Interactive Nearest Lattice Point Search in a Distributed Setting: Two Dimensions

The nearest lattice point problem in <inline-formula> <tex-math notation="LaTeX">\mathbb {R}^{n} </tex-math></inline-formula> is formulated in a distributed network with <inline-formula> <tex-math notation="LaTeX">n </tex-math></inline...

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Veröffentlicht in:IEEE transactions on communications Jg. 70; H. 8; S. 5128 - 5139
Hauptverfasser: Vaishampayan, Vinay A., Bollauf, Maiara F.
Format: Journal Article
Sprache:Englisch
Veröffentlicht: New York IEEE 01.08.2022
The Institute of Electrical and Electronics Engineers, Inc. (IEEE)
Schlagworte:
Algorithms
Communication
communication complexity
Computer networks
Costs
distributed compression
distributed function computation
Error probability
Generators
Hexagonal lattice
lattice quantization
Lattices
Matrix decomposition
nearest lattice point problem
Protocols
Quantization (signal)
ISSN:0090-6778, 1558-0857
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Zusammenfassung:The nearest lattice point problem in <inline-formula> <tex-math notation="LaTeX">\mathbb {R}^{n} </tex-math></inline-formula> is formulated in a distributed network with <inline-formula> <tex-math notation="LaTeX">n </tex-math></inline-formula> nodes. The objective is to minimize the probability that an incorrect lattice point is found, subject to a constraint on inter-node communication. Algorithms with a single as well as an unbounded number of rounds of communication are considered for the case <inline-formula> <tex-math notation="LaTeX">n=2 </tex-math></inline-formula>. For the algorithm with a single round, expressions are derived for the error probability as a function of the total number of communicated bits. We observe that the error exponent depends on the lattice structure and that zero error requires an infinite number of communicated bits. In contrast, with an infinite number of allowed communication rounds, the nearest lattice point can be determined without error with a finite average number of communicated bits and a finite average number of rounds of communication. In two dimensions, the hexagonal lattice, which is most efficient for communication and compression, is found to be the most expensive in terms of communication cost.
Bibliographie:ObjectType-Article-1
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ISSN:0090-6778
1558-0857
DOI:10.1109/TCOMM.2022.3184170