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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| Vydané v: | IEEE transactions on communications Ročník 70; číslo 8; s. 5128 - 5139 |
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| Hlavní autori: | , |
| Médium: | Journal Article |
| Jazyk: | English |
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New York
IEEE
01.08.2022
The Institute of Electrical and Electronics Engineers, Inc. (IEEE) |
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| ISSN: | 0090-6778, 1558-0857 |
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| Abstract | 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. |
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| AbstractList | 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. The nearest lattice point problem in [Formula Omitted] is formulated in a distributed network with [Formula Omitted] 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 [Formula Omitted]. 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. |
| Author | Bollauf, Maiara F. Vaishampayan, Vinay A. |
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| Cites_doi | 10.1007/978-1-4757-2249-9 10.1109/ISIT45174.2021.9518076 10.1007/BF02579403 10.1109/18.57210 10.1109/ISIT.2017.8006847 10.1109/18.915643 10.1109/18.21219 10.1109/ISIT.2017.8006848 10.1145/2785733 10.1109/18.930913 10.1109/18.86993 10.1109/TIT.2021.3097719 10.1109/TIT.1968.1054193 10.1109/JSAIT.2021.3105359 10.1002/j.1538-7305.1948.tb01340.x 10.1109/ISIT.2019.8849301 10.1109/TCOMM.2021.3070364 10.1109/18.149503 10.1109/18.32125 |
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| StartPage | 5128 |
| SubjectTerms | 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) |
| Title | Interactive Nearest Lattice Point Search in a Distributed Setting: Two Dimensions |
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