  We consider an \emph{approximate} version of the trace reconstruction
problem, where the goal is to recover an unknown string $s\in\{0,1\}^n$ from
$m$ traces (each trace is generated independently by passing $s$ through a
probabilistic insertion-deletion channel with rate $p$). We present a
deterministic near-linear time algorithm for the average-case model, where $s$
is random, that uses only \emph{three} traces. It runs in near-linear time
$\tilde O(n)$ and with high probability reports a string within edit distance
$O(\epsilon p n)$ from $s$ for $\epsilon=\tilde O(p)$, which significantly
improves over the straightforward bound of $O(pn)$.
  Technically, our algorithm computes a $(1+\epsilon)$-approximate median of
the three input traces. To prove its correctness, our probabilistic analysis
shows that an approximate median is indeed close to the unknown $s$. To achieve
a near-linear time bound, we have to bypass the well-known dynamic programming
algorithm that computes an optimal median in time $O(n^3)$.
