Prep for lecture tomorrow: recap Lecture 18

For tomorrow's lecture (#19), please recall the definition of metrics embedding into trees (problem #6 from HW4). Also, have a look at the correctness proof for low-diameter decompositions posted on Thursday, and try to prove the following extension of that result:

Extension: For a vertex $v$ and radius $r$, let $B(v,r) = \{ w | d_{vw} \leq r\}$ be the "ball" of all vertices within distance $r$ to $v$. Extend the proof from Lecture 16 to show that the probability of an edge $(u,v)$ being cut is at most

$O(1) \times (d_{uv} / r) \times \log (|B(u,2r)| / |B(u,r/4)|)$.

(Hint: The previous proof sums $1/j$ for all $j$ from $1$ to $n$ to get $H_n$. Can you show it suffices to sum over a smaller range? You may also want to consider cases depending whether or not $d_{uv}$ is at least $r/4$.)