 The best known algorithms for trace reconstruction require \exp%DIFDELCMD < \left(%%% O(n^{1/3)}%DIFDELCMD < \right) %%% samples. For population recovery under the deletion channel , Ban et . al. provided an algorithm that could learn \ell-sparse distributions over strings using \exp %DIFDELCMD < \left(%%% \big( n^{1/2} \cdot (\log n)^{O(\ell)} %DIFDELCMD < \right) %%% \big) samples.
<clarity>  The best known algorithms for trace reconstruction require \exp%DIFDELCMD < \left(%%% O(n^{1/3)}%DIFDELCMD < \right) %%% samples. For population recovery under the deletion channel , Ban et . al. provided an algorithm that could learn \ell-sparse distributions over strings using \exp %DIFDELCMD < \left(%%% \big( n^{1/2} \cdot (\log n)^{O(\ell)} %DIFDELCMD < \right) %%% \big) samples.
 The best known algorithms for trace reconstruction require \exp%DIFDELCMD < \left(%%% O(n^{1/3)}%DIFDELCMD < \right) %%% under the deletion channel , Ban et . al. provided an algorithm that could learn \ell-sparse distributions over strings using \exp %DIFDELCMD < \left(%%% \big( n^{1/2} \cdot (\log n)^{O(\ell)} %DIFDELCMD < \right) %%% \big) samples.