The well-known trace reconstruction problem is the problem of inferring an unknown source string $x \in \{0,1\}^n$ from independent ``traces'', i.e. copies of $x$ that have been corrupted by a $\delta$-deletion channel which independently deletes each bit of $x$ with probability $\delta$ and concatenates the surviving bits. The current paper considers the extreme data-limited regime in which only a single trace is provided to the reconstruction algorithm. In this setting exact reconstruction is of course impossible, and the question is to what accuracy the source string $x$ can be approximately reconstructed.
We give a detailed study of this question, providing algorithms and lower bounds for the high, intermediate, and low deletion rate regimes in both the worst-case ($x$ is arbitrary) and average-case ($x$ is drawn uniformly from $\{0,1\}^n$) models. In several cases the lower bounds we establish are matched by computationally efficient algorithms that we provide.