Network Orientation

Abstract

This paper analyses how the symmetry of a processor network influences the existence of a solution for the network orientation problem. The orientation of hypercubes and tori is the problem of assigning labels to each link of each processor, in such a way that a sense of direction is given to the network. In this paper the problem of network orientation for these two topologies is studied under the assumption that the network contains a single leader, under the assumption that the processors possess unique identities, and under the assumption that the network is anonymous. The distinction between these three models is considered fundamental in distributed computing. It is shown that orientations can be computed by deterministic algorithms only when either a leader or unique identities are available. Orientations can be computed for anonymous networks by randomized algorithms, but only when the number of processors is known. When the number of processors is not known, even randomized algorithms cannot compute orientations for anonymous processor networks. Lower bounds on the message complexity of orientation and algorithms achieving these bounds are given.

Optimal Distributed Algorithms in Unlabeled Tori and Chordal Rings

“…Since sense of direction is known to improve the communication complexity of distributed algorithms, computation of as a preprocessing phase in unlabeled topology has been studied [21,22]. So far, results have not been encouraging: any algorithm computing exchanges at least (e−1/2 N ) messages in a network with N nodes and e edges.…”

Optimal Elections in Labeled Hypercubes

“…Without a labeling of the links, the message complexity required to build the orientation of hypercubes has been proved to be at least (1/2) N(log N Ϫ 1) [20].…”

Torus orientation

“…Syrotiuk et al [23] demonstrate that there is no deterministic orientation protocol if the processes are allowed to label their communication links arbitrarily, and then investigate which labeling restrictions allow solving orientation in anonymous tori. Tel [25] proves that no terminating (deterministic or randomized) protocol exists for torus orientation if the processes (i) do not know the number of processes and (ii) do not have identifiers; solutions to torus orientation in other cases are presented in [20] and [25]. None of the solutions presented in [20], [23], and [25] is selfstabilizing.…”

“…The problem of computing a compass labeling in tori has been studied in [20], [23], and [25]. Syrotiuk et al [23] demonstrate that there is no deterministic orientation protocol if the processes are allowed to label their communication links arbitrarily, and then investigate which labeling restrictions allow solving orientation in anonymous tori.…”