This is the first in a series of Island Golf challenges. Next challenge

Given an island in ASCII-art, output an optimal path to circumnavigate it.

## Input

Your input will be a rectangular grid consisting of two characters, representing land and water. In the examples below, land is # and water is ., but you may substitute any two distinct characters you wish.

...........
...##......
..#####....
..#######..
.#########.
...#######.
...#####.#.
....####...
...........


There will always be at least one land tile. The land tiles will all be contiguous (i.e. there's only one island). The water tiles will also be contiguous (i.e. there are no lakes). The outer border of the grid will all be water tiles. Land tiles will not be connected diagonally: i.e., you will never see something like

....
.#..
..#.
....


## Output

Your code must output the same grid, with a shortest circumnavigation drawn on it. In the examples below, the circumnavigation path is drawn with o, but you may substitute any character as long as it is distinct from your land and water characters.

A circumnavigation is a simple closed curve, drawn entirely on water tiles, that fully encircles all land tiles on the grid. Diagonal connections are allowed. For instance, this is a circumnavigation of the above island (but not a shortest one):

.ooooo.....
o..##.oo...
o.#####.o..
o.#######o.
o#########o
ooo#######o
..o#####.#o
..oo####..o
....oooooo.


The length of a circumnavigation is computed as follows: For every pair of adjacent tiles on the path, if they are connected horizontally or vertically, add 1; if they are connected diagonally, add √2. The length of the above path is 22 + 7√2 (≈ 31.9).

A shortest circumnavigation is a circumnavigation with the shortest possible length. Your program should output any one path that satisfies this condition. For most islands, there will be multiple possible solutions. Here is one solution for the above island, with length 10 + 13√2 (≈ 28.4):

...oo......
..o##oo....
.o#####oo..
.o#######o.
o#########o
.o.#######o
..o#####.#o
...o####.o.
....ooooo..


## Details

Your solution may be a full program or a function. Any of the default input and output methods are acceptable.

Your input and output may be a multiline string or a list of strings. If your language has a character type distinct from single-character strings, you may substitute "list of characters" for "string" in the previous sentence. If your language needs to input the height and/or width of the grid, you may do so. Your output may (optionally) have a single trailing newline. As mentioned above, you may use any three distinct characters in place of #.o (please specify in your submission which characters you're using).

## Test cases

A. Islands with unique shortest circumnavigations:

...
.#.
...

.o.
o#o
.o.

......
.####.
......

.oooo.
o####o
.oooo.

......
......
..##..
...#..
......
......

......
..oo..
.o##o.
..o#o.
...o..
......

.......
.#####.
...#...
...#...
.#####.
.......

.ooooo.
o#####o
o..#..o
o..#..o
o#####o
.ooooo.

.......
...#...
...#...
.#####.
...#...
...#...
.......

...o...
..o#o..
.o.#.o.
o#####o
.o.#.o.
..o#o..
...o...

.......
.#####.
.##..#.
..#..#.
.......

.ooooo.
o#####o
o##..#o
.o#..#o
..oooo.


B. Example of an island with multiple possible solutions:

........
....##..
...####.
..###...
.#####..
.#####..
..##....
........


Possible outputs:

....oo..
...o##o.
..o####o
.o###.o.
o#####o.
o#####o.
.o##oo..
..oo....

....oo..
...o##o.
..o####o
.o###.o.
o#####o.
o#####o.
.o##.o..
..ooo...

....oo..
...o##o.
..o####o
.o###..o
o#####.o
o#####o.
.o##oo..
..oo....

....oo..
...o##o.
..o####o
.o###..o
o#####.o
o#####o.
.o##.o..
..ooo...


This is : the shortest code in each language wins.

• The shortest circumnavigation for the 3rd test case is the 'loaf' pattern in Conway's Game of Life! – Comrade SparklePony Mar 23 '17 at 17:08

## Mathematica (version 9), 165 bytes

The nice, short ConvexHullMesh function that Greg Martin used was only introduced in Mathematica version 10, so I thought I'd make an attempt without it, using my ancient Mathematica version 9. I managed to get it slightly shorter, though! It's a function that takes and returns a list of strings (with ., # and o as the symbols).

