Shortest path in a maze

Question

I'm developing a game similar to Pacman: consider this maze:

Each white square is a node from the maze where an object located at P, say X, is moving towards node A in the right-to-left direction. X cannot switch to its opposite direction unless it encounters a dead-end such as A. Thus the shortest path joining P and B goes through A because X cannot reverse its direction towards the rightmost-bottom node (call it C). A common A* algorithm would output:

to get to B from P first go rightward, then go upward;

which is wrong. So I thought: well, I can set the C's visited attribute to true before running A* and let the algorithm find the path. Obviously this method doesn't work for the linked maze, unless I allow it to rediscover some nodes (the question is: which nodes? How to discriminate from useless nodes?). The first thinking that crossed my mind was: use the previous method always keeping track of the last-visited cell; if the resulting path isn't empty, you are done. Otherwise, when you get to the last-visited dead-end, say Y, (this step is followed by the failing of A*) go to Y, then use standard A* to get to the goal (I'm assuming the maze is connected). My questions are: is this guaranteed to work always? Is there a more efficient algorithm, such as an A*-derived algorithm modified to this purpose? How would you tackle this problem? I would greatly appreciate an answer explaining both optimal and non-optimal search techniques (actually I don't need the shortest path, a slightly long path is good, but I'm curious if such an optimal algorithm running as efficiently as Dijkstra's algorithm exists; if it does, what is its running time compared to a non-optimal algorithm?)

EDIT For Valdo: I added 3 cells in order to generalize a bit: please tell me if I got the idea:

Answer 1

Good question. I can suggest the following approach.

Use Dijkstra (or A*) algorithm on a directed graph. Each cell in your maze should be represented by multiple (up to 4) graph nodes, each node denoting the visited cell in a specific state.

That is, in your example you may be in the cell denoted by P in one of 2 states: while going left, and while going right. Each of them is represented by a separate graph node (though spatially it's the same cell). There's also no direct link between those 2 nodes, since you can't switch your direction in this specific cell.

According to your rules you may only switch direction when you encounter an obstacle, this is where you put links between the nodes denoting the same cell in different states.

You may also think of your graph as your maze copied into 4 layers, each layer representing the state of your pacman. In the layer that represents movement to the right you put only links to the right, also w.r.t. to the geometry of your maze. In the cells with obstacles where moving right is not possible you put links to the same cells at different layers.

Update:

Regarding the scenario that you described in your sketch. It's actually correct, you've got the idea right, but it looks complicated because you decided to put links between different cells AND states.

I suggest the following diagram:

The idea is to split your inter-cell AND inter-state links. There are now 2 kinds of edges: inter-cell, marked by blue, and inter-state, marked by red.

Blue edges always connect nodes of the same state (arrow direction) between adjacent cells, whereas red edges connect different states within the same cell.

According to your rules the state change is possible where the obstacle is encountered, hence every state node is the source of either blue edges if no obstacle, or red if it encounters an obstacle (i.e. can't emit a blue edge). Hence I also painted the state nodes in blue and red.

If according to your rules state transition happens instantly, without delay/penalty, then red edges have weight 0. Otherwise you may assign a non-zero weight for them, the weight ratio between red/blue edges should correspond to the time period ratio of turn/travel.