Maze solver complexity - c++

Question

I have sevral questions about maze - solver algorithms:C
What is the time complexity of recursive (backtracking) maze-solver?(As an amount of paths in the matrix? - I can`t figure this number..)
What is the time complexity of BFS based maze-solver?(O(n^2)?) n is a dimension of the squared maze matrix?
What is the best algoritm to count the number of all possible paths from the source to the destination in the maze?
Can you propose the idea if and how an it be implemented using parallel computing (opecl/cuda)
Here is the class for my maze solver ,which has brute (recursive) and bfs based version.I implemented it and the questions are based on this maze-solver implementation
//MazeSolver.h
//#define N 5
typedef enum {BLACK,WHITE,GRAY,VISITED} color;
class MazeSolver
{
public:
MazeSolver(){}
struct Cell
{
unsigned int _x;
unsigned int _y;
Cell* _p;
Cell(unsigned int x = 0,unsigned int y = 0, Cell* p = NULL) : _x(x),_y(y),_p(p) {}
bool operator == (const Cell& c)
{
return _x == c._x && _y == c._y;
}
};
bool solveMazeBrute(color maze[][N],unsigned int n,int xS,int yS,int xD,int yD,std::list<Cell>& path);
bool solveMazeBFS(color maze[][N],unsigned int n,int xS,int yS,int xD,int yD,std::list<Cell>& path);
private:
std::queue<Cell* > _bfs;
std::vector<Cell* > _cells;
Cell* addCellBFS(color maze[][N],unsigned int n,int x,int y,Cell* p = NULL);
};
//MazeSolver.cpp
MazeSolver::Cell* MazeSolver::addCellBFS(color maze[][N],unsigned int n,int x,int y,Cell* p)
{
if (x >= 0 && x < n && y >= 0 && y < n && maze[x][y] == WHITE)
{
Cell* c = new Cell(x,y,p);
maze [x][y] = VISITED;
_bfs.push(c);
_cells.push_back(c);
return c;
}
return NULL;
}
bool MazeSolver::solveMazeBrute(color maze[][N],unsigned int n,int xS,int yS,int xD,int yD,std::list<MazeSolver::Cell>& path)
{
bool solved = false;
if (xS < 0 || xS >= n || yS < 0 || yS >= n || maze[xS][yS] == VISITED || maze[xS][yS] == BLACK)
{
return false;
}
Cell s(xS,yS);
Cell d(xD,yD);
if (s == d)
{
path.push_front(s);
return true;
}
maze[xS][yS] = VISITED;
if (solveMazeBrute(maze,n,xS + 1,yS,xD,yD,path) ||
solveMazeBrute(maze,n,xS - 1,yS,xD,yD,path) ||
solveMazeBrute(maze,n,xS,yS + 1,xD,yD,path) ||
solveMazeBrute(maze,n,xS,yS - 1,xD,yD,path))
{
path.push_front(s);
solved = true;
}
maze[xS][yS] = WHITE;
return solved;
}
bool MazeSolver::solveMazeBFS(color maze[][N],unsigned int n,int xS,int yS,int xD,int yD,std::list<Cell>& path)
{
Cell d(xD,yD);
addCellBFS(maze,n,xS,yS);
while(!_bfs.empty())
{
Cell* cur = _bfs.front();
if (*cur == d)
{
while (cur != NULL)
{
path.push_front(*cur);
cur = cur->_p;
}
return true;
}
_bfs.pop();
addCellBFS(maze,n,cur->_x - 1,cur->_y,cur);
addCellBFS(maze,n,cur->_x + 1,cur->_y,cur);
addCellBFS(maze,n,cur->_x,cur->_y - 1,cur);
addCellBFS(maze,n,cur->_x,cur->_y + 1,cur);
}
for(std::vector<Cell*>::iterator itC= _cells.begin();itC != _cells.end();++itC)
{
maze[(*itC)->_x][(*itC)->_y] = WHITE;
delete *itC;
}
return false;
}

Answer 1

Maybe we can find the target in O(n).
Let's imagine 5X5 matrix.
In each iteration we'll step single step ahead, we'll check the cell is valid and is not the end of maze, and mark it as "visited".
So, we'll start at the first cell (0,0). in next iteration we'll check the next layer, mean (0,1),(1,0), in next iteration we'll continue check the next layer (0,2),(1,1),(2,0). and so on.
So, we'll check each cell once only! and we'll find the end (the target) in n complexity.
Am i wrong?