  Designing energy-efficient all-to-all multicasting protocols is of of great
importance for multi-hop wireless networks such as wireless sensor networks and
wireless ad hoc networks. In an all-to-all multicast session, there exists a
set of wireless destination nodes, and each destination node needs to send some
data packets to all other destination nodes. We consider the problem of
building a shared multicast tree spanning the destination nodes such that the
total energy consumption of realizing an all-to-all multicast session using the
shared multicast tree is minimized. Since building such a multicast tree has
been proved to be NP-complete, we provide both centralized and distributed
approximation algorithms with provable approximation ratios for it. When the
transmission power of each wireless node is fixed, our centralized and
distributed algorithms have the approximation ratios of $4ln(\Delta+1)+7$ and
13, respectively, where $\Delta$ is the maximum node degree in the network.
When the transmission power of each wireless node is adjustable, both of our
centralized and distributed algorithms have the constant approximation ratio of
145.
