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Time, Message and Memory-Optimal Distributed Minimum Spanning Tree and Partwise Aggregation

Michael Elkin Tanya Goldenfeld

Abstract

Memory-(in)efficiency is a crucial consideration that oftentimes prevents deployment of state-of-the-art distributed algorithms in real-life modern networks. In the context of the MST problem, roughly speaking, there are three types of algorithms. The algorithm of Gallager-Humblet-Spira and its versions are memory- and message- efficient, but their running time is at least linear in the number of vertices $n$, even when the unweighted diameter $D$ is much smaller than $n$. The algorithm of Garay-Kutten-Peleg and its versions are time-efficient, but not message- or memory-efficient. The more recent algorithms of are time- and message-efficient, but are not memory-efficient. As a result, GHS-type algorithms are much more prominent in real-life applications than time-efficient ones. In this paper we develop a deterministic time-, message- and memory-efficient algorithm for the MST problem. It is also applicable to the more general partwise aggregation problem. We believe that our techniques will be useful for devising memory-efficient versions for many other distributed problems.

Time, Message and Memory-Optimal Distributed Minimum Spanning Tree and Partwise Aggregation

Abstract

Memory-(in)efficiency is a crucial consideration that oftentimes prevents deployment of state-of-the-art distributed algorithms in real-life modern networks. In the context of the MST problem, roughly speaking, there are three types of algorithms. The algorithm of Gallager-Humblet-Spira and its versions are memory- and message- efficient, but their running time is at least linear in the number of vertices , even when the unweighted diameter is much smaller than . The algorithm of Garay-Kutten-Peleg and its versions are time-efficient, but not message- or memory-efficient. The more recent algorithms of are time- and message-efficient, but are not memory-efficient. As a result, GHS-type algorithms are much more prominent in real-life applications than time-efficient ones. In this paper we develop a deterministic time-, message- and memory-efficient algorithm for the MST problem. It is also applicable to the more general partwise aggregation problem. We believe that our techniques will be useful for devising memory-efficient versions for many other distributed problems.
Paper Structure (54 sections, 3 theorems, 3 equations, 3 figures, 2 tables, 2 algorithms)

This paper contains 54 sections, 3 theorems, 3 equations, 3 figures, 2 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. Background and Main Results
  3. Related Work
  4. Memory Complexity
  5. Additional Related Work
  6. Technical Overview
  7. Structure of the Paper
  8. Preliminaries
  9. The Computational Model
  10. The Partwise Aggregation Problem
  11. MST-construction Algorithms
  12. The GKP phase
  13. Lenzen's phase
  14. Message-Efficient Algorithms
  15. Generalizing MST to MSF
  16. ...and 39 more sections

Key Result

corollary 1

A GKP phase can be carried out using $O(\log^* n)$ virtual rounds, i.e., its overall time, message and memory complexity is $O(\log^* n)$ communication cycles.

Figures (3)

  • Figure 1: Progression of time slots through the BFS Tree $T$: The yellow vertices belong to the fragment $F_1$ associated with slot $1$. The overall tree depth is $d(T)$ (here $d(T)=3$). For each slot $i=1,2,\ldots$, during round $t$ the vertices at depth $d(T)-t+i$ upcast the messages of slot $i$. If a vertex $v$ is associated with slot $i$ and has depth $d(T)-t+i$, then $v$ injects its input into the convergecast during round $t$.
  • Figure 2: The top diagram illustrates an allocation of one slot to every node. The bottom diagram illustrates an allocation of slots only to the yellow nodes.
  • Figure 3: A detailed illustration of a single iteration of step 2 of the slot set generation procedure described in Section \ref{['5.3_SlotSetGen']}.

Theorems & Definitions (44)

  • Claim 1
  • Claim 2
  • Claim 3
  • proof
  • Claim 4
  • proof
  • Claim 5
  • proof
  • corollary 1
  • Claim 6
  • ...and 34 more