}, {0, -1}, {1, -1}, {-1, 0}, {1, 0}, {-1, 1}, {0, 1}, {1, 1}}

type route int // {cost, fromy, fromx}

var zeroRoute = route{0, 0, 0}
var routes [][]route // route for each gmooh[][]

func (p pyx) destruct() (int, int) {
return p, p
}

func (r route) destruct() (int, int, int) {
return r, r, r
}

func search(y, x int) {
// Simple breadth-first search, populates routes.
// This isn't strictly Dijkstra because graph edges are not weighted.
cost := 0
routes = make([][]route, height)
for i := 0; i < width; i++ {
routes[i] = make([]route, width)
}
routes[y][x] = route{0, y, x} // zero-cost, the starting point
var next []route
for {
n := int(gmooh[y][x] - '0')
for di := 0; di < len(d); di++ {
dx, dy := d[di].destruct()
rx, ry := x+n*dx, y+n*dy
if rx >= 0 && rx < width && ry >= 0 && ry < height && gmooh[rx][ry] >= '0' {
ryx := routes[ry][rx]
if ryx == zeroRoute || ryx > cost+1 {
routes[ry][rx] = route{cost + 1, y, x}
if gmooh[ry][rx] > '0' {
next = append(next, route{cost + 1, ry, rx})
// If the graph was weighted, at this point
// that would get shuffled up into place.
}
}
}
}
if len(next) == 0 {
break
}
cost, y, x = next.destruct()
next = next[1:]
}
}

func getRoute(y, x int) []pyx {
cost := 0
res := []pyx{{y, x}}
for {
cost, y, x = routes[y][x].destruct()
if cost == 0 {
break
}
res = append(res, pyx{0, 0})
copy(res[1:], res[0:])
res = pyx{y, x}
}
return res
}

func showShortest() {
shortest := 9999
var res []pyx
for x := 0; x < width; x++ {
for y := 0; y < height; y++ {
if gmooh[y][x] == '0' {
ryx := routes[y][x]
if ryx != zeroRoute {
cost := ryx
if cost <= shortest {
if cost < shortest {
res = res[:0]
shortest = cost
}
res = append(res, pyx{y, x})
}
}
}
}
}
areis, s := "is", ""
if len(res) > 1 {
areis = "are"
s = "s"
}
fmt.Printf("There %s %d shortest route%s of %d days to safety:\n", areis, len(res), s, shortest)
for _, r := range res {
fmt.Println(getRoute(r, r))
}
}

func showUnreachable() {
var res []pyx
for x := 0; x < width; x++ {
for y := 0; y < height; y++ {
if gmooh[y][x] >= '0' && routes[y][x] == zeroRoute {
res = append(res, pyx{y, x})
}
}
}
fmt.Println("\nThe following cells are unreachable:")
fmt.Println(res)
}

func showLongest() {
longest := 0
var res []pyx
for x := 0; x < width; x++ {
for y := 0; y < height; y++ {
if gmooh[y][x] >= '0' {
ryx := routes[y][x]
if ryx != zeroRoute {
rl := ryx
if rl >= longest {
if rl > longest {
res = res[:0]
longest = rl
}
res = append(res, pyx{y, x})
}
}
}
}
}
fmt.Printf("\nThere are %d cells that take %d days to send reinforcements to:\n", len(res), longest)
for _, r := range res {
fmt.Println(getRoute(r, r))
}
}

func main() {
search(11, 11)
showShortest()

search(21, 11)
fmt.Println("\nThe shortest route from {21,11} to {1,11}:")
fmt.Println(getRoute(1, 11))

search(1, 11)
fmt.Println("\nThe shortest route from {1,11} to {21,11}:")
fmt.Println(getRoute(21, 11))

