Abstract
This paper addresses the cornerstone family of local problems in distributed computing, and investigates the curious gap between randomized and deterministic solutions under bandwidth restrictions. Our main contribution is in providing tools for derandomizing solutions to local problems, when the n nodes can only send $$O(\log n)$$-bit messages in each round of communication. Our framework mostly follows by the derandomization approach of Luby (J Comput Syst Sci 47(2):250–286, 1993) combined with the power of all to all communication. Our key results are as follows: first, we show that in the congested clique model, which allows all-to-all communication, there is a deterministic maximal independent set algorithm that runs in $$O(\log ^2 {\varDelta })$$ rounds, where $${\varDelta }$$ is the maximum degree. When $${\varDelta }=O(n^{1/3})$$, the bound improves to $$O(\log {\varDelta })$$. In addition, we deterministically construct a $$(2k-1)$$-spanner with $$O(kn^{1+1/k}\log n)$$ edges in $$O(k \log n)$$ rounds in the congested clique model.
Highlights
1.1 MotivationA cornerstone family of problems in distributed computing are the so-called local problems
We address the tension between the deterministic and randomized complexities of local problems in the congested clique model, where the communication graph is complete but the size of messages is restricted to O(log n) bits
The curious phenomenon that shows up here is that the derandomization toolbox that was developed for sequential algorithms does not seem to lend itself for the LOCAL model, but it can be used in the congested clique model
Summary
A cornerstone family of problems in distributed computing are the so-called local problems. Building upon a new lower bound technique of Brandt et al [10], they show an exponential separation between the randomized and deterministic complexity of ∆-coloring trees These results hold in the LOCAL model, which allows unbounded messages. We address the tension between the deterministic and randomized complexities of local problems in the congested clique model, where the communication graph is complete but the size of messages is restricted to O(log n) bits. The curious phenomenon that shows up here is that the derandomization toolbox that was developed for sequential algorithms does not seem to lend itself for the LOCAL model, but it can be used in the congested clique model This allows us to obtain deterministic algorithms for local problems in the congested clique model, whose complexities roughly match the complexities of their randomized counterparts in the LOCAL model. This can be contrasted with the exponential in ∆ or near-exponential in n gaps between the deterministic and randomized complexities of these problems in the LOCAL model alone
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