  We study the stochastic online metric matching problem. In this problem, $m$
servers and $n$ requests are located in a metric space, where all servers are
available upfront and requests arrive one at a time. In particular, servers are
adversarially chosen, and requests are independently drawn from a known
distribution. Upon the arrival of a new request, it needs to be immediately and
irrevocably matched to a free server, resulting in a cost of their distance.
The objective is to minimize the total matching cost.
  In this paper, we show that the problem can be reduced to a more accessible
setting where both servers and requests are drawn from the same distribution by
incurring a moderate cost. Combining our reduction with previous techniques,
for $[0, 1]^d$ with various choices of distributions, we achieve improved
competitive ratios and nearly optimal regrets in both balanced and unbalanced
markets. In particular, we give $O(1)$-competitive algorithms for $d \geq 3$ in
both balanced and unbalanced markets with smooth distributions. Our algorithms
improve on the $O((\log \log \log n)^2)$ competitive ratio of
\cite{DBLP:conf/icalp/GuptaGPW19} for balanced markets in various regimes, and
provide the first positive results for unbalanced markets.
