  For a group of users in $V$ where everyone observes a component of a discrete
multiple random source, the process that users exchange data so as to reach
omniscience, the state where everyone recovers the entire source, is called
communication for omniscience (CO). We first consider how to improve the
existing complexity $O(|V|^2 \cdot \text{SFM}(|V|)$ of minimizing the sum of
communication rates in CO, where $\text{SFM}(|V|)$ denotes the complexity of
minimizing a submodular function. We reveal some structured property in an
existing coordinate saturation algorithm: the resulting rate vector and the
corresponding partition of $V$ are segmented in $\alpha$, the estimation of the
minimum sum-rate. A parametric (PAR) algorithm is then proposed where, instead
of a particular $\alpha$, we search the critical points that fully determine
the segmented variables for all $\alpha$ so that they converge to the solution
to the minimum sum-rate problem and the overall complexity reduces to $O(|V|
\cdot \text{SFM}(|V|))$.
  For the successive omniscience (SO), we consider how to attain local
omniscience in some complimentary user subset so that the overall sum-rate for
the global omniscience still remains minimum. While the existing algorithm only
determines a complimentary user subset in $O(|V| \cdot \text{SFM}(|V|))$ time,
we show that, if a lower bound on the minimum sum-rate is applied to the
segmented variables in the PAR algorithm, not only a complimentary subset, but
also an optimal rate vector for attaining the local omniscience in it are
returned in $O(|V| \cdot \text{SFM}(|V|))$ time.
