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Overlay Network Construction: Improved Overall and Node-Wise Message Complexity

Authors Yi-Jun Chang , Yanyu Chen , Gopinath Mishra

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ACM Subject Classification
  • Theory of computation → Distributed algorithms
Keywords
  • Distributed algorithms
  • Overlay networks
  • Expander graphs

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Abstract

We consider the problem of constructing distributed overlay networks, where nodes in a reconfigurable system can create or sever connections with nodes whose identifiers they know. Initially, each node knows only its own and its neighbors' identifiers, forming a local channel, while the evolving structure is termed the global channel. The goal is to reconfigure any connected graph into a desired topology, such as a bounded-degree expander graph or a well-formed tree (WFT) with a constant maximum degree and logarithmic diameter, minimizing the total number of rounds and message complexity. This problem mirrors real-world peer-to-peer network construction, where creating robust and efficient systems is desired.

We study the overlay reconstruction problem in a network of n nodes in two models: GOSSIP-reply and HYBRID. In the GOSSIP-reply model, each node can send a message and receive a corresponding reply message in one round. In the HYBRID model, a node can send O(1) messages to each neighbor in the local channel and a total of O(log n) messages in the global channel. 

In both models, we propose protocols for WFT construction with O (n log n) message complexities using messages of O(log n) bits. In the GOSSIP-reply model, our protocol takes O(log n) rounds while in the HYBRID model, our protocol takes O(log² n) rounds. Both protocols use O (n log² n) bits of communication. 

We obtain improved bounds over prior work: 
GOSSIP-reply: A recent result by Dufoulon et al. (ITCS 2024) achieved O(log⁵ n) round complexity and O (n log⁵ n) message complexity using messages of at least Ω(log² n) bits in GOSSIP-reply. With messages of size O(log n), our protocol achieves an optimal round complexity of O(log n) and an improved message complexity of O(n log n). 
HYBRID: Götte et al. (Distributed Computing 2023) showed an optimal O(log n)-round algorithm with O(log² n) global messages per round which incurs a message complexity of Ω(m), where m is the number of edges in the initial topology. At the cost of increasing the round complexity to O(log² n) while using only O(log n) messages globally, our protocol achieves a message complexity that is independent of m. Our approach ensures that the total number of messages for node v, with degree deg(v) in the initial topology, is bounded by O(deg(v) + log n), while the algorithm of Götte et al. requires O(deg(v) + (log⁴ n)/(log log n)) messages per node.

Cite As Get BibTex

Yi-Jun Chang, Yanyu Chen, and Gopinath Mishra. Overlay Network Construction: Improved Overall and Node-Wise Message Complexity. In 45th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 360, pp. 21:1-21:21, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025) https://doi.org/10.4230/LIPIcs.FSTTCS.2025.21

Author Details

Yi-Jun Chang
  • National University of Singapore, Singapore
Yanyu Chen
  • National University of Singapore, Singapore
Gopinath Mishra
  • National University of Singapore, Singapore

Funding

The research is supported by the Ministry of Education, Singapore, under its Academic Research Fund Tier 1 (24-1323-A0001). Any opinions, findings and conclusions or recommendations expressed in this material are those of the authors and do not reflect the views of the Ministry of Education, Singapore.

Acknowledgements

We are grateful to an anonymous FSTTCS reviewer for suggesting the use of the hash functions of [King et al., 2015] to reduce the number of bits required for identifying an outgoing edge. In an earlier version of this work, this task had been carried out using the linear sketches of [Ahn et al., 2012]. The incorporation of the reviewer’s suggestion results in an O(log n)-factor improvement in the overall communication complexity.

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