""<>#&/@("o"MorphologicalTransform[MorphologicalComponents[#,Method->"ConvexHull"],Max[#(1-#[[2,2]])CrossMatrix@1]&]+"#"#/.{0->"."})&[Characters@#/.{"."->0,"#"->1}]&


Explanation:

• First, Characters@# /. {"."->0, "#"->1} turns the input into a matrix of 0s and 1s.
• "o"MorphologicalTransform[MorphologicalComponents[#,Method->"ConvexHull"],Max[#(1-#[[2,2]])CrossMatrix@1]&]+"#"# then uses Mathematica's powerful (but extremely byte-heavy…) image processing capabilities to first fill in the island's convex hull (which is the shape you'd get if you stretched a piece of string around it), and then take its boundary. We then multiply this matrix by the string "o" to get a matrix of 0s and "o"s (thanks to Mathematica's impressive adaptability about types), and add this to "#" times the matrix of the island.
• Finally, ""<>#& /@ (... /. {0->"."}) turns this matrix of "o"s, "#"s and 0s into a matrix of "o"s, "#"s and "."s, and joins each row into a string.

When we test this on example B, we get the output

{"....oo..",
"...o##o.",
"..o####o",
".o###..o",
"o#####o.",
"o#####o.",
".o##oo..",
"..oo...."}


[Edit, thanks to Greg Martin:] If we're allowed to use arrays of characters instead of lists of strings, we can trim this down to 144 bytes:

"o"MorphologicalTransform[MorphologicalComponents[#,Method->"ConvexHull"],Max[#(1-#[[2,2]])CrossMatrix@1]&]+"#"#/.{0->"."}&[#/.{"."->0,"#"->1}]&

• Nicely done! I never knew about MorphologicalComponents[#, Method->"ConvexHull"] :) You can save even more bytes by assuming that the input is already split into characters, and by returning a 2D array of characters as well. – Greg Martin Mar 23 '17 at 5:50
• @GregMartin, I didn't know about that use of MorphologicalComponents either until today! – Not a tree Mar 23 '17 at 6:41
• Mathematica novice here: how should I call this function? I tried f[{"...",".#.","..."}] and got some errors. – DLosc Mar 24 '17 at 5:54
• @DLosc, The function is the whole thing, not just f. (Well, strictly speaking it's the stuff after the semicolon.) To call the function, type the entire thing into a Mathematica window, followed by [, your input, and ], so it should look something like f@m_:=m(1-m[[2,2]]) . . . #/.{"."->0,"#"->1}]&[{"...", ".#.", "..."}] (abridged for space). – Not a tree Mar 24 '17 at 8:56
• @DLosc Well, that's because the code's broken. I think I've fixed it now, though! (I've got no idea what happened there; sorry…) – Not a tree Mar 28 '17 at 1:08

(But go upvote Notatree's solution, it's better!)

# Mathematica, 168 bytes

(x_~n~y_:=Min[Norm[x-#]&/@y];i=#;p=i~Position~#&;t=p["#"|"."]~Select~#&;(i~Part~##="o")&@@@t[#~n~t[ConvexHullMesh[Join[s=p@"#",#+{.1,.1}&/@s]]~RegionMember~#&]==1&];i)&


Pure function taking a 2D array of characters as input and returning a 2D array of characters. An easier-to-read version:

1  (x_~n~y_ := Min[Norm[x - #] & /@ y];
2  i = #; p = i~Position~# &;
3  t = p["#" | "."]~Select~# &;
4  (i~Part~## = "o") & @@@
5    t[#~n~
6      t[ConvexHullMesh[
7        Join[s = p@"#", # + {.1, .1} & /@ s]]
8      ~RegionMember~# &] == 1 &];
9  i) &


Line 1 defines a function n that produces the (smallest) distance between a point x in the plane and a set y of other points. Line 2 initializes the variable i to the input, both to resolve a currying ambiguity later and to be able to modify it to produce the eventual output; line 2 also defines a function p that returns the coordinates of all occurrences of its input in i.

On line 3, p["#" | "."] represents every single coordinate from the input map (since all its characters are either "#" or "."), so t is a function that selects only those coordinates that satisfy a yet-unspecified property. On line 4, i~Part~## = "o" is going to change a bunch of entries of i to the character "o"; those characters will be selected from the set of possible coordinates according to the stuff on lines 5-8. And line 9 just returns the answer once it's computed.

Okay, infrastructure done, now to the actual computation. ConvexHullMesh is Mathematica's built-in for computing the convex hull of a set of points (the smallest convex polygon containing those points). Morally speaking, this should "fill in" the coves and fjords of the island (which is s = p@"#"), to rule them out of our cicrumnavigation. There's a little problem with ConvexHullMesh when that set of points is all in a line (thank you, test case #2), which we resolve by appending a slightly offset version of s to itself in line 7. This output is a polygon, so lines 7-9 (t[...~RegionMember~# &]) produces a list of the points with integer coordinates in that polygon. Finally, line 5 and the end of line 9 calculate all points that are at a distance exactly 1 (hence not 0) from this set of integer points; that set becomes the circumnavigating path.

