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What Is the Minimum Number of Random Bits Required for Computability and Efficiency in Anonymous Networks?

Authors Dariusz R. Kowalski , Piotr Krysta , Shay Kutten

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Subject Classification

ACM Subject Classification
  • Theory of computation → Distributed algorithms
  • Computing methodologies → Distributed algorithms
Keywords
  • Distributed computability
  • Anonymous Networks
  • Randomness
  • Leader Election

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Abstract

Angluin (STOC'80) and Yamashita and Kameda (PODC'88) show that some useful distributed tasks are impossible (for deterministic algorithms) in a general network if nodes do not possess unique identifiers. However, any task decidable in the non-distributed context, can be solved deterministically if the network has a unique leader. Alternatively, much research has been devoted to randomized distributed algorithms in anonymous networks. We present tight upper and lower bounds for the fundamental question: How much randomness is necessary and sufficient to solve Leader Election (LE) in anonymous networks, i.e., to transform an anonymous network into a non-anonymous one? We prove that at least one random bit per node is required in some cases. Surprisingly, a single random bit is also enough, for a total of n bits, where n is the number of nodes. However, the time complexity of our (total of) n random bits algorithm for general networks turned out to be impractically high. Hence, we also developed time-efficient algorithms for the very symmetric graphs of cliques and cycles, paying only an additional cost of o(n) random bits. 
The primary steps of our algorithms are of independent interest. At first glance, it seems that using one random bit per node, any algorithm can distinguish only two sets of nodes: those with 0 and those with 1. Our algorithms manage to partition the nodes into more than two sets with high probability. In some sense, they perform the task of a "distributed pseudorandom generator", for example, one of our algorithms turns n bits, one per node, into n unique (with high probability) numbers. Even though a complete graph looks very symmetric, the algorithms explore interesting asymmetries inherent in any n permutations (of n values each), if each describes the assignment (by the adversary) of ports in a node to edges leading to neighbors.
Finally, we show how to transform any randomized algorithm that generates xn+o(n) random bits in total to one where each node generates at most x+1 bits. Our results apply to both synchronous and asynchronous networks.

Cite As Get BibTex

Dariusz R. Kowalski, Piotr Krysta, and Shay Kutten. What Is the Minimum Number of Random Bits Required for Computability and Efficiency in Anonymous Networks?. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 353, pp. 41:1-41:24, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025) https://doi.org/10.4230/LIPIcs.APPROX/RANDOM.2025.41

Author Details

Dariusz R. Kowalski
  • School of Computer and Cyber Sciences, Augusta University, GA, USA
Piotr Krysta
  • School of Computer and Cyber Sciences, Augusta University, GA, USA
Shay Kutten
  • Faculty of Data and Decision Sciences and GTEP program, Technion, Haifa, Israel

Funding

  • Kutten, Shay: Shay Kutten was supported in part by grant 1346/22 from the Israeli Science Foundation.

Acknowledgements

We thank Omer Reingold for his help in tailoring the universal exploration sequence to our needs. Shay Kutten was supported in part by grant 1346/22 from the Israeli Science Foundation. His research was carried out within the framework of Grand Technion Energy Program (GTEP). A part of the work was carried out while with Fraunhofer SIT, Darmstadt, Germany.

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