## Lecture 15(b): Illustrative Examples

Here’s an example from the Christiano et al. paper that shows that the width of the basic electrical flow oracle could be as large as ${\sqrt{m}}$. The effective resistance of the black edges is ${1}$, so half the current flows on the top red edge. If we set ${F = k+1}$ (which is the max-flow), this means a current of ${\Theta(\sqrt{m})}$ goes on the top edge.

And here’s another example (via Gary, who got this one from Olek Madry) that shows that the fancier algorithm does need ${\Omega(m^{1/3})}$ iterations, and does need to delete ${\Omega(m^{1/3})}$ edges. Again, in this example, ${m = \Theta(n)}$.

Again, each black gadget has a unit effective resistance, and if you do the calculation, the effective resistance between ${s}$ and ${t}$ tends to the golden ratio. If we set we set ${F = n^{1/3}}$ (which is almost the max-flow), this means a constant fraction of the current, or about ${\Theta(n^{1/3})}$ goes on the edge ${e_1}$. Once that is deleted, the next red edge ${e_2}$ carries a lot of current, etc. Until all red edges get deleted.