The optimality of distributive constructions of minimum weight and degree restricted spanning trees in a complete network of processors

In a previous paper we showed that the distributive construction of a spanning tree in a complete network of processors can be done in $O(n\log n)$ messages. We show in this work that if the spanning tree is required to satisfy certain properties, then the complexity of its construction increases: First we show that the construction of a minimum weight spanning tree requires, in the worst case, at least $\Omega (n^2 )$ messages, and then we show that the construction of a spanning tree where the maximum degree is at most k may require at least $\Omega ({{n^2 } / k})$ messages in the worst case. Actually, in both cases the lower bounds are shown for the number of edges used in the worst case. Moreover, the results are valid for both asynchronous and synchronous networks, and are independent of the lengths of the messages. On the other hand, there are algorithms for the above tasks which achieve these lower bounds, up to a constant factor, and use messages of $O(\log n)$ length.