search(11, 11)
showUnreachable()
showLongest()
}
```

{{out}}

```
There are 40 shortest routes of 4 days to safety:
[[11 11] [11 12] [8 9] [14 3] [11 0]]
[[11 11] [10 11] [7 8] [7 5] [12 0]]
[[11 11] [12 10] [13 10] [13 5] [13 0]]
[[11 11] [11 12] [8 9] [8 3] [6 1]]
[[11 11] [11 12] [8 9] [8 3] [8 1]]
[[11 11] [10 10] [8 8] [12 4] [9 1]]
[[11 11] [10 10] [12 8] [16 4] [13 1]]
[[11 11] [10 10] [8 8] [12 4] [15 1]]
[[11 11] [10 10] [12 8] [16 4] [16 1]]
[[11 11] [10 10] [8 8] [8 4] [6 2]]
[[11 11] [12 11] [15 8] [15 5] [18 2]]
[[11 11] [11 10] [10 9] [9 9] [3 3]]
[[11 11] [10 11] [13 8] [14 7] [18 3]]
[[11 11] [10 10] [8 10] [5 7] [2 4]]
[[11 11] [10 11] [7 8] [4 5] [3 4]]
[[11 11] [10 10] [12 8] [16 4] [19 4]]
[[11 11] [10 11] [7 8] [7 5] [2 5]]
[[11 11] [10 11] [7 11] [7 12] [1 6]]
[[11 11] [10 10] [8 8] [4 8] [1 8]]
[[11 11] [10 10] [8 10] [5 13] [1 9]]
[[11 11] [11 12] [14 9] [18 13] [22 9]]
[[11 11] [12 11] [15 8] [18 11] [22 11]]
[[11 11] [10 10] [8 12] [6 12] [0 12]]
[[11 11] [10 10] [8 10] [5 13] [1 13]]
[[11 11] [11 12] [14 9] [18 13] [22 13]]
[[11 11] [10 11] [7 8] [4 11] [1 14]]
[[11 11] [11 12] [8 9] [2 15] [1 15]]
[[11 11] [12 10] [13 10] [18 15] [21 15]]
[[11 11] [11 12] [8 9] [2 15] [1 16]]
[[11 11] [11 12] [8 9] [2 15] [2 16]]
[[11 11] [10 10] [12 8] [16 12] [20 16]]
[[11 11] [12 11] [12 14] [8 18] [3 18]]
[[11 11] [11 12] [14 15] [16 15] [19 18]]
[[11 11] [10 11] [13 11] [17 15] [20 18]]
[[11 11] [12 11] [9 14] [6 17] [4 19]]
[[11 11] [10 11] [10 14] [12 16] [16 20]]
[[11 11] [11 12] [11 15] [11 17] [7 21]]
[[11 11] [12 11] [12 14] [16 18] [13 21]]
[[11 11] [11 12] [11 15] [11 17] [15 21]]
[[11 11] [12 11] [12 14] [16 18] [16 21]]

The shortest route from {21,11} to {1,11}:
[[21 11] [20 10] [19 9] [18 9] [13 4] [6 11] [4 11] [1 11]]

The shortest route from {1,11} to {21,11}:
[[1 11] [2 10] [5 13] [9 9] [15 3] [20 8] [20 10] [21 11]]

The following cells are unreachable:
[[4 3] [2 18] [18 20]]

There are 5 cells that take 6 days to send reinforcements to:
[[11 11] [10 10] [12 8] [16 12] [20 12] [21 11] [22 12]]
[[11 11] [11 12] [14 15] [16 17] [17 16] [18 16] [20 14]]
[[11 11] [12 11] [9 14] [6 17] [4 17] [3 17] [3 19]]
[[11 11] [10 11] [7 11] [7 12] [7 18] [7 20] [6 20]]
[[11 11] [10 11] [10 14] [12 16] [12 20] [15 20] [17 20]]

```

Alternative using Floyd-Warshall for Part 2, and finding the longest shortest path between any two points.

```package main

import (
"fmt"
"math"
"strings"
)

var gmooh = strings.Split(
`.........00000.........
......00003130000......
....000321322221000....
...00231222432132200...
..0041433223233211100..
..0232231612142618530..
.003152122326114121200.
.031252235216111132210.
.022211246332311115210.
00113232262121317213200
03152118212313211411110
03231234121132221411410
03513213411311414112320
00427534125412213211400
.013322444412122123210.
.015132331312411123120.
.003333612214233913300.
..0219126511415312570..
..0021321524341325100..
...00211415413523200...
....000122111322000....
......00001120000......
.........00000.........`, "\n")

var width, height = len(gmooh), len(gmooh)

type pyx int // {y, x}

var d = []pyx{{-1, -1}, {0, -1}, {1, -1}, {-1, 0}, {1, 0}, {-1, 1}, {0, 1}, {1, 1}}

var dist, next [][]int
var pmap []pyx

const (
max = math.MaxInt32
min = -1
)