Below is the output for the large test case at the OP's link. Notice at the upper left, the unusual choices of when to go west versus southwest hint at the fact that it's shadowing an invisible line of slope -2/3 between two peninsulas (said line segment being part of the convex hull's boundary).

........................
.............o..........
...........oo#ooooooo...
..........o#.#.##...#o..
........oo.#.#.###.##o..
.......o..########.##o..
.....oo...############o.
...oo#....############o.
..o#.###.##############o
.o##.##################o
.o####################o.
.o.##################.o.
.o##################..o.
.o..################..o.
o###################..o.
o#####################o.
o.##################.o..
o####################o..
o#...##############.o...
o##...#############o....
o#.....###....#oooo.....
.oooooo#ooooooo.........
.......o................

• Does Mathematica usually represent strings as 1D arrays of characters? If not, then you'll need to take/return a 1D array of strings instead. (Also, looking forward to the explanation! I don't think I'll be able to run this without having Mathematica, right?) – DLosc Mar 23 '17 at 3:20
• Mathematica has a string data type, but it seems that an array of characters is also valid for the purposes of this site (i.e., I learned this when I started on PPCG, but I forget the legalities of why). Yes, unfortunately, Mathematica is non-free and thus not accessible to many people :( – Greg Martin Mar 23 '17 at 5:30
• @GregMartin I always try out Mathematica solutions at sandbox.open.wolframcloud.com – ovs Mar 23 '17 at 20:28
• Current consensus says that lists of single-character strings can't be used in place of a string. As far as I can tell, the "characters" in Mathematica are just single-character strings, like in Python. The situation is different in a language like Java, which has a separate char type; in that case, a char array could be used in place of a string. – DLosc Mar 24 '17 at 5:19
• Here's how I read it: The main upvoted answer was posted in 2014. The answer I linked was posted in 2016, as an attempt to clarify the ambiguity in the earlier answer. So I read the negative score on the newer answer as people saying, "No, we don't want the older answer to mean that lists of single-char strings are okay." But regardless of meta, I'm disallowing lists of single-char strings in this question (and I clarified the wording to reflect that). – DLosc Mar 24 '17 at 18:16

# Python 3, 779 bytes (indenting with tabs)

This is the whole program. It reads input from stdin and prints it to stdout. Stdin must end with EOF. Example run with the big input: https://ideone.com/XIfYY0

import itertools,sys
L=list
C=itertools.count
X=len(d[0])
Y=len(d)
R=range
def P(r):return all(d[y][x]=='.'for x,y in r)
def S(f):
for n in C(0):
if P(f(n)):l=n
else:break
for n in C(l+1):
if P(f(n)):return l,n
def f(V,a,*b):return L(eval('lambda '+a+':('+i+')',V)for i in b)
V=locals()
def D(n):
y=min(n,Y-1);x=n-y
while y>=0and x<X:yield(x,y);x+=1;y-=1
def E(n):
x=max(0,n-Y);y=x+Y-n
while y<Y and x<X:yield(x,y);x+=1;y+=1
F=f(V,'p','(p,y)for y in R(0,Y)','(x,p)for x in R(0,X)')+[D,E]
r=f(V,'x,y','x','y','x+y','x-y+Y')
B=L(map(S,F))
for x in R(0,X):
for y in R(0,Y):
z=L(zip(r,B))
if all(g(x,y)in R(a,b+1)for g,(a,b)in z)and any(g(x,y)in e for g,e in z):d[y][x]='o'
print('\n'.join(''.join(x)for x in d))


The idea is simple: it computes smallest octagonal bounds, and draws cells which are inside all of the computed bounds and intersect at least one of the edges.

• You don't really need to use sys.stdin as input. input(), getting multiline would do the job and cost less bytes – Dead Possum Mar 23 '17 at 12:18
• May be able to replace R(0,x) with R(x) – ceilingcat Mar 23 '17 at 17:33
• +1 for not using a built-in. – Robert Fraser Mar 24 '17 at 1:48
• Nice! Some more golfing tips: save 5 bytes each by using lambdas to define P and f; L(generator expression) => [generator expression]; F, r, and B appear to be used only once apiece and thus can be inlined. – DLosc Mar 27 '17 at 18:23

