Global sum on symmetric networks

TL;DR

This paper investigates the global sum problem in symmetric networks, establishing bounds on the number of steps needed for all processors to compute the sum, and proves the conjecture for graphs with diameter 2.

Contribution

It proves that for vertex symmetric graphs, the global sum can be computed in a number of steps equal to the diameter when the diameter is 2.

Findings

s is bounded below by the diameter

s is bounded above by twice the diameter

Conjecture holds for graphs with diameter 2

Abstract

We are interested in the following problem we call global sum. Each processor starts with a single real value. At each time step, every directed edge in the graph can simultaneously be used to transmit a single (bounded) number between the processors (vertices). How many time steps s are required to ensure that every processor acquires the global sum? We know that s is bounded below by the diameter and above by two times the diameter. We conjecture that for vertex symmetric graphs, s is equal to the diameter. We show this is true if the diameter is 2.