Algorithm for Maze problem.

We'll solve the problem of finding and marking a solution path using recursion.

Remember that a recursive algorithm has at least 2 parts:

    Base case(s) that determine when to stop.

    Recursive part(s) that call the same algorithm (i.e., itself) to assist in solving the problem.

Recursive parts
Because our algorithm must be recursive, we need to view the problem in terms of similar sub problems. In this case, that means we need to "find a path" in terms of "finding paths."

Let's start by assuming there is already some algorithm that finds a path from some point in a maze to the goal, call it FIND-PATH(x, y).

Also suppose that we got from the start to position x=1, y=2 in the maze (by some method):

+#####
++...#
#+####
#.####
...#.G
##...#

    What we now want to know is whether there is a path from x=1, y=2 to the goal. If there is a path to the goal from x=1, y=2, then there is a path from the start to the goal (since we already got to x=1, y=2).

To find a path from position x=1, y=2 to the goal, we can just ask FIND-PATH to try to find a path from the North, East, South, and West of x=1, y=2:

    FIND-PATH(x=1, y=1) North
    FIND-PATH(x=2, y=2) East
    FIND-PATH(x=1, y=3) South
    FIND-PATH(x=0, y=2) West

Generalizing this, we can call FIND-PATH recursively to move from any location in the maze to adjacent locations. In this way, we move through the maze.
Base cases
It's not enough to know how to use FIND-PATH recursively to advance through the maze. We also need to determine when FIND-PATH must stop.

One such base case is to stop when it reaches the goal.

The other base cases have to do with moving to invalid positions. For example, we have mentioned how to search North of the current position, but disregarded whether that North position is legal. In order words, we must ask:

    Is the position in the maze (...or did we just go outside its bounds)?
    Is the position open (...or is it blocked with an obstacle)?

Now, to our base cases and recursive parts, we must add some steps to mark positions we are trying, and to unmark positions that we tried, but from which we failed to reach the goal:

FIND-PATH(x, y)

    if (x,y outside maze) return false
    if (x,y is goal) return true
    if (x,y not open) return false
    mark x,y as part of solution path
    if (FIND-PATH(North of x,y) == true) return true
    if (FIND-PATH(East of x,y) == true) return true
    if (FIND-PATH(South of x,y) == true) return true
    if (FIND-PATH(West of x,y) == true) return true
    unmarked x,y as part of solution path
    return false

All these steps together complete a basic algorithm that finds and marks a path to the goal (if any exists) and tells us whether a path was found or not (i.e., returns true or false). This is just one such algorithm--other variations are possible.

Note: FIND-PATH will be called at least once for each position in the maze that is tried as part of a path.

Also, after going to another position (e.g., North):

    if (FIND-PATH(North of x,y)¹ == true) return true²

if a path to the goal was found, it is important that the algorithm stops. I.e., if going North of x,y finds a path (i.e., returns true¹), then from the current position (i.e., current call of FIND-PATH) there is no need to check East, South or West. Instead, FIND-PATH just need return true² to the previous call.

Path marking will be done with the '+' symbol and unmarking with the 'x' symbol.
Using Algorithm
To use FIND-PATH to find and mark a path from the start to the goal with our given representation of mazes, we just need to:

    Locate the start position (call it startx, starty).
    Call FIND-PATH(startx, starty).
    Re-mark* the start position with 'S'.

*In the algorithm, the start position (marked 'S') needs to be considered an open position and must be marked as part of the path for FIND-PATH to work correctly. That is why we re-mark it at the end.


To be continued...