## JavaScript (ES6), 369 343 bytes

f=s=>(a=s.split
.map(s=>[...s]),m=Array(8),a.map((b,i)=>b.map((c,j)=>c>'#'||[i-j,i,j+i,j,j-i,-i,-i-j,-j].map((d,k)=>d>m[k]||(m[k]=d-1)))),[o,p,q,r,t,u,v,w]=m,g=(i,j,k,l,...p)=>i-k|j-l?a[i][j]=g(i+(k>i)-(k<i),j+(l>j)-(l<j),k,l,...p):1/p[0]?g(k,l,...p):'o',g(p,p-o,p,q-p,q-r,r,r-t,r,-u,t-u,-u,u-v,w-v,-w,o-w,-w,p,p-o),a.map(b=>b.join).join
)


Explanation: The string is split into a character array (I'm unclear as to whether character array input is acceptable). The array is then iterated over and the positions of all the land squares are located. The bounding lines given by the equations x - y = o, x = p, x + y = q, y = r, y - x = t, -x = u, -x - y = v, -y = w are determined such that the maximum possible parameter is chosen where all the land lies beyond the line. This has the effect of enclosing the island in an octagon. The coordinates of the corners of the octagon are readily calculated from the parameters and the cells on its edge are filled in. The array is then joined back into a string. The reason an octagon suffices is as follows:

   /o#     /o#     /o#
|/o #   |/o #   |/ o#
*o###   * o #   *  o#
/|o #   /|o #   /| o#
|o#     |o#     |o#


Consider a corner of the octagon. At some point along the two edges the path will be limited by the land because we constructed the octagon to touch the land as closely as possible. If there is no land at the corner itself, the path could take the alternate routes as shown on the right, but that is still the same number of orthogonal and diagonal steps, so the distance is unchanged.

• What does ´...p´ do? – Robert Fraser Mar 23 '17 at 14:54
• @RobertFraser The technical name is array destructuring. In this case however, it just acts as a rest of arguments parameter. – Neil Mar 23 '17 at 14:58
• @Neil Actually, the technical name is rest parameter. The same syntax is used for the spread operator. (You use both as ...p in different places.) Destructuring is something else (though the spread operator can be used in destructuring). – Brian McCutchon Mar 23 '17 at 15:25
• @BrianMcCutchon You're right, I meant spread operator, but destructuring works in argument lists anyway. – Neil Mar 23 '17 at 15:30

# Python 3.5, 224, 263, 234 218 bytes

Golfed off another 16 bytes by getting rid of the nested function and making it a one-liner.

def h(s,k=0,i=0):w=s.find('\n')+1;x=s.find('X')-w;k=k or x;d=[1,w+1,w,w-1,-1,-w-1,-w,-w+1]*2;u=s[:k]+'o'+s[k+1:];return['X'>s[k]and i<8and(h(u,k+d[i+2],i+2)or h(u,k+d[i+1],i+1)or h(u,k+d[i],i))or'',s][s[k]>'X'and k==x]


Golfed off 29 bytes:

def f(s):
w=s.find('\n')+1;x=s.find('X')-w;d=[1,w+1,w,w-1,-1,-w-1,-w,-w+1]*2
def h(s,k,i):u=s[:k]+'o'+s[k+1:];return['X'>s[k]and i<8and(h(u,k+d[i+2],i+2)or h(u,k+d[i+1],i+1)or h(u,k+d[i],i))or'',s][s[k]>'X'and k==x]
return h(s,x,0)


Input is a single string using '~' for ocean, 'X' for land, and 'o' for the boundary. (Using 'X' saves a byte for '>' instead of '==')

def f(s):
w=s.find('\n')+1                         # width of one row
x=s.find('X')-w                          # starting point
d=[1,w+1,w,w-1,-1,-w-1,-w,-w+1]*2        # delta to add to current index to move in
# the 8 directions: E, SE, S, SW, W, NW,
# N, NE. Make it long to avoid
# lots of modulo operations in
#    the recursive calls

def h(s,k,i):                            # s is the island string, k the current
# position, i the direction index
if s[k]>'X'and k==x:                 # if back at the begining,
return s                         #   return the map

elif 'X'>s[k] and i<8:               # if there is water here, and haven't
#  looped around,
u=s[:k]+'o'+s[k+1:]              #  make a new map with an 'o' in the
#  current spot

r = h(u,k+d[i+2],i+2)            # try a 90 degree right turn
if r: return r

r = h(u,k+d[i+1],i+1)            # try a 45 degree turn
if r: return r

r= h(u,k+d[i],i)                 # try straight ahead
if r: return r

return ''                            # this path failed

return h(s,x,0)

