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Message Optimality and Message-Time Trade-offs for APSP and Beyond

Fabien Dufoulon, Shreyas Pai, Gopal Pandurangan, Sriram Pemmaraju, Peter Robinson

TL;DR

This work focuses on a fundamental graph optimization problem, All Pairs Shortest Path (APSP), whose message complexity is still unresolved and presents an algorithm that solves weighted APSP, which is message-optimal (up to logarithmic factors) for algorithms that take poly(n) rounds.

Abstract

Round complexity is an extensively studied metric of distributed algorithms. In contrast, our knowledge of the \emph{message complexity} of distributed computing problems and its relationship (if any) with round complexity is still quite limited. To illustrate, for many fundamental distributed graph optimization problems such as (exact) diameter computation, All-Pairs Shortest Paths (APSP), Maximum Matching etc., while (near) round-optimal algorithms are known, message-optimal algorithms are hitherto unknown. More importantly, the existing round-optimal algorithms are not message-optimal. This raises two important questions: (1) Can we design message-optimal algorithms for these problems? (2) Can we give message-time tradeoffs for these problems in case the message-optimal algorithms are not round-optimal? In this work, we focus on a fundamental graph optimization problem, \emph{All Pairs Shortest Path (APSP)}, whose message complexity is still unresolved. We present two main results in the CONGEST model: (1) We give a message-optimal (up to logarithmic factors) algorithm that solves weighted APSP, using $\tilde{O}(n^2)$ messages. This algorithm takes $\tilde{O}(n^2)$ rounds. (2) For any $0 \leq \varepsilon \le 1$, we show how to solve unweighted APSP in $\tilde{O}(n^{2-\varepsilon })$ rounds and $\tilde{O}(n^{2+\varepsilon })$ messages. At one end of this smooth trade-off, we obtain a (nearly) message-optimal algorithm using $\tilde{O}(n^2)$ messages (for $\varepsilon = 0$), whereas at the other end we get a (nearly) round-optimal algorithm using $\tilde{O}(n)$ rounds (for $\varepsilon = 1$). This is the first such message-time trade-off result known.

Message Optimality and Message-Time Trade-offs for APSP and Beyond

TL;DR

This work focuses on a fundamental graph optimization problem, All Pairs Shortest Path (APSP), whose message complexity is still unresolved and presents an algorithm that solves weighted APSP, which is message-optimal (up to logarithmic factors) for algorithms that take poly(n) rounds.

Abstract

Round complexity is an extensively studied metric of distributed algorithms. In contrast, our knowledge of the \emph{message complexity} of distributed computing problems and its relationship (if any) with round complexity is still quite limited. To illustrate, for many fundamental distributed graph optimization problems such as (exact) diameter computation, All-Pairs Shortest Paths (APSP), Maximum Matching etc., while (near) round-optimal algorithms are known, message-optimal algorithms are hitherto unknown. More importantly, the existing round-optimal algorithms are not message-optimal. This raises two important questions: (1) Can we design message-optimal algorithms for these problems? (2) Can we give message-time tradeoffs for these problems in case the message-optimal algorithms are not round-optimal? In this work, we focus on a fundamental graph optimization problem, \emph{All Pairs Shortest Path (APSP)}, whose message complexity is still unresolved. We present two main results in the CONGEST model: (1) We give a message-optimal (up to logarithmic factors) algorithm that solves weighted APSP, using messages. This algorithm takes rounds. (2) For any , we show how to solve unweighted APSP in rounds and messages. At one end of this smooth trade-off, we obtain a (nearly) message-optimal algorithm using messages (for ), whereas at the other end we get a (nearly) round-optimal algorithm using rounds (for ). This is the first such message-time trade-off result known.
Paper Structure (34 sections, 42 theorems, 1 figure)

This paper contains 34 sections, 42 theorems, 1 figure.

Table of Contents

  1. Introduction
  2. Model
  3. $\mathsf{CONGEST}$ model
  4. $\mathsf{BCONGEST}$ model
  5. Our Technical Contributions
  6. Message-Optimal Algorithms.
  7. Message-Time Tradeoffs.
  8. Additional Related Work
  9. Technical Preliminaries
  10. The Congestion Plus Dilation Framework
  11. Distributed Primitives: Upcast and Downcast
  12. Message-Efficient Simulation of $\mathsf{BCONGEST}$ Algorithms
  13. Low Diameter and Communication (LDC) Graph Decompositions
  14. Description of the Simulation
  15. Preprocessing.
  16. ...and 19 more sections

Key Result

Theorem 1.1

There is a $\mathsf{CONGEST}\xspace$ algorithm that, with high probability, computes exact weighted APSP in $\tilde{O}(n^2)$ rounds and with $\tilde{O}(n^2)$ message complexity, even on directed graphs and even if the edge weights are negative.

Figures (1)

  • Figure 1: Example of an $(r,d)$-LDC decomposition with five clusters. The bold inter-cluster edges are the edges in $F$ whereas the dashed intercluster edges are in $E \setminus F$. At a high level, the simulation described in Section \ref{['sec:simulation']} saves messages by not using the edges in $E \setminus F$. Each cluster has strong diameter at most $r$, and each node in a cluster has at most $d$ outgoing edges in $F$.

Theorems & Definitions (45)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3: Ghaffari ghaffarilmr
  • Theorem 1.4
  • Lemma 1.5
  • Lemma 1.6
  • Theorem 2.1
  • Corollary 2.2
  • Definition 2.3: Low Diameter and Communication (LDC) Graph Decomposition
  • Lemma 2.4
  • ...and 35 more