  The generalized $k$-server problem is an extension of the weighted $k$-server
problem, which in turn extends the classic $k$-server problem. In the
generalized $k$-server problem, each of $k$ servers $s_1, \dots, s_k$ remains
in its own metric space $M_i$. A request is a tuple $(r_1,\dots,r_k)$, where
$r_i \in M_i$, and to service it, an algorithm needs to move at least one
server $s_i$ to the point $r_i$. The objective is to minimize the total
distance traveled by all servers.
  In this paper, we focus on the generalized $k$-server problem for the case
where all $M_i$ are uniform metrics. We show an $O(k^2 \cdot \log
k)$-competitive randomized algorithm improving over a recent result by Bansal
et al. [SODA 2018], who gave an $O(k^3 \cdot \log k)$-competitive algorithm. To
this end, we define an abstract online problem, called Hydra game, and we show
that a randomized solution of low cost to this game implies a randomized
algorithm to the generalized $k$-server problem with low competitive ratio.
  We also show that no randomized algorithm can achieve competitive ratio lower
than $\Omega(k)$, thus improving the lower bound of $\Omega(k / \log^2 k)$ by
Bansal et al.