func (p pyx) destruct() (int, int) {
return p, p
}

func fwPath(u, v int) string {
res := ""
if next[u][v] != min {
path := []string{fmt.Sprintf("%v", pmap[u])}
for u != v {
u = next[u][v]
path = append(path, fmt.Sprintf("%v", pmap[u]))
}
res = strings.Join(path, "->")
}
return res
}

func showFwPath(u, v int) {
fmt.Printf("%v->%v   %2d   %s\n", pmap[u], pmap[v], dist[u][v], fwPath(u, v))
}

func floydWarshall() {
point := 0
var weights []pyx
points := make([][]int, height)
for i := 0; i < width; i++ {
points[i] = make([]int, width)
}
// First number the points.
for x := 0; x < width; x++ {
for y := 0; y < height; y++ {
if gmooh[y][x] >= '0' {
points[y][x] = point
point++
pmap = append(pmap, pyx{y, x})
}
}
}
// ...and then a set of edges (all of which have a "weight" of 1 day)
for x := 0; x < width; x++ {
for y := 0; y < height; y++ {
if gmooh[y][x] > '0' {
n := int(gmooh[y][x] - '0')
for di := 0; di < len(d); di++ {
dx, dy := d[di].destruct()
rx, ry := x+n*dx, y+n*dy
if rx >= 0 && rx < width && ry >= 0 && ry < height && gmooh[rx][ry] >= '0' {
weights = append(weights, pyx{points[y][x], points[ry][rx]})
}
}
}
}
}
// Before applying Floyd-Warshall.
vv := len(pmap)
dist = make([][]int, vv)
next = make([][]int, vv)
for i := 0; i < vv; i++ {
dist[i] = make([]int, vv)
next[i] = make([]int, vv)
for j := 0; j < vv; j++ {
dist[i][j] = max
next[i][j] = min
}
}
for k := 0; k < len(weights); k++ {
u, v := weights[k].destruct()
dist[u][v] = 1 // the weight of the edge (u,v)
next[u][v] = v
}
// Standard Floyd-Warshall implementation,
// with the optimization of avoiding processing of self/infs,
// which surprisingly makes quite a noticeable difference.
for k := 0; k < vv; k++ {
for i := 0; i < vv; i++ {
if i != k && dist[i][k] != max {
for j := 0; j < vv; j++ {
if j != i && j != k && dist[k][j] != max {
dd := dist[i][k] + dist[k][j]
if dd < dist[i][j] {
dist[i][j] = dd
next[i][j] = next[i][k]
}
}
}
}
}
}
showFwPath(points, points)
showFwPath(points, points)

var maxd, mi, mj int
for i := 0; i < vv; i++ {
for j := 0; j < vv; j++ {
if j != i {
dd := dist[i][j]
if dd != max && dd > maxd {
maxd, mi, mj = dd, i, j
}
}
}
}
fmt.Println("\nMaximum shortest distance:")
showFwPath(mi, mj)
}

func main() {
floydWarshall()
}
```

{{out}}

```
[21 11]->[1 11]    7   [21 11]->[20 10]->[19 10]->[14 10]->[10 10]->[8 8]->[4 8]->[1 11]
[1 11]->[21 11]    7   [1 11]->[2 10]->[5 13]->[9 9]->[15 3]->[20 8]->[20 10]->[21 11]

Maximum shortest distance:
[7 3]->[20 14]    9   [7 3]->[8 4]->[10 6]->[11 7]->[15 11]->[16 11]->[17 12]->[17 16]->[18 16]->[20 14]

```

## Julia

Uses the LightGraphs package.

```
using LightGraphs

const grid = reshape(Vector{UInt8}(replace("""
00000
00003130000
000321322221000
00231222432132200
0041433223233211100
0232231612142618530
003152122326114121200
031252235216111132210
022211246332311115210
00113232262121317213200
03152118212313211411110
03231234121132221411410
03513213411311414112320
00427534125412213211400
013322444412122123210
015132331312411123120
003333612214233913300
0219126511415312570
0021321524341325100
00211415413523200
000122111322000
00001120000
00000         """, "\n" => "")), 23, 23)

const board = map(c -> c == UInt8(' ') ? -1 : c - UInt8('0'), grid)
const startingpoints = [i for i in 1:529 if board[i] > 0]
const safety = [i for i in 1:529 if board[i] == 0]
const legalendpoints = [i for i in 1:529 if board[i] >= 0]

