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 $��(\log n)$ 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 $W\geq (\log n)^{��_n(1)}$. (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 $\varepsilon,��>0$ and $W=n^��$, any (randomized) data structure that dynamically maintains the weight of an MSF of a graph $G$ with edge weights in $[1,W]$ and $W = ��(\varepsilon m^*)$ within a multiplicative factor of $(1+\varepsilon)$ takes $��(\log n)$ time per operation.

Partial results have been reported in arXiv:1907.04745