This paper considers Gaussian half-duplex diamond n-relay networks, where a source communicates with a destination by hopping information through one layer of n non-communicating relays that operate in half-duplex. The main focus consists of investigating the following question: What is the contribution of a single relay on the approximate capacity of the entire network? In particular, approximate capacity refers to a quantity that approximates the Shannon capacity within an additive gap which only depends on n, and is independent of the channel parameters. This paper answers the above question by providing a fundamental bound on the ratio between the approximate capacity of the highest-performing single relay and the approximate capacity of the entire network, for any number n. Surprisingly, it is shown that such a ratio guarantee is f = 1/(2+2 (2\pil/(n+2))), that is a sinusoidal function of n, which decreases as n increases. It is also shown that the aforementioned ratio guarantee is tight, i.e., there exist Gaussian half-duplex diamond n-relay networks, where the highest-performing relay has an approximate capacity equal to an f fraction of the approximate capacity of the entire network.
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Manuscript received January 20, 2020; revised August 2, 2020 and December 18, 2020; accepted February 28, 2021. Date of publication March 23, 2021; date of current version August 12, 2021. This work was supported by NSF under Award 1907785. This article was presented in part at the 2020 IEEE International Symposium on Information Theory. The associate editor coordinating the review of this article and approving it for publication was R. Tandon. (Corresponding author: Sarthak Jain.) The authors are with the Electrical and Computer Engineering Department, University of Minnesota, Twin Cities, Minneapolis, MN 55455 USA (e-mail: email@example.com; firstname.lastname@example.org; email@example.com).
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- approximate capacity
- diamond network
- relay selection