We study the cost of distributed MST construction in the setting where each edge has a latency and a capacity, along with the weight. Edge latencies capture the delay on the links of the communication network, while capacity captures their throughput (in this case, the rate at which messages can be sent). Depending on how the edge latencies relate to the edge weights, we provide several tight bounds on the time and messages required to construct an MST. When edge weights exactly correspond with the latencies, we show that, perhaps interestingly, the bottleneck parameter in determining the running time of an algorithm is the total weight $W$ of the MST (rather than the total number of nodes $n$, as in the standard CONGEST model). That is, we show a tight bound of $tilde{Theta}(D + sqrt{W/c})$ rounds, where $D$ refers to the latency diameter of the graph, $W$ refers to the total weight of the constructed MST and edges have capacity $c$. The proposed algorithm sends $tilde{O}(m+W)$ messages, where $m$, the total number of edges in the network graph under consideration, is a known lower bound on message complexity for MST construction. We also show that $Omega(W)$ is a lower bound for fast MST constructions. When the edge latencies and the corresponding edge weights are unrelated, and either can take arbitrary values, we show that (unlike the sub-linear time algorithms in the standard CONGEST model, on small diameter graphs), the best time complexity that can be achieved is $tilde{Theta}(D+n/c)$. However, if we restrict all edges to have equal latency $ell$ and capacity $c$ while having possibly different weights (weights could deviate arbitrarily from $ell$), we give an algorithm that constructs an MST in $tilde{O}(D + sqrt{nell/c})$ time. In each case, we provide nearly matching upper and lower bounds.
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