  The Sensitivity Conjecture and the Log-rank Conjecture are among the most
important and challenging problems in concrete complexity. Incidentally, the
Sensitivity Conjecture is known to hold for monotone functions, and so is the
Log-rank Conjecture for $f(x \wedge y)$ and $f(x\oplus y)$ with monotone
functions $f$, where $\wedge$ and $\oplus$ are bit-wise AND and XOR,
respectively. In this paper, we extend these results to functions $f$ which
alternate values for a relatively small number of times on any monotone path
from $0^n$ to $1^n$. These deepen our understandings of the two conjectures,
and contribute to the recent line of research on functions with small
alternating numbers.
