insight - Algorithms and Data Structures - # Approximating All-Pairs Shortest Paths in Congested Clique

Core Concepts

This paper presents new randomized algorithms for approximating All-Pairs Shortest Paths (APSP) in the Congested Clique model, achieving an O(1)-approximation in O(log log log n) rounds, and providing a trade-off between the number of rounds and the approximation quality.

Abstract

The paper focuses on the problem of computing All-Pairs Shortest Paths (APSP) in the Congested Clique model, where a fully connected communication network of n nodes exchange Θ(log n)-bit messages in synchronous rounds.
The key contributions are:
An O(1)-approximation algorithm for weighted undirected APSP that takes O(log log log n) rounds, improving exponentially on the previous best algorithms.
A trade-off between the number of rounds and the approximation quality, allowing an O(log^(2-t) n)-approximation in O(t) rounds, for any constant t ≥ 1. This includes the ability to get an O((log n)^(1/2t))-approximation in O(1) rounds.
The main technical ingredients are:
A lemma that transforms an O(a)-approximation for APSP into an O(√a)-approximation in O(1) rounds.
A fast O(1)-round algorithm for constructing a k-nearest hopset, which allows each node to compute distances to its k closest nodes efficiently.
A fast O(1)-round algorithm for computing the distances to the k nearest nodes, given a k-nearest hopset.
A skeleton graph construction that allows reducing the APSP problem on the original graph to an APSP problem on a much smaller graph.
By combining these building blocks, the paper presents the improved APSP approximation algorithms in the Congested Clique model.

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by Hong Duc Bui... at **arxiv.org** 05-07-2024

Deeper Inquiries

The techniques developed in this paper can be extended to other distributed graph problems by leveraging the concepts of hopsets, skeleton graphs, and fast computation of nearest nodes. For example, algorithms for computing Minimum Spanning Trees (MST), Maximum Flow, or even graph clustering could benefit from similar approaches. By adapting the idea of constructing sparse graphs that approximate the original graph's properties, these algorithms could be optimized for faster computation in distributed settings. Additionally, the concept of filtering and matrix multiplication could be applied to problems like graph traversal or community detection, where identifying nearest neighbors or paths of specific lengths is crucial.

The Congested Clique model has limitations in terms of scalability and applicability to real-world distributed systems. One major limitation is the assumption of synchronous communication and the fully connected network structure, which may not reflect the communication patterns in practical distributed systems. To adapt the algorithms for other distributed computing models like the Message Passing Model or the CONGEST model, modifications would be needed to account for asynchronous communication, limited message sizes, and different network topologies. Additionally, the algorithms would need to be optimized for fault tolerance and load balancing, which are critical aspects in real-world distributed systems.

The trade-off between the number of rounds and the approximation quality provided by these algorithms could be particularly useful in scenarios where real-time decision-making is required based on approximate graph computations. For example, in network routing protocols where quick decisions need to be made based on estimated shortest paths, having a fast algorithm that provides a reasonably accurate approximation in a minimal number of rounds can be beneficial. Similarly, in large-scale data processing systems where iterative graph algorithms are used, the ability to control the trade-off between computation time and accuracy can lead to more efficient resource utilization and faster processing of complex graph structures.

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