## Aug 19, 2011

### An example of the breadth-first search

You can use the breadth-first search to process every state like the depth-first search.

When you want to search the shortest path for search space like a maze, the breadth-first search is a good solution for you because you have to search the same state so many times with the depth first-search.

The breadth-first search has a weak point. It needs memory that is proportional to a number of state.

This is an example of the breadth-first search:

- Description
You have a maze which size is M * N. The maze consists of path and wall.
You can move 4 direction (up, down, right and left) in one turn.
Solve the shortest path from the start to the goal.
- Constraint
N, M >= 100
- Sample Input
10 10
#S######.#
......#..#
.#.##.##.#
.#........
##.##.####
....#....#
.#######.#
....#.....
.####.###.
....#...G#
- Sample Output
22
My solution is like this. You can find it on my github repository.
```#include
#include
#include
using namespace std;

const int INF = 1000000;
const int MAX_N = 100;
const int MAX_M = 100;

typedef pair P;
char maze[MAX_N][MAX_M+1];  // maze
int n, m;   // size of maze
int sx, sy; // start position
int gx, gy; // goal position

//shortest path for each point
int d[MAX_N][MAX_M];

//vector for for moving direction
int dx = {1, 0, -1, 0};
int dy = {0, 1, 0, -1};

//solve the shortest path from (sx, sy) to (gx, gy)
int bfs() {
queue q;
//initialize every point with INF before search
for(int i = 0; i < n; i++) {
for (int j = 0; j < m; j++) {
d[i][j] = INF;
}
}
q.push(P(sx,sy));
d[sx][sy] = 0;

while(q.size()) {
P tmp = q.front();
q.pop();
//if you find goal then you finish search
if ((tmp.first == gx)&&(tmp.second == gy)){
break;
}
for(int i = 0; i < 4; i++) {
int nx = tmp.first + dx[i];
int ny = tmp.second + dy[i];

if ((0 <= nx) && (nx < n) &&
(0 <= ny) && (ny < m) &&
(maze[nx][ny] != '#') &&
(d[nx][ny] == INF)) {
// if d[nx][ny] == INF then it is a position that
q.push(P(nx,ny));
d[nx][ny] = d[tmp.first][tmp.second] + 1;
}
}
}
return d[gx][gy];
}

void solve() {
int res = bfs();
cout << res << endl;
}

int main(int argc, char* argv[]) {
if (argc != 2) {
cout << "./a.out ";
return 1;
}
ifstream ifs(argv);
if (!ifs) {
cout << "can't open file:" << argv << endl;
return 1;
}
string buf;
getline(ifs, buf);
sscanf(buf.c_str(), "%d %d", &n, &m);

for(int i = 0; i < n; i++) {
getline(ifs, buf);
for (int j = 0; j < m; j++) {
maze[i][j] = buf[j];
if (maze[i][j] == 'S') {
sx = i;
sy = j;
}
if (maze[i][j] == 'G') {
gx = i;
gy = j;
}
}
}

solve();

return 0;
}
```