Is it possible to solve the shortest path problem by hardware? And is it worth it? Okay, I recently thought of an idea. Suppose given any map you create an equivalent circuit where there are same number of vertices (intersection of wires) and  length of the path is proportional to resistance between the wires, the start point is attached to a battery and exit is grounded. By measuring the flow of electricity and basically following where the maximum current (of course some a different algorithm will be used when the current is same in two different paths) goes through one will figure out the shortest path.   
My Question: Is there any flaw in my thinking? 
2nd Question: Is it worth building such a machine? 
(one which creates equivalent circuit of a map)
Thanks
 A: As Rahul Narain points out in the comments, your transformation doesn't work, because the combined resistance of two paths between the same two vertices is in general lower than the minimum of the two individual resistances.
But there are ways to build a machine which solves the shortes path problem quickly, if you allow the machine size to grow with the size of the problem. Here's one approach that works in theory, although probably not in practice.

Assign numbers to all the edges, and represent edge $i$ by a wire whose length is proportional to the length assigned to the edge, and whose resistance is $2^i$. Connect one vertex to a signal generator and another vertex to ground via some resistor, and also to an oscilloscope. Now let the signal generator output a short impulse, and watch the oscilloscope. From the latency between the time at which the impuls starts and the time at which it appears on the oscilloscope you can determine the shortest path's length. From the amplitude of that first impulse you can determine the sum of resistances along that path. Since the individual resistances are all powers of $2$, looking at the binary representation of the resistance tells you which edges are included in the shortest path.