k, ret = board[i], Int[]
row, col = divrem(i - 1, 23) .+ 1
col > k && push!(ret, i - k)
23 - col >= k && push!(ret, i + k)
row > k && push!(ret, i - 23 * k)
row + k <= 23 && push!(ret, i + 23 * k)
row > k && col > k && push!(ret, i - 24 * k)
row + k <= 23 && (23 - col >= k) && push!(ret, i + 24 * k)
row > k && (23 - col >= k) && push!(ret, i - 22 * k)
row + k <= 23 && col > k && push!(ret, i + 22 * k)
ret
end

const graph = SimpleDiGraph(529)

for i in 1:529
if board[i] > 0
if board[p] >= 0
end
end
end
end

"""
allnpaths(graph, a, b, vec, n)

Return a vector of int vectors, each of which is a path from a to a member of
vec and where n is the length of each path and the nodes in a path do not repeat.
"""
function allnpaths(graph, a, vec, n)
ret = [[a]]
for j in 2:n
nextret = Vector{Vector{Int}}()
for path in ret, x in neighbors(graph, path[end])
if !(x in path) && (j < n || x in vec)
push!(nextret, [path; x])
end
end
ret = nextret
end
return (ret == [[a]] && a != b) ? [] : ret
end

function pathtostring(path)
ret = ""
for node in path
c = CartesianIndices(board)[node]
ret *= "(\$(c-1), \$(c-1)) "
end
ret
end

function pathlisting(paths)
join([pathtostring(p) for p in paths], "\n")
end

println("Part 1:")
let
start = 23 * 11 + 12
pathsfromcenter = dijkstra_shortest_paths(graph, start)
safepaths = filter(p -> length(p) > 1, enumerate_paths(pathsfromcenter, safety))
safelen = mapreduce(length, min, safepaths)
paths = unique(allnpaths(graph, start, safety, safelen))
println("The \$(length(paths)) shortest paths to safety are:\n",
pathlisting(paths))
end

println("\nPart 2:")
let
p = enumerate_paths(bellman_ford_shortest_paths(graph, 21 * 23 + 12), 23 + 12)
println("One shortest route from (21, 11) to (1, 11): ", pathtostring(p))

p = enumerate_paths(bellman_ford_shortest_paths(graph, 23 + 12), 21 * 23 + 12)
println("\nOne shortest route from (1, 11) to (21, 11): ", pathtostring(p))

allshortpaths = [enumerate_paths(bellman_ford_shortest_paths(graph, 23 + 12), p) for p in startingpoints]
maxlen, idx = findmax(map(length, allshortpaths))
println("\nLongest Shortest Route (length \$(maxlen - 1)) is: ", pathtostring(allshortpaths[idx]))
end

println("\nExtra Credit Questions:")
let
println("\nIs there any cell in the country that can not be reached from HQ (11, 11)?")
frombase = bellman_ford_shortest_paths(graph, 11 * 23 + 12)
unreached = Int[]
for pt in legalendpoints
path = enumerate_paths(frombase, pt)
if isempty(path) && pt != 11 * 23 + 12
push!(unreached, pt)
end
end
print("There are \$(length(unreached)): ")
println(pathtostring(unreached))

println("\nWhich cells will it take longest to send reinforcements to from HQ (11, 11)?")
p = [enumerate_paths(frombase, x) for x in legalendpoints]
maxlen = mapreduce(length, max, p)
allmax = [path for path in p if length(path) == maxlen]
println("There are \$(length(allmax)) of length \$(maxlen - 1):")
println(pathlisting(allmax))
end

```

{{out}}

