From a graph-theoretical perspective, the problem of constructing multicast distribution paths, modeled as 'Steiner trees', is NP-complete. So, many heuristics-based algorithms are available to generate near-optimal trees. Typically, an algorithm first assigns the edge costs for all links, and then examines various candidate paths for interconnecting a given set of nodes. This strategy does not work well for the evolving multimedia application configurations (such as audio-video and image distributions) where it is often necessary to construct multiple distribution paths, viz., one per media data stream. This is because the overlapping of multiple tree segments over a common link forces the link cost to change, based on the delay and bandwidth characteristics of data streams flowing over these segments. Accordingly, algorithms that hitherto have assumed non-varying link costs during various phases of a run now need to take into account the variability of link costs as candidate trees with different levels of path overlapping are examined in a given run. In other words, as an algorithm runs by examining various paths, the link costs change. The paper embarks on a study of heuristics-based algorithms to tackle this 'modified Steiner tree' problem. The algorithms allow more cost-efficient routing of data than feasible otherwise with a classical treatment of the 'Steiner tree' problem.