Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Efficient Counting and Simulation in Content-Oblivious Rings

Jérémie Chalopin, Yi-Jun Chang, Giuseppe Antonio Di Luna, Haoran Zhou

Abstract

In the content-oblivious (CO) model (proposed by Censor-Hillel et al.), processes inhabit an asynchronous network and communicate only by exchanging pulses. A series of works has clarified the computational power of this model. In particular, it was shown that, when a leader is present and the network is 2-edge-connected, content-oblivious communication can simulate classical asynchronous message passing. Subsequent results extended this equivalence to leaderless oriented and unoriented rings, and, under non-uniform assumptions, to general 2-edge-connected networks. The simulator of Censor-Hillel et al. requires $O(n^3b+n^3\log n)$ pulses to emulate the send of a single $b$-bit message, making it impractical even on modest-size networks. We focus on message-efficient computation in CO networks. We study the fundamental problem of counting in ring topologies, both because knowing the exact network size is a basic prerequisite for many distributed tasks and because counting immediately implies a broad class of aggregation primitives. We give an algorithm that counts using $O(n^{1.5})$ pulses in anonymous rings with a leader, an $O(n\log^2 n)$ algorithm for counting in rings with IDs. Moreover, we show that any counting algorithm in CO requires $Ω(n\log n)$ pulses. Interestingly, in the course of this investigation, we design a simulator for classic message passing: in one simulated round, each process can send a $b$-bit message to each of its neighbors using only $O(b)$ pulses per process. The simulator extends to general 2-edge-connected networks, after a pre-processing step that requires $O(n^{8}\log n)$ pulses, where $n$ is the number of processes, allowing thus efficient simulation of asynchronous message passing in general 2-edge-connected networks.

Efficient Counting and Simulation in Content-Oblivious Rings

Abstract

In the content-oblivious (CO) model (proposed by Censor-Hillel et al.), processes inhabit an asynchronous network and communicate only by exchanging pulses. A series of works has clarified the computational power of this model. In particular, it was shown that, when a leader is present and the network is 2-edge-connected, content-oblivious communication can simulate classical asynchronous message passing. Subsequent results extended this equivalence to leaderless oriented and unoriented rings, and, under non-uniform assumptions, to general 2-edge-connected networks. The simulator of Censor-Hillel et al. requires pulses to emulate the send of a single -bit message, making it impractical even on modest-size networks. We focus on message-efficient computation in CO networks. We study the fundamental problem of counting in ring topologies, both because knowing the exact network size is a basic prerequisite for many distributed tasks and because counting immediately implies a broad class of aggregation primitives. We give an algorithm that counts using pulses in anonymous rings with a leader, an algorithm for counting in rings with IDs. Moreover, we show that any counting algorithm in CO requires pulses. Interestingly, in the course of this investigation, we design a simulator for classic message passing: in one simulated round, each process can send a -bit message to each of its neighbors using only pulses per process. The simulator extends to general 2-edge-connected networks, after a pre-processing step that requires pulses, where is the number of processes, allowing thus efficient simulation of asynchronous message passing in general 2-edge-connected networks.

Paper Structure

This paper contains 41 sections, 28 theorems, 1 equation, 1 table, 9 algorithms.

Table of Contents

  1. Introduction
  2. Contributions
  3. Self-decomposable aggregation.
  4. Algorithm simulation for content-oblivious rings.
  5. Efficient aggregation and counting with unique IDs.
  6. Counting with anonymity.
  7. Finding the minimum and computing the multiset of the input.
  8. Algorithm simulation for general 2-edge-connected networks.
  9. Related work
  10. System model
  11. IDs and Anonymity.
  12. Asynchronous System.
  13. Content-Oblivious Algorithms.
  14. Oriented Rings.
  15. Quiescent Termination.
  16. ...and 26 more sections

Key Result

Theorem 1

Let $f$ be a self-decomposable aggregation function defined on multisets over a universe $U$. Consider a content-oblivious ring $C$ of $n$ processes with a designated leader, where each process $p$ is equipped with a distinct identifier of at most $\lambda$ bits and holds an element $x_p \in U$. Ass

Theorems & Definitions (29)

  • Theorem 1: Efficient aggregation
  • Corollary 2: Efficient counting
  • Theorem 3: Lower bound
  • Theorem 4: Counting with anonymity
  • Theorem 5
  • Theorem 6
  • Theorem 7: Algorithm simulation for 2-edge-connected networks
  • Definition 8: Composable ending property
  • Theorem 9
  • Theorem 9: Counting with anonymity
  • ...and 19 more