```
Part 1:
The 71 shortest paths to safety are:
(11, 11) (10, 10) (8, 8) (4, 8) (1, 8)
(11, 11) (10, 10) (8, 8) (8, 4) (6, 2)
(11, 11) (10, 10) (8, 8) (12, 4) (9, 1)
(11, 11) (10, 10) (8, 8) (12, 4) (15, 1)
(11, 11) (10, 10) (8, 10) (5, 7) (2, 4)
(11, 11) (10, 10) (8, 10) (5, 13) (1, 9)
(11, 11) (10, 10) (8, 10) (5, 13) (1, 13)
(11, 11) (10, 10) (8, 12) (6, 12) (0, 12)
(11, 11) (10, 10) (12, 8) (8, 4) (6, 2)
(11, 11) (10, 10) (12, 8) (12, 4) (9, 1)
(11, 11) (10, 10) (12, 8) (12, 4) (15, 1)
(11, 11) (10, 10) (12, 8) (16, 4) (13, 1)
(11, 11) (10, 10) (12, 8) (16, 4) (16, 1)
(11, 11) (10, 10) (12, 8) (16, 4) (19, 4)
(11, 11) (10, 10) (12, 8) (16, 12) (20, 16)
(11, 11) (10, 11) (7, 8) (4, 5) (3, 4)
(11, 11) (10, 11) (7, 8) (4, 8) (1, 8)
(11, 11) (10, 11) (7, 8) (4, 11) (1, 8)
(11, 11) (10, 11) (7, 8) (4, 11) (1, 14)
(11, 11) (10, 11) (7, 8) (7, 5) (2, 5)
(11, 11) (10, 11) (7, 8) (7, 5) (12, 0)
(11, 11) (10, 11) (7, 11) (6, 12) (0, 12)
(11, 11) (10, 11) (7, 11) (7, 12) (1, 6)
(11, 11) (10, 11) (10, 14) (12, 16) (16, 20)
(11, 11) (10, 11) (13, 8) (14, 7) (18, 3)
(11, 11) (10, 11) (13, 11) (17, 15) (20, 18)
(11, 11) (10, 12) (9, 12) (7, 12) (1, 6)
(11, 11) (11, 10) (10, 9) (9, 9) (3, 3)
(11, 11) (11, 10) (11, 9) (9, 9) (3, 3)
(11, 11) (11, 12) (8, 9) (2, 9) (1, 8)
(11, 11) (11, 12) (8, 9) (2, 9) (1, 9)
(11, 11) (11, 12) (8, 9) (2, 15) (1, 14)
(11, 11) (11, 12) (8, 9) (2, 15) (1, 15)
(11, 11) (11, 12) (8, 9) (2, 15) (1, 16)
(11, 11) (11, 12) (8, 9) (2, 15) (2, 16)
(11, 11) (11, 12) (8, 9) (8, 3) (6, 1)
(11, 11) (11, 12) (8, 9) (8, 3) (8, 1)
(11, 11) (11, 12) (8, 9) (14, 3) (11, 0)
(11, 11) (11, 12) (8, 12) (6, 12) (0, 12)
(11, 11) (11, 12) (8, 15) (9, 16) (2, 16)
(11, 11) (11, 12) (11, 9) (9, 9) (3, 3)
(11, 11) (11, 12) (11, 15) (11, 17) (7, 21)
(11, 11) (11, 12) (11, 15) (11, 17) (15, 21)
(11, 11) (11, 12) (14, 9) (18, 5) (19, 4)
(11, 11) (11, 12) (14, 9) (18, 13) (22, 9)
(11, 11) (11, 12) (14, 9) (18, 13) (22, 13)
(11, 11) (11, 12) (14, 12) (16, 12) (20, 16)
(11, 11) (11, 12) (14, 15) (16, 15) (19, 18)
(11, 11) (12, 10) (11, 9) (9, 9) (3, 3)
(11, 11) (12, 10) (13, 10) (13, 5) (13, 0)
(11, 11) (12, 10) (13, 10) (18, 5) (19, 4)
(11, 11) (12, 10) (13, 10) (18, 15) (21, 15)
(11, 11) (12, 10) (13, 11) (17, 15) (20, 18)
(11, 11) (12, 11) (9, 14) (6, 17) (4, 19)
(11, 11) (12, 11) (12, 8) (8, 4) (6, 2)
(11, 11) (12, 11) (12, 8) (12, 4) (9, 1)
(11, 11) (12, 11) (12, 8) (12, 4) (15, 1)
(11, 11) (12, 11) (12, 8) (16, 4) (13, 1)
(11, 11) (12, 11) (12, 8) (16, 4) (16, 1)
(11, 11) (12, 11) (12, 8) (16, 4) (19, 4)
(11, 11) (12, 11) (12, 8) (16, 12) (20, 16)
(11, 11) (12, 11) (12, 14) (8, 18) (3, 18)
(11, 11) (12, 11) (12, 14) (16, 18) (13, 21)
(11, 11) (12, 11) (12, 14) (16, 18) (16, 21)
(11, 11) (12, 11) (12, 14) (16, 18) (19, 18)
(11, 11) (12, 11) (15, 8) (15, 5) (18, 2)
(11, 11) (12, 11) (15, 8) (18, 5) (19, 4)
(11, 11) (12, 11) (15, 8) (18, 11) (22, 11)
(11, 11) (12, 11) (15, 11) (16, 12) (20, 16)
(11, 11) (12, 11) (15, 14) (16, 15) (19, 18)
(11, 11) (12, 12) (13, 11) (17, 15) (20, 18)

