The largest sawtooth subsequence

A sequence of numbers is called a sawtooth sequence if every element (except the first and last) is either greater than both its neighbors or less than both its neighbors. For example, the sequence \textbf{1}, \textbf{2}, \textbf{1}, \textbf{3}, \textbf{2} is a sawtooth sequence, while \textbf{1}, \textbf{2}, \textbf{3}, \textbf{1}, \textbf{2} is not, because \textbf{1} < \textbf{2} < \textbf{3}. Any sequence consisting of a single element is considered a sawtooth sequence. A sequence with two elements is a sawtooth sequence if its two elements are not equal.