We can avoid reading the whole page for each comparison, even in the worst case.

Say that they have a binary tree. Each left edge is labelled 0, and each
right edge is labelled 1. Whenever we read a bit from the page we follow the
edge with the label of the bit. So for example, if we look for the zero page
in the tree then we navigate to the leftmost child node.

This is clearly very inefficient, as this simple algorithms needs exactly
(4096*8) branches to find any page. However we read the page exactly once
for all comparisons. And we can optimize it, e.g. if we have a chain of
nodes with only single children we can merge them into a single edge.

Another approach would be to make use of the fact that if say, we know all
child pages of the current node start with "Foo" we don't need to compare
the first 3 characters anymore. As such, we'd usually only have to access
each cache-line once.