Part 2:
One shortest route from (21, 11) to (1, 11): (21, 11) (21, 12) (19, 14) (14, 14) (12, 14) (8, 18) (3, 13) (1, 11)

One shortest route from (1, 11) to (21, 11): (1, 11) (2, 10) (5, 13) (9, 9) (15, 3) (20, 8) (20, 10) (21, 11)

Longest Shortest Route (length 9) is: (1, 11) (2, 10) (5, 13) (9, 9) (15, 15) (16, 14) (16, 17) (17, 16) (18, 16) (20, 14)

Extra Credit Questions:

Is there any cell in the country that can not be reached from HQ (11, 11)?
There are 3: (2, 18) (4, 3) (18, 20)

Which cells will it take longest to send reinforcements to from HQ (11, 11)?
There are 5 of length 6:
(11, 11) (12, 11) (9, 14) (6, 17) (4, 17) (4, 18) (3, 19)
(11, 11) (10, 11) (7, 11) (7, 12) (7, 18) (7, 20) (6, 20)
(11, 11) (11, 12) (11, 15) (13, 17) (15, 19) (15, 20) (17, 20)
(11, 11) (11, 12) (14, 15) (16, 17) (17, 16) (18, 16) (20, 14)
(11, 11) (12, 12) (13, 11) (17, 15) (20, 12) (21, 11) (22, 12)

```

## Phix

Using a simple breadth-first search. Parts 1 and 2 and extra credit.

```constant gmooh = split("""
.........00000.........
......00003130000......
....000321322221000....
...00231222432132200...
..0041433223233211100..
..0232231612142618530..
.003152122326114121200.
.031252235216111132210.
.022211246332311115210.
00113232262121317213200
03152118212313211411110
03231234121132221411410
03513213411311414112320
00427534125412213211400
.013322444412122123210.
.015132331312411123120.
.003333612214233913300.
..0219126511415312570..
..0021321524341325100..
...00211415413523200...
....000122111322000....
......00001120000......
.........00000.........""",'\n')

constant width = length(gmooh),
height = length(gmooh),
d = {{-1,-1},{0,-1},{+1,-1},
{-1, 0},       {+1, 0},
{-1,+1},{0,+1},{+1,+1}}

sequence routes -- {cost,fromy,fromx} for each gmooh[][].

procedure search(integer y, x)
-- simple breadth-first search, populates routes
-- (this isn't strictly dijkstra, because graph edges are not weighted)
integer cost = 0
sequence route = {{y,x}},
next = {}
routes = repeat(repeat(0,width),height)
routes[y,x] = {0,y,x} -- zero-cost the starting point
while 1 do
integer n = gmooh[y,x]-'0'
for di=1 to length(d) do
integer {dx,dy} = d[di]
integer {rx,ry} = {x+n*dx,y+n*dy}
if rx>=1 and rx<=width
and ry>=1 and ry<=height
and gmooh[ry,rx]>='0' then
object ryx = routes[ry,rx]
if ryx=0
or ryx>cost+1 then
routes[ry,rx] = {cost+1,y,x}
if gmooh[ry,rx]>'0' then
next = append(next,{cost+1,ry,rx})
-- (if the graph was weighted, at this point
--   that would get shuffled up into place.)
end if
end if
end if
end for
if length(next)=0 then exit end if
{cost,y,x} = next
next = next[2..\$]
end while
end procedure

function get_route(sequence yx)
integer {y,x} = yx, cost
sequence res = {{y,x}}
while 1 do
{cost,y,x} = routes[y,x]
if cost=0 then exit end if
res = prepend(res,{y,x})
end while
return res
end function

procedure show_shortest_routes_to_safety()
integer shortest = 9999
sequence res = {}
for x=1 to width do
for y=1 to height do
if gmooh[y,x]='0' then
object ryx = routes[y,x]
if ryx!=0 then
integer cost = ryx
if cost<=shortest then
if cost<shortest then
res = {}
shortest = cost
end if
res = append(res,{y,x})
end if
end if
end if
end for
end for
string {areis,s} = iff(length(res)>1?{"are","s"}:{"is",""})
printf(1,"There %s %d shortest route%s of %d days to safety:\n",{areis,length(res),s,shortest})
for i=1 to length(res) do
?get_route(res[i])
end for
end procedure

