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# Computing Power of Hybrid Models in Synchronous Networks

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## Cite As

Pierre Fraigniaud, Pedro Montealegre, Pablo Paredes, Ivan Rapaport, Martín Ríos-Wilson, and Ioan Todinca. Computing Power of Hybrid Models in Synchronous Networks. In 26th International Conference on Principles of Distributed Systems (OPODIS 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 253, pp. 20:1-20:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)
https://doi.org/10.4230/LIPIcs.OPODIS.2022.20

## Abstract

During the last two decades, a small set of distributed computing models for networks have emerged, among which LOCAL, CONGEST, and Broadcast Congested Clique (BCC) play a prominent role. We consider hybrid models resulting from combining these three models. That is, we analyze the computing power of models allowing to, say, perform a constant number of rounds of CONGEST, then a constant number of rounds of LOCAL, then a constant number of rounds of BCC, possibly repeating this figure a constant number of times. We specifically focus on 2-round models, and we establish the complete picture of the relative powers of these models. That is, for every pair of such models, we determine whether one is (strictly) stronger than the other, or whether the two models are incomparable. The separation results are obtained by approaching communication complexity through an original angle, which may be of an independent interest. The two players are not bounded to compute the value of a binary function, but the combined outputs of the two players are constrained by this value. In particular, we introduce the XOR-Index problem, in which Alice is given a binary vector x ∈ {0,1}ⁿ together with an index i ∈ [n], Bob is given a binary vector y ∈ {0,1}ⁿ together with an index j ∈ [n], and, after a single round of 2-way communication, Alice must output a boolean out_A, and Bob must output a boolean out_B, such that out_A ∧ out_B = x_j⊕ y_i. We show that the communication complexity of XOR-Index is Ω(n) bits.

## Subject Classification

##### ACM Subject Classification
• Theory of computation → Distributed algorithms
##### Keywords
• hybrid model
• synchronous networks
• LOCAL
• CONGEST

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