Problem 6 (3 points). Let $L_n$ be a directed line on the vertices {1,2,..., n} with directed edges ($i$, $i$ + 1) for every $i$ ? {1,2,..., n-1}. A 2-spanner for $L_n$ is a graph S obtained by adding directed edges (from the smaller to the larger indices, so that the directions of the paths are preserved) to $L_n$ such that for every pair $i$, $j$ ? {1,2,..., n} with $i$ < $j$ there is a path of length at most 2 in S. Show an algorithm that builds a 2-spanner for $L_n$ with O($n$ log $n$) edges. Prove that your algorithm indeed produces a 2-spanner.
