Allowing each node to communicate only once in a distributed system: shared whiteboard models

Abstract

In this paper we study distributed algorithms on massive graphs where links represent a particular relationship between nodes (for instance, nodes may represent phone numbers and links may indicate telephone calls). Since such graphs are massive they need to be processed in a distributed and streaming way. When computing graph-theoretic properties, nodes become natural units for distributed computation. Links do not necessarily represent communication channels between the computing units and therefore do not restrict the communication flow. Our goal is to model and analyze the computational power of such distributed systems where one computing unit is assigned to each node. Communication takes place on a whiteboard where each node is allowed to write at most one message. Every node can read the contents of the whiteboard and, when activated, can write one small message based on its local knowledge. When the protocol terminates its output is computed from the final contents of the whiteboard. We describe four synchronization models for accessing the whiteboard. We show that message size and synchronization power constitute two orthogonal hierarchies for these systems. We exhibit problems that separate these models, i.e., that can be solved in one model but not in a weaker one, even with increased message size. These problems are related to maximal independent set and connectivity. We also exhibit problems that require a given message size independently of the synchronization model.

Distributedly Testing Cycle-Freeness

“…Beside introducing complexity classes related to local distributed testing, and studying the relationships between these classes, [3] focused attention to verification, and on the size of the certificates involved in the verification. Our paper is also very much related to [4,5], which use models that resemble local distributed testing, but where nodes are restricted to perform just one round of communication before outputting a value on O(log n) bits. In the restricted setting of [4,5], even checking the presence of a 4-cycle in the network may not be feasible.…”

“…Our paper is also very much related to [4,5], which use models that resemble local distributed testing, but where nodes are restricted to perform just one round of communication before outputting a value on O(log n) bits. In the restricted setting of [4,5], even checking the presence of a 4-cycle in the network may not be feasible. Instead, in local distributed testing, the number of communication rounds is just restricted to be constant, but one aims at producing smaller output values, e.g., on O(1) bits.…”

“…For 1-sided error, Ω( √ n) queries are required [15], and this is sufficient [8]. If one does not allow errors, it is folklore that, even in connected graphs, cycle-freeness cannot be distributedly decided 4 . It is however known [18] that cycle-freeness in connected graphs can be verified distributedly with the help of O(log n)-bit additional information (i.e., certificates) per node.…”

“…The central authority then accepts if and only if the sum of the outputs equals 2(n − 1). 4 To see why, assume there exists a local algorithm A deciding cycle-freeness locally.…”

Brief Announcement

“…This setting is equivalent to the multi-party, number-inhand computation model, where communication takes place in a shared whiteboard [1,2,3,4,5,7]: writing a message M on the whiteboard is equivalent to broadcasting M.…”