procedure show_unreachable()
sequence res = {}
for x=1 to width do
for y=1 to height do
if gmooh[y,x]>='0'
and routes[y,x]=0 then
res = append(res,{y,x})
end if
end for
end for
puts(1,"The following cells are unreachable:\n")
?res
end procedure

procedure show_longest()
integer longest = 0
sequence res = {}
for x=1 to width do
for y=1 to height do
if gmooh[y,x]>='0' then
object ryx = routes[y,x]
if ryx!=0 then
integer rl = ryx
if rl>=longest then
if rl>longest then
res = {}
longest = rl
end if
res  = append(res,{y,x})
end if
end if
end if
end for
end for
printf(1,"There are %d cells that take %d days to send reinforcements to\n",{length(res),longest})
for i=1 to length(res) do
?get_route(res[i])
end for
end procedure

procedure main()
search(12,12)
show_shortest_routes_to_safety()

search(22,12)
puts(1,"The shortest route from 22,12 to 2,12:\n")
?get_route({2,12})

search(2,12)
puts(1,"The shortest route from 2,12 to 22,12:\n")
?get_route({22,12})

search(12,12)

show_unreachable()
show_longest()

end procedure
main()
```

{{out}} Note: Phix indexes are 1-based and therefore so too are these results.

```
There are 40 shortest routes of 4 days to safety:
{{12,12},{12,13},{9,10},{15,4},{12,1}}
{{12,12},{11,12},{8,9},{8,6},{13,1}}
{{12,12},{13,11},{14,11},{14,6},{14,1}}
{{12,12},{12,13},{9,10},{9,4},{7,2}}
{{12,12},{12,13},{9,10},{9,4},{9,2}}
{{12,12},{11,11},{9,9},{13,5},{10,2}}
{{12,12},{11,11},{13,9},{17,5},{14,2}}
{{12,12},{11,11},{9,9},{13,5},{16,2}}
{{12,12},{11,11},{13,9},{17,5},{17,2}}
{{12,12},{11,11},{9,9},{9,5},{7,3}}
{{12,12},{13,12},{16,9},{16,6},{19,3}}
{{12,12},{12,11},{11,10},{10,10},{4,4}}
{{12,12},{11,12},{14,9},{15,8},{19,4}}
{{12,12},{11,11},{9,11},{6,8},{3,5}}
{{12,12},{11,12},{8,9},{5,6},{4,5}}
{{12,12},{11,11},{13,9},{17,5},{20,5}}
{{12,12},{11,12},{8,9},{8,6},{3,6}}
{{12,12},{11,12},{8,12},{8,13},{2,7}}
{{12,12},{11,11},{9,9},{5,9},{2,9}}
{{12,12},{11,11},{9,11},{6,14},{2,10}}
{{12,12},{12,13},{15,10},{19,14},{23,10}}
{{12,12},{13,12},{16,9},{19,12},{23,12}}
{{12,12},{11,11},{9,13},{7,13},{1,13}}
{{12,12},{11,11},{9,11},{6,14},{2,14}}
{{12,12},{12,13},{15,10},{19,14},{23,14}}
{{12,12},{11,12},{8,9},{5,12},{2,15}}
{{12,12},{12,13},{9,10},{3,16},{2,16}}
{{12,12},{13,11},{14,11},{19,16},{22,16}}
{{12,12},{12,13},{9,10},{3,16},{2,17}}
{{12,12},{12,13},{9,10},{3,16},{3,17}}
{{12,12},{11,11},{13,9},{17,13},{21,17}}
{{12,12},{13,12},{13,15},{9,19},{4,19}}
{{12,12},{12,13},{15,16},{17,16},{20,19}}
{{12,12},{11,12},{14,12},{18,16},{21,19}}
{{12,12},{13,12},{10,15},{7,18},{5,20}}
{{12,12},{11,12},{11,15},{13,17},{17,21}}
{{12,12},{12,13},{12,16},{12,18},{8,22}}
{{12,12},{13,12},{13,15},{17,19},{14,22}}
{{12,12},{12,13},{12,16},{12,18},{16,22}}
{{12,12},{13,12},{13,15},{17,19},{17,22}}
The shortest route from 22,12 to 2,12:
{{22,12},{21,11},{20,10},{19,10},{14,5},{7,12},{5,12},{2,12}}
The shortest route from 2,12 to 22,12:
{{2,12},{3,11},{6,14},{10,10},{16,4},{21,9},{21,11},{22,12}}
The following cells are unreachable:
{{5,4},{3,19},{19,21}}
There are 5 cells that take 6 days to send reinforcements to
{{12,12},{11,11},{13,9},{17,13},{21,13},{22,12},{23,13}}
{{12,12},{12,13},{15,16},{17,18},{18,17},{19,17},{21,15}}
{{12,12},{13,12},{10,15},{7,18},{5,18},{4,18},{4,20}}
{{12,12},{11,12},{8,12},{8,13},{8,19},{8,21},{7,21}}
{{12,12},{11,12},{11,15},{13,17},{13,21},{16,21},{18,21}}

```

Alternative using Floyd-Warshall for Part 2, and finding the longest shortest path between any two points.

