Lower bounds on the loading of multiple bus networks for binary tree algorithms

A multiple bus network (MBN) connects a set of processors via set of buses. Two important parameters of an MBN are its loading (largest number of connections on a bus) and its degree (largest number of connections to a processor). These parameters determine the cost, speed, and implementability of the MBN. The smallest degree that any useful MBN can have is 2. In this paper, we study the relationship between running time, degree, and loading of degree-2 MBNs running a fundamental class of algorithms called binary tree algorithms. (A binary tree algorithm reduces 2ⁿ inputs at the leaves of a balanced-binary tree to a single result at the root of the tree.) Specifically, we establish a nontrivial Ω(n/logn) loading lower bound for any degree-2 MBN running a 2ⁿ input binary tree algorithm optimally in n steps. We show that this bound does not hold if the restriction on the degree or the running time is relaxed. That is, optimal-time, degree-3, constant loading MBNs and suboptimal-time, degree-2, constant loading MBNs exist for binary tree algorithms. We also derive a lower bound on the additional time (beyond the optimal) needed to run binary tree algorithms on a degree-2, loading-L MBN, for any L>3.