• @DLosc fixed. (should I delete the old answer?) – RootTwo Mar 30 '17 at 3:14
• Nice! (Yes, you should remove the old answer--if anyone wants to see it, they can look at the post's revision history.) – DLosc Mar 30 '17 at 3:18

# C# 7 - 414369 327 bytes

Edit: switched to 1D looping, computing i and j on the fly

Edit: changed input method, collapsed lookup table, and switched to well defined initial bounds ...and removed the pointless space in the last outer for-loop

using C=System.Console;class P{static void Main(){var D=C.In.ReadToEnd().Replace("\r","");int W=D.IndexOf('\n')+1,H=D.Length,z=H,k,q,c;int P()=>z%W*(k%3-1)+z/W*(k/3-1)+H;var B=new int[9];for(;z-->0;)for(k=9;k-->0&D[z]%7<1;)if(B[k]<=P())B[k]=P()+1;for(;++z<H;C.Write(q>9?'o':D[z]))for(q=k=9;k-->0;)q*=(c=P()-B[k])>0?0:c<0?1:2;}}


Try It Online

Complete program, takes input to standard in, prints it to standard out, uses #, ., and o. For each cell, it computes a 'profile' (which is the distance over 8 directions (it appears to compute a ninth for convenience, but this is always 0), and records a maximum of each of these. It then writes out the whole map again, and replaces any cell which is both on a boundary and not outside any with a 'o'. The commented code below explains how it all works.

As per my answer to Save the Geese from Extinction, this produces the smallest octagon (valid circumnavigation with largest area) which bounds the island.

Note: that for once in my life I'm using something from the current decade, and this code requires C# 7 to compile. If you do not have C# 7, there is one line that will need to be replaced, which is clearly marked in the code.

Example usage and output:

type t7.txt | IslandGolf1.exe

.........ooooooooooo....
........o....#......o...
.......o...#.#.##...#o..
......o....#.#.###.##.o.
.....o....########.##..o
....o.....############.o
...o.#....############.o
..o#.###.##############o
.o##.##################o
o.####################.o
o..##################..o
o.##################...o
o...################...o
o###################...o
o#####################.o
o.##################..o.
o####################o..
o#...##############.o...
o##...#############o....
o#.....###....#...o.....
.o.....#.........o......
..ooooooooooooooo.......


Formatted and commented code:

using C=System.Console;

class P
{
static void Main()
{
// \n 10
// # 35
// . 46
// o 111

int W=D.IndexOf('\n')+1, // width
H=D.Length, // length
z=H, // position in map (decomposed into i and j by and for P)
k, // bound index
q, // bound distance, and later cell condition (0 -> outside, 8 -> inside, >8 -> on boudary)
c; // (free), comparison store

// 'indexes' into a profile for the point z at index k
// effectively {i=z%W,j=z/W,-i,-j,i+j,j-i,-i-j,i-j,0}[k] (inside order is a bit different) (0 const is always treated as 'inside bounds')
// each non-zero-const entry describes the distance in one of the 8 directions: we want to maximise these to find the 'outer bounds'
// the non-zero-const bounds describe 8 lines, together an octogen
int P()=>z%W*(k%3-1)+z/W*(k/3-1)+H; // new C#7 local method syntax (if you don't have C#7, you can test this code with the line below instead)
//k=0;System.Func<int>P=()=>z%W*(k%3-1)+z/W*(k/3-1)+H; // old lambda syntax (must pre-assign k to make static checker happy)

var B=new int[9]; // our current bounds, each is initially null (must only call P() when on a #)
// B[k] starts off a 0, P() has a +H term, and W+(H/W)<H for W >= 3, so B[k] is assigned the first time we compare it (H-i-j always > 0)

for(;z-->0;) // for each cell
for(k=9;k-->0& // for each bound
D[z]%7<1;) // if this cell is #
if(B[k]<=P())B[k]=P()+1; // update bound if necessary (add one so that we define the bound _outside_ the hashes)
// z=-1
for(;++z<H; // for each cell
C.Write(q>9?'o':D[z])) // print the cell (if q > 9, then we are on the bounds, otherwise, spit out whatever we were before)
// check we are not 'outside' any of the bounds, and that we are 'on' atleast one of them
for(q=k=9;k-->0;) // for each bound
q*=(c=P()-B[k])>0?0: // outside bound (q=0)    (??0 is cheaper than (int) or .Value)
c<0?1: // inside (preserve q)
2; // on bound (if q != 0, then q becomes > 9)
}
}

• largest octagon? or smallest? – Display Name Mar 23 '17 at 16:42
• @SargeBorsch thanks, fixed the text. – VisualMelon Mar 23 '17 at 16:46