## 2 Related Work and Our Contributions  
In this section, we first present the existing work in the literature on algorithms related to aggregation delay for data gathering trees in sensor networks. We then motivate the need for analyzing the aggregation delay as a function of the type of the data gathering trees and thereby also exploring the diameter-aggregation delay tradeoff. Most of the work conducted in the literature focus on deriving theoretical bounds for the minimum and maximum aggregation delay as a function of the network parameters such as the number of nodes, maximum node degree, diameter of the network, transmission range of the nodes, _etc._, without considering the type of data gathering trees that are determined. For example, the paper [8] proposes O(\(\Delta+D\)/2) as the upper bound for aggregation delay where \(\Delta\) is the maximum node degree and \(D\) is the diameter of the network (in terms of the number of hops), and in [10], the authors showed that the upper bound for aggregation latency in a multi-sink scenario is O(\(\Delta+k*D\)/2), where \(k\) is the number of sinks. In [21], the authors show that the lower bound for the aggregation delay is max\(\left\{D\text{/2},\,\log_{2}^{n}\right\},\) where \(n\) is the number of nodes in the network. However, the aforementioned works do not analyze the possible differences in the minimum or maximum aggregation delays due to the type of the underlying data gathering trees involved. To the best of our knowledge, ours is the first such work to show that for a given network, the minimum aggregation delay principally depends on the type of the data gathering tree (bottleneck node weight-based or bottleneck link weight-based) and could be significantly different for the two categories of data gathering trees. Likewise, the focus of tradeoff analysis has been on the aggregation delay--energy consumption tradeoff (e.g., [22; 23; 24]) as a function of the underlying interference models, availability of the channels and the channel scheduling schemes used. To the best of our knowledge, ours is the first such work to analyze the diameter-aggregation delay tradeoff as a function of the type of the underlying data gathering tree (bottleneck node weight-based or bottleneck link weight-based) involved. Some of the works in the literature (e.g., [25]) focus on developing distributed algorithms to minimize the aggregation delay as much as possible (given a set of available channels). A common trend among these distributed algorithms is to first construct a spanning tree of the network based on a link or node criterion, determine a connected dominating set (CDS) [26] of the nodes (a CDS is a set of nodes such that every node in the network is either in this set or is a neighbor of the nodes in this set) in the spanning tree, form a data gathering tree of the nodes that are part of the CDS and construct a TDMA-schedule of the nodes that are part of the CDS to aggregate data according to the availability of the channels. Our approach in the paper is different. Given a graph with node weights or link weights, we directly construct a bottleneck node weight or bottleneck link weight-based data gathering tree of the network. We make use of the observation made in [27] that the intermediate nodes of a bottleneck node weight or bottleneck link weight based DG tree are also the nodes constituting a CDS constructed with the objective of optimizing the bottleneck node weight or bottleneck link weight. Our contributions in this paper are twofold: First, we propose a benchmarking algorithm that, given a DG tree, (with sufficient number of communication channels with unique CDMA codes) determines the minimum aggregation delay at every intermediate node (including that of the root node) of the DG tree. To the best of our knowledge, we have not come across a benchmarking algorithm that gives the minimum aggregation delay for any intermediate node _vis-a-vis_ its position in the DG tree.