  We consider a multi-level aggregation problem in a weighted rooted tree,
studied recently by Bienkowski et al. (2015). In this problem requests arrive
over time at the nodes of the tree, and each request specifies a deadline. A
request is served by sending it to the root before its deadline at a cost equal
to the weight of the path from the node in which it resides to the root.
However, requests from different nodes can be aggregated, and served together,
so as to save on cost. The cost of serving an aggregated set of requests is
equal to the weight of the subtree spanning the nodes in which the requests
reside. Thus, the problem is to find a competitive online aggregation algorithm
that minimizes the total cost of the aggregated requests. This problem arises
naturally in many scenarios, including multicasting, supply-chain management
and sensor networks. It is also related to the well studied TCP-acknowledgement
problem and the online joint replenishment problem.
  We present an online $O(D)$-competitive algorithm for the problem, where $D$
is the depth, or number of levels, of the aggregation tree. This result
improves upon the $D^2 2^D$-competitive algorithm obtained recently by
Bienkowski et al. (2015).