```--(same constants as above: gmooh, width, height, d)
constant inf = 1e300*1e300

sequence dist, next, pmap = {}

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

procedure show_fw_path(integer u, v)
printf(1,"{%d,%d}->{%d,%d}   %2d   %s\n",pmap[u]&pmap[v]&{dist[u,v],fw_path(u,v)})
end procedure

procedure FloydWarshall()
integer point = 0
sequence weights = {},
points = repeat(repeat(0,width),height)
-- First number the points...
for x=1 to width do
for y=1 to height do
if gmooh[y,x]>='0' then
point += 1
points[y,x] = point
pmap = append(pmap,{y,x})
end if
end for
end for
-- ...and then a set of edges (all of which have a "weight" of 1 day)
for x=1 to width do
for y=1 to height do
if gmooh[y,x]>'0' then
integer n = gmooh[y,x]-'0'
for di=1 to length(d) do
integer {dx,dy} = d[di]
integer {rx,ry} = {x+n*dx,y+n*dy}
if rx>=1 and rx<=width
and ry>=1 and ry<=height
and gmooh[ry,rx]>='0' then
--                      weights = append(weights,{points[y,x],points[ry,rx],1})
weights = append(weights,{points[y,x],points[ry,rx]})
end if
end for
end if
end for
end for
-- Before applying Floyd-Warshall
integer V = length(pmap)
dist = repeat(repeat(inf,V),V)
next = repeat(repeat(null,V),V)
for k=1 to length(weights) do
--      integer {u,v,w} = weights[k]
integer {u,v} = weights[k]
--      dist[u,v] := w  -- the weight of the edge (u,v)
dist[u,v] := 1  -- the weight of the edge (u,v)
next[u,v] := v
end for
-- standard Floyd-Warshall implementation,
-- with the optimisation of avoiding processing of self/infs,
-- which surprisingly makes quite a noticeable difference.
for k=1 to V do
for i=1 to V do
if i!=k and dist[i,k]!=inf then
for j=1 to V do
if j!=i and j!=k and dist[k,j]!=inf then
atom d = dist[i,k] + dist[k,j]
if d<dist[i,j] then
dist[i,j] := d
next[i,j] := next[i,k]
end if
end if
end for
end if
end for
end for
show_fw_path(points[22,12],points[2,12])
show_fw_path(points[2,12],points[22,12])

integer maxd = 0, mi, mj
for i=1 to V do
for j=1 to V do
if j!=i then
atom d = dist[i,j]
if d!=inf and d>maxd then
{maxd,mi,mj} = {d,i,j}
end if
end if
end for
end for
printf(1,"Maximum shortest distance:\n")
show_fw_path(mi,mj)

end procedure

FloydWarshall()
```

{{out}}

```
{22,12}->{2,12}    7   {22,12}->{21,11}->{20,11}->{15,11}->{11,11}->{9,9}->{5,9}->{2,12}
{2,12}->{22,12}    7   {2,12}->{3,11}->{6,14}->{10,10}->{16,4}->{21,9}->{21,11}->{22,12}
Maximum shortest distance:
{8,4}->{21,15}   9   {8,4}->{9,5}->{11,7}->{12,8}->{16,12}->{17,12}->{18,13}->{18,17}->{19,17}->{21,15}

```