Since ~W~ and ~H~ are fairly small and each vertex has at most six neighbors, we can construct a graph which would consist of "black" and "white" vertices. Each vertex in the maze will be represented by two vertices (black and white) in the graph. If a vertex is reached after passing a white circle, then it is white, if after passing a black circle – it is black.

While reading the input data, we connect "black" vertices with adjacent "white" ones if there exists an edge with white circle connecting them, and vice versa. No two vertices of the same "color" are connected directly which implies that moving in this graph satisfies the rule of alternating circles. We then perform breadth-first search starting from entry vertices (both "black" and "white"). The answer is the smaller value of distances of "black" and "white" exit vertices.

The task is easy if one finds a clever way to represent the graph.