In this problem, Farmer John is building a fence on a 2D grid. He wants to count the number of gates he needs to build in order to make sure that he can travel from any location on the farm to any other one. Gates are necessary to cross between two squares which are separated by a fence.

Because the fence is small, we can build the fence directly and then count the number of distinct regions Farmer John's farm is broken up into, say using a "floodfill" approach. If it is broken up into $N$ distinct regions, then Farmer John needs to build $N-1$ gates, since a gate can only connect make it possible to travel between two regions.

In terms of storing the fence, we opt to use a two-dimensional array of booleans to track which locations have fence built on them. Instead of having fence pieces be unit length, we double the length of the fence. Therefore, distinct squares of the fence are at points $(x, y)$ where both $x$ and $y$ are odd, and to check if there is a fence between those two points, you simply inspect the point in between.

Here is my Java code demonstrating this.

import java.io.*; import java.util.*; public class gates { public static void main(String[] args) throws IOException { BufferedReader br = new BufferedReader(new FileReader("gates.in")); PrintWriter pw = new PrintWriter(new BufferedWriter(new FileWriter("gates.out"))); int n = Integer.parseInt(br.readLine()); String s = br.readLine(); int currX = 1002; int currY = 1002; boolean[][] isFence = new boolean[2005][2005]; isFence[currX][currY] = true; for(int i = 0; i < s.length(); i++) { int dirX = 0, dirY = 0; if(s.charAt(i) == 'N') { dirX = -1; } else if(s.charAt(i) == 'S') { dirX = 1; } else if(s.charAt(i) == 'W') { dirY = -1; } else { dirY = 1; } for(int a = 0; a < 2; a++) { currX += dirX; currY += dirY; isFence[currX][currY] = true; } } int ret = -1; int[] dx = new int[]{-1,1,0,0}; int[] dy = new int[]{0,0,-1,1}; for(int i = 0; i < isFence.length; i++) { for(int j = 0; j < isFence[i].length; j++) { if(isFence[i][j]) { continue; } ret++; LinkedList<Point> q = new LinkedList<Point>(); q.add(new Point(i, j)); isFence[i][j] = true; while(!q.isEmpty()) { Point curr = q.removeFirst(); for(int k = 0; k < dx.length; k++) { int nx = curr.x + dx[k]; int ny = curr.y + dy[k]; if(nx >= 0 && nx < isFence.length && ny >= 0 && ny < isFence[nx].length && !isFence[nx][ny]) { isFence[nx][ny] = true; q.add(new Point(nx, ny)); } } } } } pw.println(ret); pw.close(); } static class Point { public int x,y; public Point(int xIn, int yIn) { x = xIn; y = yIn; } } }(Alternative Solution Analysis by Lazar Ilic)

In this problem, Farmer John creates a number of discrete regions in his farm. Adding a gate between two previously disconnected regions will combine them so the number of regions will be reduced by one. Thus the total number of gates needed is the number of initially disconnected regions minus one. It is important we note that the region outside all the fences counts as a region: indeed if there were no fenced in region we would need no gates, if we had one fenced in region we would need one gate, etc. As it happens there is a result on planar graphs called Euler's formula which states that $v-e+f=2$, where $v$, $e$, and, $f$ are the number of vertices, edges, and faces (regions) respectively. Furthermore the end state of the farm will be a planar graph where fences are edges and endpoints of fences are vertices since, by definition, no two fences will intersect at any point that is not a vertex. So, in our problem we want to find $f-1=e-v+1$. This enables us to cast the problem as one of vertices and edges instead of faces. Indeed all that remains is for us to count the number of distinct vertices and edges. In the code below I used the HashSet to avoid redundancies and used a few variables to keep track of the $(x,y)$ values of the current point where Farmer John resides and of the previous point. Then we add the vertex and the edge (where each is a uniquely defined string with edges are composed of the vertices ordered in terms of their South/West to North/East ends) and move along through the loop.

import java.io.*; import java.util.*; public class gates { public static void main(String[] args) throws Exception{ Scanner in=new Scanner(new File("gates.in")); PrintStream out=new PrintStream(new File("gates.out")); int a=in.nextInt(); String b=in.next(); int x=0; int y=0; int xpre; int ypre; Set edges=new HashSet(); Set vertices=new HashSet(); vertices.add(x+" "+y); for(int c=0;c<a;c++) { xpre=x; ypre=y; if(b.charAt(c)=='N') { x=xpre; y=ypre+1; } else if(b.charAt(c)=='S') { x=xpre; y=ypre-1; } else if(b.charAt(c)=='E') { x=xpre+1; y=ypre; } else { x=xpre-1; y=ypre; } vertices.add(x+" "+y); if(b.charAt(c)=='N' || b.charAt(c)=='E') { edges.add(xpre+" "+ypre+" "+x+" "+y); } else { edges.add(x+" "+y+" "+xpre+" "+ypre); } } out.println(edges.size()-vertices.size()+1); } }