Before revision:  We present a new algorithm for sampling  arbitrary motifs which, up to   (\log n) factors, for any motif H whose decomposition contains at least two components or at least one star, is always preferable .
After revision:  We present a new algorithm for sampling and approximately counting arbitrary motifs which, up to  (\log n) factors, for any motif H whose decomposition contains at least two components or at least one star, is always preferable .