In the words of (http://www.cs.utah.edu/~suresh/5962/lectures/17.pdf, section 17.2), "Each $f(x)$ can be interpreted as deﬁning a hyperplane in $R^n$. Thus, tracing a path through the tree computes the intersection of the half-planes deﬁned by the nodes touched by the path."
I fail to visualize how path tracing is done? I would be glad to see it explained through the presentation of the path in a 2-dimensional space.
I do understand that $x_1,x_2,...,x_n$ is a point in the $R^n$ dimensional space. But I don't get how Figure 17.1 in (http://www.cs.utah.edu/~suresh/5962/lectures/17.pdf) helped in proving the lower bounds of Element Uniqueness as $\Omega(n\log n)$. I also don't get the implication of $\#F$ being connected components; why can't they simply be called solutions?
Unfortunately, reading online resources did not help me much understand the aforementioned concepts.
Thank you in advance.