Trees

Now, consider a tree of PEs with network processors. When the results are not to be returned all PEs must stop computing simultaneously. We assume that the order of processor activation is given, data is distributed to a receiver in one chunk, and then redistributed to its descendants. $P_1$ is the originator and the root of the tree. Each node of data distribution tree (except leaves) becomes an originator for its descendants. Let $succ(i)$ denote the set of the successors of $P_i$ (not necessarily immediate). Let $pred(i)$ denote a PE activated immediately before $P_i$ by the same node $P_j$ as $P_i$ (i.e. both $P_i$ and $pred(i)$ received their data from $P_j$). When $P_i$ was the first PE activated by $P_j$ then $pred(i)=P_j$. For each pair of PEs activated one after another we can formulate a modified version of equations (5):

$$\alpha_{pred(i)}A_{pred(i)}=%% S_{i}+(\alpha_{i}+\sum_{j\in ... ...(i)}\alpha_j)C_{i}+\alpha_iA_{i} {\rm\hspace{4mm}} i=2,\dots,m$$ (9)