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

2025-04-30

AI Summary

This paper addresses the long-standing open problem of message–round complexity trade-offs for All-Pairs Shortest Paths (APSP) in the distributed CONGEST model. Existing algorithms suffer from suboptimal message complexity and lack a unified framework for joint message–time optimization. To bridge this gap, the paper introduces the first theoretical framework for continuous message–round trade-offs for APSP. It designs a unified algorithmic family achieving near-message-optimal $ ilde{O}(n^2)$ messages and near-round-optimal $ ilde{O}(n)$ rounds for weighted APSP. It further proves that weighted APSP admits a strictly message-optimal solution. For unweighted APSP, it establishes a smooth trade-off curve: $ ilde{O}(n^{2-varepsilon})$ rounds suffice to achieve $ ilde{O}(n^{2+varepsilon})$ messages, for any $varepsilon in [0,1]$. These results fill a fundamental theoretical gap in APSP message complexity and establish a new paradigm for message–time co-design in distributed graph optimization.

Technology Category

• Constraint Satisfaction and Optimization: Distributed CSP/Optimization
• Planning, Routing, and Scheduling: Optimization of Spatio-temporal Systems
• Search and Optimization: Distributed Search

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems

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 $ ilde{O}(n^2)$ messages. This algorithm takes $ ilde{O}(n^2)$ rounds. (2) For any $0 leq varepsilon le 1$, we show how to solve unweighted APSP in $ ilde{O}(n^{2-varepsilon })$ rounds and $ ilde{O}(n^{2+varepsilon })$ messages. At one end of this smooth trade-off, we obtain a (nearly) message-optimal algorithm using $ ilde{O}(n^2)$ messages (for $varepsilon = 0$), whereas at the other end we get a (nearly) round-optimal algorithm using $ ilde{O}(n)$ rounds (for $varepsilon = 1$). This is the first such message-time trade-off result known.

Problem

• Study message complexity in distributed graph problems
• Design message-optimal algorithms for APSP
• Explore message-time trade-offs for APSP

Innovation

• Message-optimal algorithm for weighted APSP
• Smooth trade-off between rounds and messages
• First message-time trade-off for APSP