We study the problem of the amount of knowledge about a communication network that must be given to its nodes in order to efficiently disseminate information. While previous results about communication in networks used particular partial information available to nodes, such as the knowledge of the neighborhood or the knowledge of the network topology within some radius, our approach is quantitative: we investigate the minimum total number of bits of information (minimum oracle size) that has to be available to nodes in order to perform efficient communication. It turns out that the minimum oracle size for which a distributed task can be accomplished efficiently, can serve as a measure of the difficulty of this task. We use this measure to make a quantitative distinction between the difficulty of two apparently similar fundamental communication primitives: the broadcast and the wakeup. In both of them a distinguished node, called the source, has a message, which has to be transmitted to all other nodes of the network. In the wakeup, only nodes that already got the source message (i.e., are awake) can send messages to their neighbors, thus waking them up. In the broadcast, all nodes can send control messages even before getting the source message, thus potentially facilitating its future dissemination. In both cases we are interested in accomplishing the communication task with optimal message complexity, i.e., using a number of messages linear in the number of nodes. We show that the minimum oracle size permitting the * Supported by the projects PairAPair of the ACI Masses de Données, and FRAGILE of the ACI Sécurité Informatique. Additional support from the INRIA project "Grand Large". † Supported in part by NSERC discovery grant and by the Research Chair in Distributed Computing of the Université du Québec en Outaouais. Permission to make digital or hard copies of all or part of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage and that copies bear this notice and the full citation on the first page. To copy otherwise, to republish, to post on servers or to redistribute to lists, requires prior specific permission and/or a fee. wakeup with a linear number of messages in a n-node network, is Θ(n log n), while the broadcast with a linear number of messages can be achieved with an oracle of size O(n). We also show that the latter oracle size is almost optimal: no oracle of size o(n) can permit to broadcast with a linear number of messages. Thus an efficient wakeup requires strictly more information about the network than an efficient broadcast.
A different approach to the problem of evaluating the impact of information on the performance of network algorithms was adopted in REF where the framework of network algorithms with advice has been introduced and used to study the task of broadcasting with a linear number of messages, in the message passing model.
Oracle size: a new measure of difficulty for communication tasks
PODC '06
Computer Science