  We propose a modified decomposition algorithm (MDA) to solve the asymptotic
communication for omniscience (CO) problem where the communication rates could
be real or fractional. By starting with a lower estimation of the minimum
sum-rate, the MDA algorithm iteratively updates the estimation by the optimizer
of a Dilworth truncation problem until the minimum is reached with a
corresponding optimal rate vector. We also propose a fusion method
implementation of the coordinate-wise saturation capacity algorithm
(CoordSatCapFus) for solving the Dilworth truncation problem, where the
minimization is done over a fused user set with a cardinality smaller than the
original one. We show that the MDA algorithm is less complex than the existing
ones. In addition, we show that the non-asymptotic CO problem, where the
communication rates are integral, can be solved by one more call of the
CoordSatCapfus algorithm. By choosing a proper linear ordering of the user
indices in the MDA algorithm, the optimal rate vector is also the one with the
minimum weighted sum-rate.
