  In the online non-metric variant of the facility location problem, there is a
given graph consisting of a set $F$ of facilities (each with a certain opening
cost), a set $C$ of potential clients, and weighted connections between them.
The online part of the input is a sequence of clients from $C$, and in response
to any requested client, an online algorithm may open an additional subset of
facilities and must connect the given client to an open facility.
  We give an online, polynomial-time deterministic algorithm for this problem,
with a competitive ratio of $O(\log |F| \cdot (\log |C| + \log \log |F|))$. The
result is optimal up to loglog factors. Our algorithm improves over the
$O((\log |C| + \log |F|) \cdot (\log |C| + \log \log |F|))$-competitive
construction that first reduces the facility location instance to a set cover
one and then later solves such instance using the deterministic algorithm by
Alon et al. [TALG 2006]. This is an asymptotic improvement in a typical
scenario where $|F| \ll |C|$.
  We achieve this by a more direct approach: we design an algorithm for a
fractional relaxation of the non-metric facility location problem with
clustered facilities. To handle the constraints of such non-covering LP, we
combine the dual fitting and multiplicative weight updates approach. By
maintaining certain additional monotonicity properties of the created
fractional solution, we can handle the dependencies between facilities and
connections in a rounding routine.
  Our result, combined with the algorithm by Naor et al. [FOCS 2011] yields the
first deterministic algorithm for the online node-weighted Steiner tree
problem. The resulting competitive ratio is $O(\log k \cdot \log^2 \ell)$ on
graphs of $\ell$ nodes and $k$ terminals.
