  We give two fully dynamic algorithms that maintain a
$(1+\varepsilon)$-approximation of the weight $M$ of a minimum spanning forest
(MSF) of an $n$-node graph $G$ with edges weights in $[1,W]$, for any
$\varepsilon>0$.
  (1) Our deterministic algorithm takes $O({W^2 \log W}/{\varepsilon^3})$
worst-case update time, which is $O(1)$ if both $W$ and $\varepsilon$ are
constants. Note that there is a lower bound by Patrascu and Demaine (SIAM J.
Comput. 2006) which shows that it takes $\Omega(\log n)$ time per operation to
maintain the exact weight of an MSF that holds even in the unweighted case,
i.e. for $W=1$. We further show that any deterministic data structure that
dynamically maintains the $(1+\varepsilon)$-approximate weight of an MSF
requires super constant time per operation, if $W\geq (\log n)^{\omega_n(1)}$.
  (2) Our randomized (Monte-Carlo style) algorithm works with high probability
and runs in worst-case $O(\log W/ \varepsilon^{4})$ update time if $W=
O({(m^*)^{1/6}}/{\log^{2/3} n})$, where $m^*$ is the minimum number of edges in
the graph throughout all the updates. It works even against an adaptive
adversary. This implies a randomized algorithm with worst-case $o(\log n)$
update time, whenever $W=\min\{O((m^*)^{1/6}/\log^{2/3} n), 2^{o({\log n})}\}$
and $\varepsilon$ is constant. We complement this result by showing that for
any constant $\varepsilon,\alpha>0$ and $W=n^{\alpha}$, any (randomized) data
structure that dynamically maintains the weight of an MSF of a graph $G$ with
edge weights in $[1,W]$ and $W = \Omega(\varepsilon m^*)$ within a
multiplicative factor of $(1+\varepsilon)$ takes $\Omega(\log n)$ time per
operation.
