Why is the complexity of A* exponential in memory?

Question

Wikipedia says on A* complexity the following (link here):

More problematic than its time complexity is A*’s memory usage. In the worst case, it must

Answer 1

I am not an expert, but I studied the Wikipedia article for a while and my explanation would be this one (hope i have understood it well :)

Say, we have a 4x4 matrix of nodes.
A,B,C,D are the directions we can take at a given time (North,South,East,West)
The A* algorithm starts searching.

A
Queue: B,C,D
AA
Queue: B,C,D,AB,AC,AD
AAA-->Goal
Queue: B,C,D,AB,AC,AD,AAB,AAC,AAD
The goal is reached but there are still other possibilities to consider.

D
Queue: B,C,AB,AC,AD,AAB,AAC,AAD
DC
Queue: B,C,AB,AC,AD,AAB,AAC,AAD,DA,DB,DD
DCA
Queue: B,C,AB,AC,AD,AAB,AAC,AAD,DA,DB,DD,DCB,DCC,DCD
DCAB-->Goal
Queue: B,C,AB,AC,AD,AAB,AAC,AAD,DA,DB,DD,DCB,DCC,DCD,DCAA,DCAC,DCAD
Etc etc

As you can see, for every step taken, three more nodes are added to the queue.
Since A* follows only acyclic paths [1], the maximum number of steps per route is 15.
The max number of possible routes in this case is 3^15, or directions^nodes.
Since every route has 15 steps,the worst case steps taken is 15*3^15.
In the absolute worst case, every step ever taken is "wrong".
In that case 3*15*3^15 nodes are in the queue before finding the answer.
So the worst case amount of nodes that needs to be kept in memory is a constant, to the power of the number of nodes available. In other words the memory use is exponential to the amount of nodes.

[1] http://www.autonlab.org/tutorials/astar08.pdf, slide 15