Lower Bounds on the Loading of Degree-2 Multiple Bus Networks for Binary-Tree Algorithms

A binary-tree algorithm, Bin(n), proceeds level-bylevel from the leaves of a 2-leaf balanced binary tree to its root. This paper deals with running binary-tree algorithms on multiple bus networks (MBNs) in which processors communicate via buses. Every “binary-tree MBN” has a degree (maximum number of buses connected to a processor) of at least 2. There exists a degree-2 MBN [15] for Bin(n) that has a loading (maximum number of processors connected to a bus) of Θ(n). For any MBN that runs Bin(n) optimally, the loading was recently proved to be Ω(n 1 2 ) [3]. In this paper, we narrow the gap between the results in [3, 15] by deriving a tighter lower bound of Ω(n 2 3 ). We also establish a tradeoff between the speed and loading of degree-2 binary-tree MBNs.