  In a recent breakthrough, Paz and Schwartzman (SODA'17) presented a
single-pass ($2+\epsilon$)-approximation algorithm for the maximum weight
matching problem in the semi-streaming model. Their algorithm uses $O(n\log^2
n)$ bits of space, for any constant $\epsilon>0$.
  We present two simplified and more intuitive analyses, for essentially the
same algorithm, which also improve the space complexity to the optimal bound of
$O(n\log n)$ bits --- this is optimal as the output matching requires
$\Omega(n\log n)$ bits. Our analyses rely on a simple use of the primal-dual
method and a simple accounting method.
